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### Linear Algebra (ang)

Linear Algebra
Jim Hefferon

1 3

2 1

1 3

2 1
x1 · 1 3

2 1

x1 · 1 x1 · 3

2 1
6 8

2 1

6 8

2 1

Notation R, R+ , Rn N C {. . . . . .} (a .. b), [a .. b] ... V, W, U v, w 0, 0V B, D En = e1 , . . . , en β, δ RepB (v) Pn Mn×m [S] M ⊕N V ∼W = h, g H, G t, s T, S RepB,D (h) hi,j |T | R(h), N (h) R∞ (h), N∞ (h) real numbers, reals greater than 0, n-tuples of reals natural numbers: {0, 1, 2, . . .} complex numbers set of . . . such that . . . interval (open or closed) of reals between a and b sequence; like a set but order matters vector spaces vectors zero vector, zero vector of V bases standard basis for Rn basis vectors matrix representing the vector set of n-th degree polynomials set of n×m matrices span of the set S direct sum of subspaces isomorphic spaces homomorphisms, linear maps matrices transformations; maps from a space to itself square matrices matrix representing the map h matrix entry from row i, column j determinant of the matrix T rangespace and nullspace of the map h generalized rangespace and nullspace Lower case Greek alphabet name alpha beta gamma delta epsilon zeta eta theta character α β γ δ ζ η θ name iota kappa lambda mu nu xi omicron pi character ι κ λ µ ν ξ o π name rho sigma tau upsilon phi chi psi omega character ρ σ τ υ φ χ ψ ω

Cover. This is Cramer’s Rule for the system x1 + 2x2 = 6, 3x1 + x2 = 8. The size of the ﬁrst box is the determinant shown (the absolute value of the size is the area). The size of the second box is x1 times that, and equals the size of the ﬁnal box. Hence, x1 is the ﬁnal determinant divided by the ﬁrst determinant.

Preface

me, and I may possibly incorporate them into future editions. I am very glad for bug reports. I save them and try to periodically issue updates; people who contribute in this way are acknowledged in the text’s source ﬁles. For people reading this book on their own. This book’s emphasis on motivation and development make it a good choice for self-study. But, while a professional instructor can judge what pace and topics suit a class, if you are an independent student then perhaps you would ﬁnd some advice helpful. Here are two timetables for a semester. The ﬁrst focuses on core material. Wednesday Friday week Monday 1 One.I.1 One.I.1, 2 One.I.2, 3 2 One.I.3 One.II.1 One.II.2 One.III.2 Two.I.1 3 One.III.1, 2 4 Two.I.2 Two.II Two.III.1 5 Two.III.1, 2 Two.III.2 exam 6 Two.III.2, 3 Two.III.3 Three.I.1 7 Three.I.2 Three.II.1 Three.II.2 8 Three.II.2 Three.II.2 Three.III.1 9 Three.III.1 Three.III.2 Three.IV.1, 2 10 Three.IV.2, 3, 4 Three.IV.4 exam 11 Three.IV.4, Three.V.1 Three.V.1, 2 Four.I.1, 2 12 Four.I.3 Four.II Four.II 13 Four.III.1 Five.I Five.II.1 Five.II.3 review 14 Five.II.2 The second timetable is more ambitious (it supposes that you know One.II, the elements of vectors, usually covered in third semester calculus). week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Monday One.I.1 One.I.3 Two.I.1 Two.III.1 Two.III.4 Three.I.2 Three.III.1 Three.IV.2 Three.V.1 Three.VI.2 Four.I.2 Four.II Five.II.1, 2 Five.III.2 Wednesday One.I.2 One.III.1, 2 Two.I.2 Two.III.2 Three.I.1 Three.II.1 Three.III.2 Three.IV.3 Three.V.2 Four.I.1 Four.I.3 Four.II, Four.III.1 Five.II.3 Five.IV.1, 2 Friday One.I.3 One.III.2 Two.II Two.III.3 exam Three.II.2 Three.IV.1, 2 Three.IV.4 Three.VI.1 exam Four.I.4 Four.III.2, 3 Five.III.1 Five.IV.2

more time elsewhere. You might also try picking one or two topics that appeal to you from the end of each chapter. You’ll get more from these if you have access to computer software that can do the big calculations. The most important advice is: do many exercises. I have marked a good sample with ’s. (The answers are available.) You should be aware, however, that few inexperienced people can write correct proofs. Try to ﬁnd a knowledgeable person to work with you on this. Finally, if I may, a caution for all students, independent or not: I cannot overemphasize how much the statement that I sometimes hear, “I understand the material, but it’s only that I have trouble with the problems” reveals a lack of understanding of what we are up to. Being able to do things with the ideas is their point. The quotes below express this sentiment admirably. They state what I believe is the key to both the beauty and the power of mathematics and the sciences in general, and of linear algebra in particular (I took the liberty of formatting them as verse). I know of no better tactic than the illustration of exciting principles by well-chosen particulars. –Stephen Jay Gould If you really wish to learn then you must mount the machine and become acquainted with its tricks by actual trial. –Wilbur Wright Jim Hefferon Mathematics, Saint Michael’s College Colchester, Vermont USA 05439 http://joshua.smcvt.edu 2008-Aug-13

Author’s Note. Inventing a good exercise, one that enlightens as well as tests, is a creative act, and hard work. The inventor deserves recognition. But for some reason texts have traditionally not given attributions for questions. I have changed that here where I was sure of the source. I would greatly appreciate hearing from anyone who can help me to correctly attribute others of the questions. vi

Contents
Chapter One: Linear Systems I Solving Linear Systems . . . . . . . . . . 1 Gauss’ Method . . . . . . . . . . . . . 2 Describing the Solution Set . . . . . . 3 General = Particular + Homogeneous II Linear Geometry of n-Space . . . . . . . 1 Vectors in Space . . . . . . . . . . . . 2 Length and Angle Measures∗ . . . . . III Reduced Echelon Form . . . . . . . . . . 1 Gauss-Jordan Reduction . . . . . . . . 2 Row Equivalence . . . . . . . . . . . . Topic: Computer Algebra Systems . . . . . Topic: Input-Output Analysis . . . . . . . . Topic: Accuracy of Computations . . . . . . Topic: Analyzing Networks . . . . . . . . . . Chapter Two: Vector Spaces I Deﬁnition of Vector Space . . . . . . 1 Deﬁnition and Examples . . . . . . 2 Subspaces and Spanning Sets . . . II Linear Independence . . . . . . . . . 1 Deﬁnition and Examples . . . . . . III Basis and Dimension . . . . . . . . . 1 Basis . . . . . . . . . . . . . . . . . 2 Dimension . . . . . . . . . . . . . . 3 Vector Spaces and Linear Systems 4 Combining Subspaces∗ . . . . . . . Topic: Fields . . . . . . . . . . . . . . . . Topic: Crystals . . . . . . . . . . . . . . Topic: Voting Paradoxes . . . . . . . . . Topic: Dimensional Analysis . . . . . . . vii 1 1 2 11 20 32 32 38 46 46 52 62 64 68 72 79 80 80 91 101 101 112 112 118 123 130 140 142 146 152

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Chapter Three: Maps Between Spaces I Isomorphisms . . . . . . . . . . . . . . . . 1 Definition and Examples . . . . . . . . . 2 Dimension Characterizes Isomorphism . II Homomorphisms . . . . . . . . . . . . . . 1 Deﬁnition . . . . . . . . . . . . . . . . . 2 Rangespace and Nullspace . . . . . . . . III Computing Linear Maps . . . . . . . . . . 1 Representing Linear Maps with Matrices 2 Any Matrix Represents a Linear Map∗ . IV Matrix Operations . . . . . . . . . . . . . 1 Sums and Scalar Products . . . . . . . . 2 Matrix Multiplication . . . . . . . . . . 3 Mechanics of Matrix Multiplication . . . 4 Inverses . . . . . . . . . . . . . . . . . . V Change of Basis . . . . . . . . . . . . . . . 1 Changing Representations of Vectors . . 2 Changing Map Representations . . . . . VI Projection . . . . . . . . . . . . . . . . . . 1 Orthogonal Projection Into a Line∗ . . . 2 Gram-Schmidt Orthogonalization∗ . . . 3 Projection Into a Subspace∗ . . . . . . . Topic: Line of Best Fit . . . . . . . . . . . . . Topic: Geometry of Linear Maps . . . . . . . Topic: Markov Chains . . . . . . . . . . . . . Topic: Orthonormal Matrices . . . . . . . . . Chapter Four: Determinants I Definition . . . . . . . . . . . . . . . . 1 Exploration∗ . . . . . . . . . . . . . 2 Properties of Determinants . . . . . 3 The Permutation Expansion . . . . . 4 Determinants Exist∗ . . . . . . . . . II Geometry of Determinants . . . . . . . 1 Determinants as Size Functions . . . III Other Formulas . . . . . . . . . . . . . 1 Laplace’s Expansion∗ . . . . . . . . . Topic: Cramer’s Rule . . . . . . . . . . . . Topic: Speed of Calculating Determinants Topic: Projective Geometry . . . . . . . . Chapter Five: Similarity I Complex Vector Spaces . . . . . . . 1 Factoring and Complex Numbers; 2 Complex Representations . . . . II Similarity . . . . . . . . . . . . . . viii

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159 159 159 168 176 176 183 195 195 205 212 212 214 222 231 238 238 242 250 250 254 260 269 274 281 287 293 294 294 299 303 312 319 319 326 326 331 334 337 349 349 350 351 353

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1 Deﬁnition and Examples . . . . . . 2 Diagonalizability . . . . . . . . . . 3 Eigenvalues and Eigenvectors . . . III Nilpotence . . . . . . . . . . . . . . . 1 Self-Composition∗ . . . . . . . . . 2 Strings∗ . . . . . . . . . . . . . . . IV Jordan Form . . . . . . . . . . . . . . 1 Polynomials of Maps and Matrices∗ 2 Jordan Canonical Form∗ . . . . . . Topic: Method of Powers . . . . . . . . . Topic: Stable Populations . . . . . . . . Topic: Linear Recurrences . . . . . . . . Appendix Propositions . . . . . . . . . . Quantiﬁers . . . . . . . . . . Techniques of Proof . . . . . Sets, Functions, and Relations

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353 355 359 367 367 370 381 381 388 401 405 407 A-1 A-1 A-3 A-5 A-7

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Note: starred subsections are optional.

ix

Chapter One

Linear Systems
I Solving Linear Systems

Systems of linear equations are common in science and mathematics. These two examples from high school science [Onan] give a sense of how they arise. The ﬁrst example is from Physics. Suppose that we are given three objects, one with a mass known to be 2 kg, and are asked to ﬁnd the unknown masses. Suppose further that experimentation with a meter stick produces these two balances.
40 50 25 50

h

c
15

2

c
25

2

h

Since the sum of moments on the left of each balance equals the sum of moments on the right (the moment of an object is its mass times its distance from the balance point), the two balances give this system of two equations. 40h + 15c = 100 25c = 50 + 50h The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene C7 H8 and nitric acid HNO3 to produce trinitrotoluene C7 H5 O6 N3 along with the byproduct water (conditions have to be controlled very well, indeed — trinitrotoluene is better known as TNT). In what proportion should those components be mixed? The number of atoms of each element present before the reaction x C7 H8 + y HNO3 −→ z C7 H 5 O 6 N 3 + w H 2 O

must equal the number present afterward. Applying that principle to the ele1

2 ments C, H, N, and O in turn gives this system. 7x = 7z 8x + 1y = 5z + 2w 1y = 3z 3y = 6z + 1w

Chapter One. Linear Systems

To ﬁnish each of these examples requires solving a system of equations. In each, the equations involve only the ﬁrst power of the variables. This chapter shows how to solve any such system.

I.1 Gauss’ Method
1.1 Deﬁnition A linear equation in variables x1 , x2 , . . . , xn has the form a1 x1 + a2 x2 + a3 x3 + · · · + an xn = d where the numbers a1 , . . . , an ∈ R are the equation’s coeﬃcients and d ∈ R is the constant. An n-tuple (s1 , s2 , . . . , sn ) ∈ Rn is a solution of, or satisﬁes, that equation if substituting the numbers s1 , . . . , sn for the variables gives a true statement: a1 s1 + a2 s2 + . . . + an sn = d. A system of linear equations a1,1 x1 + a1,2 x2 + · · · + a1,n xn = d1 a2,1 x1 + a2,2 x2 + · · · + a2,n xn = d2 . . . am,1 x1 + am,2 x2 + · · · + am,n xn = dm has the solution (s1 , s2 , . . . , sn ) if that n-tuple is a solution of all of the equations in the system. 1.2 Example The ordered pair (−1, 5) is a solution of this system. 3x1 + 2x2 = 7 −x1 + x2 = 6 In contrast, (5, −1) is not a solution. Finding the set of all solutions is solving the system. No guesswork or good fortune is needed to solve a linear system. There is an algorithm that always works. The next example introduces that algorithm, called Gauss’ method. It transforms the system, step by step, into one with a form that is easily solved.

Section I. Solving Linear Systems 1.3 Example To solve this system 3x3 = 9 x1 + 5x2 − 2x3 = 2 1 =3 3 x1 + 2x2 we repeatedly transform it until it is in a form that is easy to solve.
swap row 1 with row 3

3

−→

1 3 x1 x1

+ 2x2 =3 + 5x2 − 2x3 = 2 3x3 = 9

multiply row 1 by 3

−→

x1 + 6x2 =9 x1 + 5x2 − 2x3 = 2 3x3 = 9 x1 + 6x2 = 9 −x2 − 2x3 = −7 3x3 = 9

add −1 times row 1 to row 2

−→

The third step is the only nontrivial one. We’ve mentally multiplied both sides of the ﬁrst row by −1, mentally added that to the old second row, and written the result in as the new second row. Now we can ﬁnd the value of each variable. The bottom equation shows that x3 = 3. Substituting 3 for x3 in the middle equation shows that x2 = 1. Substituting those two into the top equation gives that x1 = 3 and so the system has a unique solution: the solution set is { (3, 1, 3) }. Most of this subsection and the next one consists of examples of solving linear systems by Gauss’ method. We will use it throughout this book. It is fast and easy. But, before we get to those examples, we will ﬁrst show that this method is also safe in that it never loses solutions or picks up extraneous solutions. 1.4 Theorem (Gauss’ method) If a linear system is changed to another by one of these operations (1) an equation is swapped with another (2) an equation has both sides multiplied by a nonzero constant (3) an equation is replaced by the sum of itself and a multiple of another then the two systems have the same set of solutions. Each of those three operations has a restriction. Multiplying a row by 0 is not allowed because obviously that can change the solution set of the system. Similarly, adding a multiple of a row to itself is not allowed because adding −1 times the row to itself has the eﬀect of multiplying the row by 0. Finally, swapping a row with itself is disallowed to make some results in the fourth chapter easier to state and remember (and besides, self-swapping doesn’t accomplish anything).

4

Chapter One. Linear Systems

Proof. We will cover the equation swap operation here and save the other two

cases for Exercise 29. Consider this swap of row i with row j. a1,1 x1 + a1,2 x2 + · · · a1,1 x1 + a1,2 x2 + · · · a1,n xn = d1 . . . aj,1 x1 + aj,2 x2 + · · · ai,1 x1 + ai,2 x2 + · · · ai,n xn = di . . −→ . ai,1 x1 + ai,2 x2 + · · · aj,1 x1 + aj,2 x2 + · · · aj,n xn = dj . . . am,1 x1 + am,2 x2 + · · · am,1 x1 + am,2 x2 + · · · am,n xn = dm a1,n xn = d1 . . . aj,n xn = dj . . . ai,n xn = di . . . am,n xn = dm

The n-tuple (s1 , . . . , sn ) satisﬁes the system before the swap if and only if substituting the values, the s’s, for the variables, the x’s, gives true statements: a1,1 s1 +a1,2 s2 +· · ·+a1,n sn = d1 and . . . ai,1 s1 +ai,2 s2 +· · ·+ai,n sn = di and . . . aj,1 s1 + aj,2 s2 + · · · + aj,n sn = dj and . . . am,1 s1 + am,2 s2 + · · · + am,n sn = dm . In a requirement consisting of statements and-ed together we can rearrange the order of the statements, so that this requirement is met if and only if a1,1 s1 + a1,2 s2 + · · · + a1,n sn = d1 and . . . aj,1 s1 + aj,2 s2 + · · · + aj,n sn = dj and . . . ai,1 s1 + ai,2 s2 + · · · + ai,n sn = di and . . . am,1 s1 + am,2 s2 + · · · + am,n sn = dm . This is exactly the requirement that (s1 , . . . , sn ) solves the system after the row swap. QED 1.5 Deﬁnition The three operations from Theorem 1.4 are the elementary reduction operations, or row operations, or Gaussian operations. They are swapping, multiplying by a scalar or rescaling, and pivoting. When writing out the calculations, we will abbreviate ‘row i’ by ‘ρi ’. For instance, we will denote a pivot operation by kρi + ρj , with the row that is changed written second. We will also, to save writing, often list pivot steps together when they use the same ρi . 1.6 Example A typical use of Gauss’ method is to solve this system. x+ y =0 2x − y + 3z = 3 x − 2y − z = 3 The ﬁrst transformation of the system involves using the ﬁrst row to eliminate the x in the second row and the x in the third. To get rid of the second row’s 2x, we multiply the entire ﬁrst row by −2, add that to the second row, and write the result in as the new second row. To get rid of the third row’s x, we multiply the ﬁrst row by −1, add that to the third row, and write the result in as the new third row.
−2ρ1 +ρ2 −ρ1 +ρ3

x+

−→

y =0 −3y + 3z = 3 −3y − z = 3

Section I. Solving Linear Systems

5

(Note that the two ρ1 steps −2ρ1 + ρ2 and −ρ1 + ρ3 are written as one operation.) In this second system, the last two equations involve only two unknowns. To ﬁnish we transform the second system into a third system, where the last equation involves only one unknown. This transformation uses the second row to eliminate y from the third row.
−ρ2 +ρ3

x+

−→

y −3y +

=0 3z = 3 −4z = 0

Now we are set up for the solution. The third row shows that z = 0. Substitute that back into the second row to get y = −1, and then substitute back into the ﬁrst row to get x = 1. 1.7 Example For the Physics problem from the start of this chapter, Gauss’ method gives this. 40h + 15c = 100 −50h + 25c = 50
5/4ρ1 +ρ2

−→

40h +

15c = 100 (175/4)c = 175

So c = 4, and back-substitution gives that h = 1. (The Chemistry problem is solved later.) 1.8 Example The reduction x+ y+ z=9 2x + 4y − 3z = 1 3x + 6y − 5z = 0
−2ρ1 +ρ2 −3ρ1 +ρ3

−→

x+ y+ z= 9 2y − 5z = −17 3y − 8z = −27 x+ y+ 2y − z= 9 5z = −17 −(1/2)z = −(3/2)

−(3/2)ρ2 +ρ3

−→

shows that z = 3, y = −1, and x = 7. As these examples illustrate, Gauss’ method uses the elementary reduction operations to set up back-substitution. 1.9 Deﬁnition In each row, the ﬁrst variable with a nonzero coeﬃcient is the row’s leading variable. A system is in echelon form if each leading variable is to the right of the leading variable in the row above it (except for the leading variable in the ﬁrst row). 1.10 Example The only operation needed in the examples above is pivoting. Here is a linear system that requires the operation of swapping equations. After the ﬁrst pivot x− y =0 2x − 2y + z + 2w = 4 y + w=0 2z + w = 5 x−y
−2ρ1 +ρ2

−→

=0 z + 2w = 4 y + w=0 2z + w = 5

6

Chapter One. Linear Systems

the second equation has no leading y. To get one, we look lower down in the system for a row that has a leading y and swap it in.
ρ2 ↔ρ3

−→

x−y y

=0 + w=0 z + 2w = 4 2z + w = 5

(Had there been more than one row below the second with a leading y then we could have swapped in any one.) The rest of Gauss’ method goes as before. x−y y + z+ = 0 w= 0 2w = 4 −3w = −3

−2ρ3 +ρ4

−→

Back-substitution gives w = 1, z = 2 , y = −1, and x = −1. Strictly speaking, the operation of rescaling rows is not needed to solve linear systems. We have included it because we will use it later in this chapter as part of a variation on Gauss’ method, the Gauss-Jordan method. All of the systems seen so far have the same number of equations as unknowns. All of them have a solution, and for all of them there is only one solution. We ﬁnish this subsection by seeing for contrast some other things that can happen. 1.11 Example Linear systems need not have the same number of equations as unknowns. This system x + 3y = 1 2x + y = −3 2x + 2y = −2 has more equations than variables. Gauss’ method helps us understand this system also, since this
−2ρ1 +ρ2 −2ρ1 +ρ3

x+

−→

3y = 1 −5y = −5 −4y = −4

shows that one of the equations is redundant. Echelon form
−(4/5)ρ2 +ρ3

x+

−→

3y = 1 −5y = −5 0= 0

gives y = 1 and x = −2. The ‘0 = 0’ is derived from the redundancy. That example’s system has more equations than variables. Gauss’ method is also useful on systems with more variables than equations. Many examples are in the next subsection.

Section I. Solving Linear Systems

7

Another way that linear systems can diﬀer from the examples shown earlier is that some linear systems do not have a unique solution. This can happen in two ways. The ﬁrst is that it can fail to have any solution at all. 1.12 Example Contrast the system in the last example with this one. x + 3y = 1 2x + y = −3 2x + 2y = 0
−2ρ1 +ρ2 −2ρ1 +ρ3

x+

−→

3y = 1 −5y = −5 −4y = −2

Here the system is inconsistent: no pair of numbers satisﬁes all of the equations simultaneously. Echelon form makes this inconsistency obvious.
−(4/5)ρ2 +ρ3

x+

−→

3y = 1 −5y = −5 0= 2

The solution set is empty. 1.13 Example The prior system has more equations than unknowns, but that is not what causes the inconsistency — Example 1.11 has more equations than unknowns and yet is consistent. Nor is having more equations than unknowns necessary for inconsistency, as is illustrated by this inconsistent system with the same number of equations as unknowns. x + 2y = 8 2x + 4y = 8
−2ρ1 +ρ2

−→

x + 2y = 8 0 = −8

The other way that a linear system can fail to have a unique solution is to have many solutions. 1.14 Example In this system x+ y=4 2x + 2y = 8 any pair of numbers satisfying the ﬁrst equation automatically satisﬁes the second. The solution set {(x, y) x + y = 4} is inﬁnite; some of its members are (0, 4), (−1, 5), and (2.5, 1.5). The result of applying Gauss’ method here contrasts with the prior example because we do not get a contradictory equation.
−2ρ1 +ρ2

−→

x+y=4 0=0

Don’t be fooled by the ‘0 = 0’ equation in that example. It is not the signal that a system has many solutions.

8

Chapter One. Linear Systems

1.15 Example The absence of a ‘0 = 0’ does not keep a system from having many diﬀerent solutions. This system is in echelon form x+y+z=0 y+z=0 has no ‘0 = 0’, and yet has inﬁnitely many solutions. (For instance, each of these is a solution: (0, 1, −1), (0, 1/2, −1/2), (0, 0, 0), and (0, −π, π). There are inﬁnitely many solutions because any triple whose ﬁrst component is 0 and whose second component is the negative of the third is a solution.) Nor does the presence of a ‘0 = 0’ mean that the system must have many solutions. Example 1.11 shows that. So does this system, which does not have many solutions — in fact it has none — despite that when it is brought to echelon form it has a ‘0 = 0’ row. 2x − 2z = 6 y+ z=1 2x + y − z = 7 3y + 3z = 0 2x
−ρ1 +ρ3

−→

− 2z = 6 y+ z=1 y+ z=1 3y + 3z = 0 − 2z = 6 y+ z= 1 0= 0 0 = −3

2x
−ρ2 +ρ3 −3ρ2 +ρ4

−→

We will ﬁnish this subsection with a summary of what we’ve seen so far about Gauss’ method. Gauss’ method uses the three row operations to set a system up for back substitution. If any step shows a contradictory equation then we can stop with the conclusion that the system has no solutions. If we reach echelon form without a contradictory equation, and each variable is a leading variable in its row, then the system has a unique solution and we ﬁnd it by back substitution. Finally, if we reach echelon form without a contradictory equation, and there is not a unique solution (at least one variable is not a leading variable) then the system has many solutions. The next subsection deals with the third case — we will see how to describe the solution set of a system with many solutions. Exercises
1.16 Use Gauss’ method to ﬁnd the unique solution for each system. x −z=0 2x + 3y = 13 =1 (a) (b) 3x + y x − y = −1 −x + y + z = 4 1.17 Use Gauss’ method to solve each system or conclude ‘many solutions’ or ‘no solutions’.

Section I. Solving Linear Systems

9

(a) 2x + 2y = 5 (b) −x + y = 1 (c) x − 3y + z = 1 x − 4y = 0 x+y=2 x + y + 2z = 14 (f ) 2x + z+w= 5 (d) −x − y = 1 (e) 4y + z = 20 y − w = −1 −3x − 3y = 2 2x − 2y + z = 0 3x − z−w= 0 x +z= 5 4x + y + 2z + w = 9 x + y − z = 10 1.18 There are methods for solving linear systems other than Gauss’ method. One often taught in high school is to solve one of the equations for a variable, then substitute the resulting expression into other equations. That step is repeated until there is an equation with only one variable. From that, the ﬁrst number in the solution is derived, and then back-substitution can be done. This method takes longer than Gauss’ method, since it involves more arithmetic operations, and is also more likely to lead to errors. To illustrate how it can lead to wrong conclusions, we will use the system x + 3y = 1 2x + y = −3 2x + 2y = 0 from Example 1.12. (a) Solve the ﬁrst equation for x and substitute that expression into the second equation. Find the resulting y. (b) Again solve the ﬁrst equation for x, but this time substitute that expression into the third equation. Find this y. What extra step must a user of this method take to avoid erroneously concluding a system has a solution? 1.19 For which values of k are there no solutions, many solutions, or a unique solution to this system? x− y=1 3x − 3y = k 1.20 This system is not linear, in some sense, 2 sin α − cos β + 3 tan γ = 3 4 sin α + 2 cos β − 2 tan γ = 10 6 sin α − 3 cos β + tan γ = 9 and yet we can nonetheless apply Gauss’ method. Do so. Does the system have a solution? 1.21 What conditions must the constants, the b’s, satisfy so that each of these systems has a solution? Hint. Apply Gauss’ method and see what happens to the right side. [Anton] (a) x − 3y = b1 (b) x1 + 2x2 + 3x3 = b1 3x + y = b2 2x1 + 5x2 + 3x3 = b2 x + 7y = b3 x1 + 8x3 = b3 2x + 4y = b4 1.22 True or false: a system with more unknowns than equations has at least one solution. (As always, to say ‘true’ you must prove it, while to say ‘false’ you must produce a counterexample.) 1.23 Must any Chemistry problem like the one that starts this subsection — a balance the reaction problem — have inﬁnitely many solutions? 1.24 Find the coeﬃcients a, b, and c so that the graph of f (x) = ax2 + bx + c passes through the points (1, 2), (−1, 6), and (2, 3).

10

Chapter One. Linear Systems
1.25 Gauss’ method works by combining the equations in a system to make new equations. (a) Can the equation 3x−2y = 5 be derived, by a sequence of Gaussian reduction steps, from the equations in this system? x+y=1 4x − y = 6 (b) Can the equation 5x−3y = 2 be derived, by a sequence of Gaussian reduction steps, from the equations in this system? 2x + 2y = 5 3x + y = 4 (c) Can the equation 6x − 9y + 5z = −2 be derived, by a sequence of Gaussian reduction steps, from the equations in the system? 2x + y − z = 4 6x − 3y + z = 5 1.26 Prove that, where a, b, . . . , e are real numbers and a = 0, if ax + by = c has the same solution set as ax + dy = e then they are the same equation. What if a = 0? 1.27 Show that if ad − bc = 0 then ax + by = j cx + dy = k has a unique solution. 1.28 In the system ax + by = c dx + ey = f each of the equations describes a line in the xy-plane. By geometrical reasoning, show that there are three possibilities: there is a unique solution, there is no solution, and there are inﬁnitely many solutions. 1.29 Finish the proof of Theorem 1.4. 1.30 Is there a two-unknowns linear system whose solution set is all of R2 ? 1.31 Are any of the operations used in Gauss’ method redundant? That is, can any of the operations be synthesized from the others? 1.32 Prove that each operation of Gauss’ method is reversible. That is, show that if two systems are related by a row operation S1 → S2 then there is a row operation to go back S2 → S1 .

? 1.33 A box holding pennies, nickels and dimes contains thirteen coins with a total value of 83 cents. How many coins of each type are in the box? [Anton] ? 1.34 Four positive integers are given. Select any three of the integers, ﬁnd their arithmetic average, and add this result to the fourth integer. Thus the numbers 29, 23, 21, and 17 are obtained. One of the original integers is:

Section I. Solving Linear Systems

11

(a) 19 (b) 21 (c) 23 (d) 29 (e) 17 [Con. Prob. 1955] ? 1.35 Laugh at this: AHAHA + TEHE = TEHAW. It resulted from substituting a code letter for each digit of a simple example in addition, and it is required to identify the letters and prove the solution unique. [Am. Math. Mon., Jan. 1935] ? 1.36 The Wohascum County Board of Commissioners, which has 20 members, recently had to elect a President. There were three candidates (A, B, and C); on each ballot the three candidates were to be listed in order of preference, with no abstentions. It was found that 11 members, a majority, preferred A over B (thus the other 9 preferred B over A). Similarly, it was found that 12 members preferred C over A. Given these results, it was suggested that B should withdraw, to enable a runoﬀ election between A and C. However, B protested, and it was then found that 14 members preferred B over C! The Board has not yet recovered from the resulting confusion. Given that every possible order of A, B, C appeared on at least one ballot, how many members voted for B as their ﬁrst choice? [Wohascum no. 2] ? 1.37 “This system of n linear equations with n unknowns,” said the Great Mathematician, “has a curious property.” “Good heavens!” said the Poor Nut, “What is it?” “Note,” said the Great Mathematician, “that the constants are in arithmetic progression.” “It’s all so clear when you explain it!” said the Poor Nut. “Do you mean like 6x + 9y = 12 and 15x + 18y = 21?” “Quite so,” said the Great Mathematician, pulling out his bassoon. “Indeed, the system has a unique solution. Can you ﬁnd it?” “Good heavens!” cried the Poor Nut, “I am baﬄed.” Are you? [Am. Math. Mon., Jan. 1963]

I.2 Describing the Solution Set
A linear system with a unique solution has a solution set with one element. A linear system with no solution has a solution set that is empty. In these cases the solution set is easy to describe. Solution sets are a challenge to describe only when they contain many elements. 2.1 Example This system has many solutions because in echelon form 2x +z=3 x−y−z=1 3x − y =4
−(1/2)ρ1 +ρ2 −(3/2)ρ1 +ρ3

2x

−→

+ z= 3 −y − (3/2)z = −1/2 −y − (3/2)z = −1/2 + z= 3 −y − (3/2)z = −1/2 0= 0

−ρ2 +ρ3

2x

−→

not all of the variables are leading variables. The Gauss’ method theorem showed that a triple satisﬁes the ﬁrst system if and only if it satisﬁes the third. Thus, the solution set {(x, y, z) 2x + z = 3 and x − y − z = 1 and 3x − y = 4}

12

Chapter One. Linear Systems

can also be described as {(x, y, z) 2x + z = 3 and −y − 3z/2 = −1/2}. However, this second description is not much of an improvement. It has two equations instead of three, but it still involves some hard-to-understand interaction among the variables. To get a description that is free of any such interaction, we take the variable that does not lead any equation, z, and use it to describe the variables that do lead, x and y. The second equation gives y = (1/2) − (3/2)z and the ﬁrst equation gives x = (3/2) − (1/2)z. Thus, the solution set can be described as {(x, y, z) = ((3/2) − (1/2)z, (1/2) − (3/2)z, z) z ∈ R}. For instance, (1/2, −5/2, 2) is a solution because taking z = 2 gives a ﬁrst component of 1/2 and a second component of −5/2. The advantage of this description over the ones above is that the only variable appearing, z, is unrestricted — it can be any real number. 2.2 Deﬁnition The non-leading variables in an echelon-form linear system are free variables. In the echelon form system derived in the above example, x and y are leading variables and z is free. 2.3 Example A linear system can end with more than one variable free. This row reduction x+ y+ z− w= 1 y − z + w = −1 3x + 6z − 6w = 6 −y + z − w = 1 x+
−3ρ1 +ρ3

−→

y+ z− w= 1 y − z + w = −1 −3y + 3z − 3w = 3 −y + z − w = 1

3ρ2 +ρ3 ρ2 +ρ4

−→

x+y+z−w= 1 y − z + w = −1 0= 0 0= 0

ends with x and y leading, and with both z and w free. To get the description that we prefer we will start at the bottom. We ﬁrst express y in terms of the free variables z and w with y = −1 + z − w. Next, moving up to the top equation, substituting for y in the ﬁrst equation x + (−1 + z − w) + z − w = 1 and solving for x yields x = 2 − 2z + 2w. Thus, the solution set is {2 − 2z + 2w, −1 + z − w, z, w) z, w ∈ R}. We prefer this description because the only variables that appear, z and w, are unrestricted. This makes the job of deciding which four-tuples are system solutions into an easy one. For instance, taking z = 1 and w = 2 gives the solution (4, −2, 1, 2). In contrast, (3, −2, 1, 2) is not a solution, since the ﬁrst component of any solution must be 2 minus twice the third component plus twice the fourth.

Section I. Solving Linear Systems 2.4 Example After this reduction 2x − 2y =0 z + 3w = 2 3x − 3y =0 x − y + 2z + 6w = 4 2x − 2y
−(3/2)ρ1 +ρ3 −(1/2)ρ1 +ρ4

13

−→

=0 z + 3w = 2 0=0 2z + 6w = 4 =0 z + 3w = 2 0=0 0=0

2x − 2y
−2ρ2 +ρ4

−→

x and z lead, y and w are free. The solution set is {(y, y, 2 − 3w, w) y, w ∈ R}. For instance, (1, 1, 2, 0) satisﬁes the system — take y = 1 and w = 0. The fourtuple (1, 0, 5, 4) is not a solution since its ﬁrst coordinate does not equal its second. We refer to a variable used to describe a family of solutions as a parameter and we say that the set above is parametrized with y and w. (The terms ‘parameter’ and ‘free variable’ do not mean the same thing. Above, y and w are free because in the echelon form system they do not lead any row. They are parameters because they are used in the solution set description. We could have instead parametrized with y and z by rewriting the second equation as w = 2/3 − (1/3)z. In that case, the free variables are still y and w, but the parameters are y and z. Notice that we could not have parametrized with x and y, so there is sometimes a restriction on the choice of parameters. The terms ‘parameter’ and ‘free’ are related because, as we shall show later in this chapter, the solution set of a system can always be parametrized with the free variables. Consequently, we shall parametrize all of our descriptions in this way.) 2.5 Example This is another system with inﬁnitely many solutions. x + 2y =1 2x +z =2 3x + 2y + z − w = 4
−2ρ1 +ρ2 −3ρ1 +ρ3

x+

−→

2y =1 −4y + z =0 −4y + z − w = 1 2y −4y + z =1 =0 −w = 1

−ρ2 +ρ3

x+

−→

The leading variables are x, y, and w. The variable z is free. (Notice here that, although there are inﬁnitely many solutions, the value of one of the variables is ﬁxed — w = −1.) Write w in terms of z with w = −1 + 0z. Then y = (1/4)z. To express x in terms of z, substitute for y into the ﬁrst equation to get x = 1 − (1/2)z. The solution set is {(1 − (1/2)z, (1/4)z, z, −1) z ∈ R}. We ﬁnish this subsection by developing the notation for linear systems and their solution sets that we shall use in the rest of this book. 2.6 Deﬁnition An m×n matrix is a rectangular array of numbers with m rows and n columns. Each number in the matrix is an entry,

14

Chapter One. Linear Systems

Matrices are usually named by upper case roman letters, e.g. A. Each entry is denoted by the corresponding lower-case letter, e.g. ai,j is the number in row i and column j of the array. For instance, A= 1 3 2.2 4 5 −7

has two rows and three columns, and so is a 2 × 3 matrix. (Read that “twoby-three”; the number of rows is always stated ﬁrst.) The entry in the second row and ﬁrst column is a2,1 = 3. Note that the order of the subscripts matters: a1,2 = a2,1 since a1,2 = 2.2. (The parentheses around the array are a typographic device so that when two matrices are side by side we can tell where one ends and the other starts.) Matrices occur throughout this book. We shall use Mn×m to denote the collection of n×m matrices. 2.7 Example We can abbreviate this linear system x1 + 2x2 =4 x2 − x3 = 0 x1 + 2x3 = 4 with this matrix.  1 2 0 1 1 0 0 −1 2  4 0 4

The vertical bar just reminds a reader of the diﬀerence between the coeﬃcients on the systems’s left hand side and the constants on the right. When a bar is used to divide a matrix into parts, we call it an augmented matrix. In this notation, Gauss’ method goes this way.       1 2 0 4 1 2 0 4 1 2 0 4 −ρ1 +ρ3 2ρ2 +ρ3 0 1 −1 0 −→ 0 1 −1 0 −→ 0 1 −1 0 1 0 2 4 0 −2 2 0 0 0 0 0 The second row stands for y − z = 0 and the ﬁrst row stands for x + 2y = 4 so the solution set is {(4 − 2z, z, z) z ∈ R}. One advantage of the new notation is that the clerical load of Gauss’ method — the copying of variables, the writing of +’s and =’s, etc. — is lighter. We will also use the array notation to clarify the descriptions of solution sets. A description like {(2 − 2z + 2w, −1 + z − w, z, w) z, w ∈ R} from Example 2.3 is hard to read. We will rewrite it to group all the constants together, all the coeﬃcients of z together, and all the coeﬃcients of w together. We will write them vertically, in one-column wide matrices.       2 −2 2 −1  1  −1 {  +   · z +   · w z, w ∈ R} 0 1 0 0 0 1

Section I. Solving Linear Systems

15

For instance, the top line says that x = 2 − 2z + 2w. The next section gives a geometric interpretation that will help us picture the solution sets when they are written in this way. 2.8 Deﬁnition A vector (or column vector ) is a matrix with a single column. A matrix with a single row is a row vector . The entries of a vector are its components. Vectors are an exception to the convention of representing matrices with capital roman letters. We use lower-case roman or greek letters overlined with an arrow: a, b, . . . or α, β, . . . (boldface is also common: a or α). For instance, this is a column vector with a third component of 7.   1 v = 3 7 2.9 Deﬁnition The linear equation a1 x1 + a2 x2 + · · · + an xn = d with unknowns x1 , . . . , xn is satisﬁed by   s1 . s= .  . sn if a1 s1 + a2 s2 + · · · + an sn = d. A vector satisﬁes a linear system if it satisﬁes each equation in the system. The style of description of solution sets that we use involves adding the vectors, and also multiplying them by real numbers, such as the z and w. We need to deﬁne these operations. 2.10 Deﬁnition The vector sum of u and v is this.       u1 v1 u1 + v1  .  .   . . u+v = . + . =  . . . un vn u n + vn In general, two matrices with the same number of rows and the same number of columns add in this way, entry-by-entry. 2.11 Deﬁnition The scalar multiplication of the real number r and the vector v is this.     v1 rv1 .  .  r·v =r· . = .  . . vn rvn In general, any matrix is multiplied by a real number in this entry-by-entry way.

16

Chapter One. Linear Systems

Scalar multiplication can be written in either order: r · v or v · r, or without the ‘·’ symbol: rv. (Do not refer to scalar multiplication as ‘scalar product’ because that name is used for a diﬀerent operation.) 2.12 Example         2 3 2+3 5 3 + −1 = 3 − 1 = 2 1 4 1+4 5    7 1  4   28   7· = −1  −7  −21 −3 

Notice that the deﬁnitions of vector addition and scalar multiplication agree where they overlap, for instance, v + v = 2v. With the notation deﬁned, we can now solve systems in the way that we will use throughout this book. 2.13 Example This system 2x + y − w =4 y + w+u=4 x − z + 2w =0 reduces in  2 1 0 1 1 0 this way. 0 0 −1 −1 1 2 0 1 0  4 4 0
−(1/2)ρ1 +ρ3

−→

(1/2)ρ2 +ρ3

−→

  2 1 0 −1 0 4 0 1 0 1 1 4 0 −1/2 −1 5/2 0 −2   4 2 1 0 −1 0 0 1 0 1 1 4 0 0 −1 3 1/2 0

The solution set is {(w + (1/2)u, 4 − w − u, 3w + (1/2)u, w, u) w, u ∈ R}. We write that in vector form.         x 0 1 1/2  y  4 −1  −1           z  = 0 +  3  w + 1/2 u w, u ∈ R} {        w 0  1   0  u 0 0 1 Note again how well vector notation sets oﬀ the coeﬃcients of each parameter. For instance, the third row of the vector form shows plainly that if u is held ﬁxed then z increases three times as fast as w. That format also shows plainly that there are inﬁnitely many solutions. For example, we can ﬁx u as 0, let w range over the real numbers, and consider the ﬁrst component x. We get inﬁnitely many ﬁrst components and hence inﬁnitely many solutions.

Section I. Solving Linear Systems

17

Another thing shown plainly is that setting both w and u to zero gives that this     0 x  y  4      z  = 0     w 0 0 u is a particular solution of the linear system. 2.14 Example In the same way, this system x− y+ z=1 3x + z=3 5x − 2y + 3z = 5 reduces  1 −1 1 3 0 1 5 −2 3

  1 1 −3ρ1 +ρ2 3 −→ 0 −5ρ1 +ρ3 5 0

−1 3 3

1 −2 −2

  1 1 −ρ2 +ρ3 0 −→ 0 0 0

−1 3 0

1 −2 0

 1 0 0

to a one-parameter solution set.     1 −1/3 {0 +  2/3  z z ∈ R} 0 1 Before the exercises, we pause to point out some things that we have yet to do. The ﬁrst two subsections have been on the mechanics of Gauss’ method. Except for one result, Theorem 1.4 — without which developing the method doesn’t make sense since it says that the method gives the right answers — we have not stopped to consider any of the interesting questions that arise. For example, can we always describe solution sets as above, with a particular solution vector added to an unrestricted linear combination of some other vectors? The solution sets we described with unrestricted parameters were easily seen to have inﬁnitely many solutions so an answer to this question could tell us something about the size of solution sets. An answer to that question could also help us picture the solution sets, in R2 , or in R3 , etc. Many questions arise from the observation that Gauss’ method can be done in more than one way (for instance, when swapping rows, we may have a choice of which row to swap with). Theorem 1.4 says that we must get the same solution set no matter how we proceed, but if we do Gauss’ method in two diﬀerent ways must we get the same number of free variables both times, so that any two solution set descriptions have the same number of parameters? Must those be the same variables (e.g., is it impossible to solve a problem one way and get y and w free or solve it another way and get y and z free)?

18

Chapter One. Linear Systems

In the rest of this chapter we answer these questions. The answer to each is ‘yes’. The ﬁrst question is answered in the last subsection of this section. In the second section we give a geometric description of solution sets. In the ﬁnal section of this chapter we tackle the last set of questions. Consequently, by the end of the ﬁrst chapter we will not only have a solid grounding in the practice of Gauss’ method, we will also have a solid grounding in the theory. We will be sure of what can and cannot happen in a reduction. Exercises
2.15 Find the indicated entry of the matrix, if it is deﬁned. A= 1 2 3 −1 1 4

(a) a2,1 (b) a1,2 (c) a2,2 (d) a3,1 2.16 Give the size of each matrix. 1 1 1 0 4 5 10 1 (a) (b) −1 (c) 2 1 5 10 5 3 −1 2.17 Do the indicated vector operation, if it is deﬁned. 2 3 1 3 4 2 3 (a) 1 + 0 (b) 5 (c) 5 − 1 (d) 7 +9 −1 1 5 1 4 1 1 1 3 2 1 1 (e) + 2 (f ) 6 1 − 4 0 + 2 1 2 3 1 3 5 2.18 Solve each system using matrix notation. Express the solution using vectors. (a) 3x + 6y = 18 (b) x + y = 1 (c) x1 + x3 = 4 x + 2y = 6 x − y = −1 x1 − x2 + 2x3 = 5 4x1 − x2 + 5x3 = 17 (d) 2a + b − c = 2 (e) x + 2y − z =3 (f ) x +z+w=4 2a +c=3 2x + y +w=4 2x + y −w=2 a−b =0 x− y+z+w=1 3x + y + z =7 2.19 Solve each system using matrix notation. Give each solution set in vector notation. (a) 2x + y − z = 1 (b) x − z =1 (c) x − y + z =0 4x − y =3 y + 2z − w = 3 y +w=0 x + 2y + 3z − w = 7 3x − 2y + 3z + w = 0 −y −w=0 (d) a + 2b + 3c + d − e = 1 3a − b + c + d + e = 3 2.20 The vector is in the set. What value of the parameters produces that vector? 5 1 (a) ,{ k k ∈ R} −5 −1 −1 −2 3 2 , { 1 i + 0 j i, j ∈ R} (b) 1 0 1

Section I. Solving Linear Systems
2 1 0 −4 , { 1 m + 0 n m, n ∈ R} 1 0 2 2.21 Decide if the vector is in the set. 3 −6 (a) ,{ k k ∈ R} −1 2 (c)

19

5 5 ,{ j j ∈ R} 4 −4 1 0 2 1 ,{ 3 + −1 r r ∈ R} (c) 3 −7 −1 −3 2 1 (d) 0 , { 0 j + −1 k j, k ∈ R} 1 1 1 2.22 Parametrize the solution set of this one-equation system. x1 + x2 + · · · + xn = 0 2.23 (a) Apply Gauss’ method to the left-hand side to solve x + 2y − w=a 2x +z =b x+ y + 2w = c for x, y, z, and w, in terms of the constants a, b, and c. (b) Use your answer from the prior part to solve this. x + 2y − w= 3 2x +z = 1 x+ y + 2w = −2 2.24 Why is the comma needed in the notation ‘ai,j ’ for matrix entries? 2.25 Give the 4×4 matrix whose i, j-th entry is (a) i + j; (b) −1 to the i + j power. 2.26 For any matrix A, the transpose of A, written Atrans , is the matrix whose columns are the rows of A. Find the transpose of each of these. 1 1 2 3 2 −3 5 10 (a) (b) (c) (d) 1 4 5 6 1 1 10 5 0 2 2.27 (a) Describe all functions f (x) = ax + bx + c such that f (1) = 2 and f (−1) = 6. (b) Describe all functions f (x) = ax2 + bx + c such that f (1) = 2. 2.28 Show that any set of ﬁve points from the plane R2 lie on a common conic section, that is, they all satisfy some equation of the form ax2 + by 2 + cxy + dx + ey + f = 0 where some of a, . . . , f are nonzero. 2.29 Make up a four equations/four unknowns system having (a) a one-parameter solution set; (b) a two-parameter solution set; (c) a three-parameter solution set. ? 2.30 (a) Solve the system of equations. ax + y = a2 x + ay = 1 For what values of a does the system fail to have solutions, and for what values of a are there inﬁnitely many solutions? (b)

20
(b) Answer the above question for the system. ax + y = a3 x + ay = 1

Chapter One. Linear Systems

[USSR Olympiad no. 174] ? 2.31 In air a gold-surfaced sphere weighs 7588 grams. It is known that it may contain one or more of the metals aluminum, copper, silver, or lead. When weighed successively under standard conditions in water, benzene, alcohol, and glycerine its respective weights are 6588, 6688, 6778, and 6328 grams. How much, if any, of the forenamed metals does it contain if the speciﬁc gravities of the designated substances are taken to be as follows? Aluminum 2.7 Alcohol 0.81 Copper 8.9 Benzene 0.90 Gold 19.3 Glycerine 1.26 Lead 11.3 Water 1.00 Silver 10.8 [Math. Mag., Sept. 1952]

I.3 General = Particular + Homogeneous
The prior subsection has many descriptions of solution sets. They all ﬁt a pattern. They have a vector that is a particular solution of the system added to an unrestricted combination of some other vectors. The solution set from Example 2.13 illustrates.       0 1/2 1 4  −1  −1       { 0 + w  3  + u 1/2 w, u ∈ R}       0  0  1 0 1 0
particular solution unrestricted combination

The combination is unrestricted in that w and u can be any real numbers — there is no condition like “such that 2w − u = 0” that would restrict which pairs w, u can be used to form combinations. That example shows an inﬁnite solution set conforming to the pattern. We can think of the other two kinds of solution sets as also ﬁtting the same pattern. A one-element solution set ﬁts in that it has a particular solution, and the unrestricted combination part is a trivial sum (that is, instead of being a combination of two vectors, as above, or a combination of one vector, it is a combination of no vectors). A zero-element solution set ﬁts the pattern since there is no particular solution, and so the set of sums of that form is empty. We will show that the examples from the prior subsection are representative, in that the description pattern discussed above holds for every solution set.

Section I. Solving Linear Systems

21

3.1 Theorem For any linear system there are vectors β1 , . . . , βk such that the solution set can be described as {p + c1 β1 + · · · + ck βk c1 , . . . , ck ∈ R} where p is any particular solution, and where the system has k free variables. This description has two parts, the particular solution p and also the unrestricted linear combination of the β’s. We shall prove the theorem in two corresponding parts, with two lemmas. We will focus ﬁrst on the unrestricted combination part. To do that, we consider systems that have the vector of zeroes as one of the particular solutions, so that p + c1 β1 + · · · + ck βk can be shortened to c1 β1 + · · · + ck βk . 3.2 Deﬁnition A linear equation is homogeneous if it has a constant of zero, that is, if it can be put in the form a1 x1 + a2 x2 + · · · + an xn = 0. (These are ‘homogeneous’ because all of the terms involve the same power of their variable — the ﬁrst power — including a ‘0x0 ’ that we can imagine is on the right side.) 3.3 Example With any linear system like 3x + 4y = 3 2x − y = 1 we associate a system of homogeneous equations by setting the right side to zeros. 3x + 4y = 0 2x − y = 0 Our interest in the homogeneous system associated with a linear system can be understood by comparing the reduction of the system 3x + 4y = 3 2x − y = 1
−(2/3)ρ1 +ρ2

−→

3x +

4y = 3 −(11/3)y = −1

with the reduction of the associated homogeneous system. 3x + 4y = 0 2x − y = 0
−(2/3)ρ1 +ρ2

−→

3x +

4y = 0 −(11/3)y = 0

Obviously the two reductions go in the same way. We can study how linear systems are reduced by instead studying how the associated homogeneous systems are reduced. Studying the associated homogeneous system has a great advantage over studying the original system. Nonhomogeneous systems can be inconsistent. But a homogeneous system must be consistent since there is always at least one solution, the vector of zeros.

22

Chapter One. Linear Systems

3.4 Deﬁnition A column or row vector of all zeros is a zero vector , denoted 0. There are many diﬀerent zero vectors, e.g., the one-tall zero vector, the two-tall zero vector, etc. Nonetheless, people often refer to “the” zero vector, expecting that the size of the one being discussed will be clear from the context. 3.5 Example Some homogeneous systems have the zero vector as their only solution. 3x + 2y + z = 0 6x + 4y =0 y+z=0
−2ρ1 +ρ2

3x + 2y +

−→

z=0 3x + 2y + z=0 ρ2 ↔ρ3 −2z = 0 −→ y+ z=0 y+ z=0 −2z = 0

3.6 Example Some homogeneous systems have many solutions. One example is the Chemistry problem from the ﬁrst page of this book. 7x − 7z =0 8x + y − 5z − 2w = 0 y − 3z =0 3y − 6z − w = 0 7x
−(8/7)ρ1 +ρ2

−→

− 7z =0 y + 3z − 2w = 0 y − 3z =0 3y − 6z − w = 0 − y+ 7z =0 3z − 2w = 0 −6z + 2w = 0 −15z + 5w = 0

7x
−ρ2 +ρ3 −3ρ2 +ρ4

−→

7x
−(5/2)ρ3 +ρ4

−→

− 7z =0 y + 3z − 2w = 0 −6z + 2w = 0 0=0

The solution set:

  1/3  1  {  w w ∈ R} 1/3 1

has many vectors besides the zero vector (if we interpret w as a number of molecules then solutions make sense only when w is a nonnegative multiple of 3). We now have the terminology to prove the two parts of Theorem 3.1. The ﬁrst lemma deals with unrestricted combinations. 3.7 Lemma For any homogeneous linear system there exist vectors β1 , . . . , βk such that the solution set of the system is {c1 β1 + · · · + ck βk c1 , . . . , ck ∈ R} where k is the number of free variables in an echelon form version of the system.

Section I. Solving Linear Systems

23

Before the proof, we will recall the back substitution calculations that were done in the prior subsection. Imagine that we have brought a system to this echelon form. x + 2y − z + 2w = 0 −3y + z =0 −w = 0 We next perform back-substitution to express each variable in terms of the free variable z. Working from the bottom up, we get ﬁrst that w is 0 · z, next that y is (1/3) · z, and then substituting those two into the top equation x + 2((1/3)z) − z + 2(0) = 0 gives x = (1/3) · z. So, back substitution gives a parametrization of the solution set by starting at the bottom equation and using the free variables as the parameters to work row-by-row to the top. The proof below follows this pattern. Comment: That is, this proof just does a veriﬁcation of the bookkeeping in back substitution to show that we haven’t overlooked any obscure cases where this procedure fails, say, by leading to a division by zero. So this argument, while quite detailed, doesn’t give us any new insights. Nevertheless, we have written it out for two reasons. The ﬁrst reason is that we need the result — the computational procedure that we employ must be veriﬁed to work as promised. The second reason is that the row-by-row nature of back substitution leads to a proof that uses the technique of mathematical induction.∗ This is an important, and non-obvious, proof technique that we shall use a number of times in this book. Doing an induction argument here gives us a chance to see one in a setting where the proof material is easy to follow, and so the technique can be studied. Readers who are unfamiliar with induction arguments should be sure to master this one and the ones later in this chapter before going on to the second chapter.
Proof. First use Gauss’ method to reduce the homogeneous system to echelon

form. We will show that each leading variable can be expressed in terms of free variables. That will ﬁnish the argument because then we can use those free variables as the parameters. That is, the β’s are the vectors of coeﬃcients of the free variables (as in Example 3.6, where the solution is x = (1/3)w, y = w, z = (1/3)w, and w = w). We will proceed by mathematical induction, which has two steps. The base step of the argument will be to focus on the bottom-most non-‘0 = 0’ equation and write its leading variable in terms of the free variables. The inductive step of the argument will be to argue that if we can express the leading variables from the bottom t rows in terms of free variables, then we can express the leading variable of the next row up — the t + 1-th row up from the bottom — in terms of free variables. With those two steps, the theorem will be proved because by the base step it is true for the bottom equation, and by the inductive step the fact that it is true for the bottom equation shows that it is true for the next one up, and then another application of the inductive step implies it is true for third equation up, etc.

24

Chapter One. Linear Systems

For the base step, consider the bottom-most non-‘0 = 0’ equation (the case where all the equations are ‘0 = 0’ is trivial). We call that the m-th row: am,
m

x

m

+ am,

m +1

x

m +1

+ · · · + am,n xn = 0

where am, m = 0. (The notation here has ‘ ’ stand for ‘leading’, so am, m means “the coeﬃcient, from the row m of the variable leading row m”.) Either there are variables in this equation other than the leading one x m or else there are not. If there are other variables x m +1 , etc., then they must be free variables because this is the bottom non-‘0 = 0’ row. Move them to the right and divide by am, m x
m

= (−am,

m +1

/am,

m

)x

m +1

+ · · · + (−am,n /am,

m

)xn

to express this leading variable in terms of free variables. If there are no free variables in this equation then x m = 0 (see the “tricky point” noted following this proof). For the inductive step, we assume that for the m-th equation, and for the (m − 1)-th equation, . . . , and for the (m − t)-th equation, we can express the leading variable in terms of free variables (where 0 ≤ t < m). To prove that the same is true for the next equation up, the (m − (t + 1))-th equation, we take each variable that leads in a lower-down equation x m , . . . , x m−t and substitute its expression in terms of free variables. The result has the form am−(t+1),
m−(t+1)

x

m−(t+1)

+ sums of multiples of free variables = 0

where am−(t+1), m−(t+1) = 0. We move the free variables to the right-hand side and divide by am−(t+1), m−(t+1) , to end with x m−(t+1) expressed in terms of free variables. Because we have shown both the base step and the inductive step, by the principle of mathematical induction the proposition is true. QED We say that the set {c1 β1 + · · · + ck βk c1 , . . . , ck ∈ R} is generated by or spanned by the set of vectors {β1 , . . . , βk }. There is a tricky point to this deﬁnition. If a homogeneous system has a unique solution, the zero vector, then we say the solution set is generated by the empty set of vectors. This ﬁts with the pattern of the other solution sets: in the proof above the solution set is derived by taking the c’s to be the free variables and if there is a unique solution then there are no free variables. This proof incidentally shows, as discussed after Example 2.4, that solution sets can always be parametrized using the free variables. The next lemma ﬁnishes the proof of Theorem 3.1 by considering the particular solution part of the solution set’s description. 3.8 Lemma For a linear system, where p is any particular solution, the solution set equals this set. {p + h h satisﬁes the associated homogeneous system}

Section I. Solving Linear Systems

25

Proof. We will show mutual set inclusion, that any solution to the system is

in the above set and that anything in the set is a solution to the system.∗ For set inclusion the ﬁrst way, that if a vector solves the system then it is in the set described above, assume that s solves the system. Then s − p solves the associated homogeneous system since for each equation index i, ai,1 (s1 − p1 ) + · · · + ai,n (sn − pn ) = (ai,1 s1 + · · · + ai,n sn ) − (ai,1 p1 + · · · + ai,n pn ) = di − di =0

where pj and sj are the j-th components of p and s. We can write s − p as h, where h solves the associated homogeneous system, to express s in the required p + h form. For set inclusion the other way, take a vector of the form p + h, where p solves the system and h solves the associated homogeneous system, and note that it solves the given system: for any equation index i, ai,1 (p1 + h1 ) + · · · + ai,n (pn + hn ) = (ai,1 p1 + · · · + ai,n pn ) + (ai,1 h1 + · · · + ai,n hn ) = di + 0 = di where hj is the j-th component of h.
QED

The two lemmas above together establish Theorem 3.1. We remember that theorem with the slogan “General = Particular + Homogeneous”. 3.9 Example This system illustrates Theorem 3.1. x + 2y − z = 1 2x + 4y =2 y − 3z = 0 Gauss’ method
−2ρ1 +ρ2

−→

x + 2y − z = 1 x + 2y − z = 1 ρ2 ↔ρ3 2z = 0 −→ y − 3z = 0 y − 3z = 0 2z = 0

shows that the general solution is a singleton set.   1 {0} 0

26

Chapter One. Linear Systems

That single vector is, of course, a particular solution. The associated homogeneous system reduces via the same row operations x + 2y − z = 0 2x + 4y =0 y − 3z = 0 to also give a singleton set.   0 {0} 0 As the theorem states, and as discussed at the start of this subsection, in this single-solution case the general solution results from taking the particular solution and adding to it the unique solution of the associated homogeneous system. 3.10 Example Also discussed there is that the case where the general solution set is empty ﬁts the ‘General = Particular+Homogeneous’ pattern. This system illustrates. Gauss’ method x + z + w = −1 2x − y + w= 3 x + y + 3z + 2w = 1
−2ρ1 +ρ2 −ρ1 +ρ3 −2ρ1 +ρ2 ρ2 ↔ρ3

−→

−→

x + 2y − z = 0 y − 3z = 0 2z = 0

x

−→

+ z + w = −1 −y − 2z − w = 5 y + 2z + w = 2

shows that it has no solutions. The associated homogeneous system, of course, has a solution. x + z+ w=0 2x − y + w=0 x + y + 3z + 2w = 0
−2ρ1 +ρ2 ρ2 +ρ3 −ρ1 +ρ3

x

−→

−→

+ z+w=0 −y − 2z − w = 0 0=0

In fact, the solution set of the homogeneous system is inﬁnite.     −1 −1   −2   z + −1 w z, w ∈ R} {  0 1 0 1 However, because no particular solution of the original system exists, the general solution set is empty — there are no vectors of the form p + h because there are no p ’s. 3.11 Corollary Solution sets of linear systems are either empty, have one element, or have inﬁnitely many elements.
Proof. We’ve seen examples of all three happening so we need only prove that

those are the only possibilities. First, notice a homogeneous system with at least one non-0 solution v has inﬁnitely many solutions because the set of multiples sv is inﬁnite — if s = 1 then sv − v = (s − 1)v is easily seen to be non-0, and so sv = v.

Section I. Solving Linear Systems Now, apply Lemma 3.8 to conclude that a solution set {p + h h solves the associated homogeneous system}

27

is either empty (if there is no particular solution p), or has one element (if there is a p and the homogeneous system has the unique solution 0), or is inﬁnite (if there is a p and the homogeneous system has a non-0 solution, and thus by the prior paragraph has inﬁnitely many solutions). QED This table summarizes the factors aﬀecting the size of a general solution. number of solutions of the associated homogeneous system one unique solution no solutions inﬁnitely many inﬁnitely many solutions no solutions

particular solution exists?

yes no

The factor on the top of the table is the simpler one. When we perform Gauss’ method on a linear system, ignoring the constants on the right side and so paying attention only to the coeﬃcients on the left-hand side, we either end with every variable leading some row or else we ﬁnd that some variable does not lead a row, that is, that some variable is free. (Of course, “ignoring the constants on the right” is formalized by considering the associated homogeneous system. We are simply putting aside for the moment the possibility of a contradictory equation.) A nice insight into the factor on the top of this table at work comes from considering the case of a system having the same number of equations as variables. This system will have a solution, and the solution will be unique, if and only if it reduces to an echelon form system where every variable leads its row, which will happen if and only if the associated homogeneous system has a unique solution. Thus, the question of uniqueness of solution is especially interesting when the system has the same number of equations as variables. 3.12 Deﬁnition A square matrix is nonsingular if it is the matrix of coeﬃcients of a homogeneous system with a unique solution. It is singular otherwise, that is, if it is the matrix of coeﬃcients of a homogeneous system with inﬁnitely many solutions. 3.13 Example The systems from Example 3.3, Example 3.5, and Example 3.9 each have an associated homogeneous system with a unique solution. Thus these matrices are nonsingular.     3 2 1 1 2 −1 3 4 6 −4 0 2 4 0  2 −1 0 1 1 0 1 −3

28

Chapter One. Linear Systems

The Chemistry problem from Example 3.6 is a homogeneous system with more than one solution so its matrix is singular.   7 0 −7 0 8 1 −5 −2   0 1 −3 0  0 3 −6 −1 3.14 Example The ﬁrst of these matrices is nonsingular while the second is singular 1 2 1 2 3 4 3 6 because the ﬁrst of these homogeneous systems has a unique solution while the second has inﬁnitely many solutions. x + 2y = 0 3x + 4y = 0 x + 2y = 0 3x + 6y = 0

We have made the distinction in the deﬁnition because a system (with the same number of equations as variables) behaves in one of two ways, depending on whether its matrix of coeﬃcients is nonsingular or singular. A system where the matrix of coeﬃcients is nonsingular has a unique solution for any constants on the right side: for instance, Gauss’ method shows that this system x + 2y = a 3x + 4y = b has the unique solution x = b − 2a and y = (3a − b)/2. On the other hand, a system where the matrix of coeﬃcients is singular never has a unique solution — it has either no solutions or else has inﬁnitely many, as with these. x + 2y = 1 3x + 6y = 2 x + 2y = 1 3x + 6y = 3

Thus, ‘singular’ can be thought of as connoting “troublesome”, or at least “not ideal”. The above table has two factors. We have already considered the factor along the top: we can tell which column a given linear system goes in solely by considering the system’s left-hand side — the constants on the right-hand side play no role in this factor. The table’s other factor, determining whether a particular solution exists, is tougher. Consider these two 3x + 2y = 5 3x + 2y = 5 3x + 2y = 5 3x + 2y = 4

with the same left sides but diﬀerent right sides. Obviously, the ﬁrst has a solution while the second does not, so here the constants on the right side

Section I. Solving Linear Systems

29

decide if the system has a solution. We could conjecture that the left side of a linear system determines the number of solutions while the right side determines if solutions exist, but that guess is not correct. Compare these two systems 3x + 2y = 5 4x + 2y = 4 3x + 2y = 5 3x + 2y = 4

with the same right sides but diﬀerent left sides. The ﬁrst has a solution but the second does not. Thus the constants on the right side of a system don’t decide alone whether a solution exists; rather, it depends on some interaction between the left and right sides. For some intuition about that interaction, consider this system with one of the coeﬃcients left as the parameter c. x + 2y + 3z = 1 x+ y+ z=1 cx + 3y + 4z = 0 If c = 2 this system has no solution because the left-hand side has the third row as a sum of the ﬁrst two, while the right-hand does not. If c = 2 this system has a unique solution (try it with c = 1). For a system to have a solution, if one row of the matrix of coeﬃcients on the left is a linear combination of other rows, then on the right the constant from that row must be the same combination of constants from the same rows. More intuition about the interaction comes from studying linear combinations. That will be our focus in the second chapter, after we ﬁnish the study of Gauss’ method itself in the rest of this chapter. Exercises
3.15 Solve each system. Express the solution set using vectors. Identify the particular solution and the solution set of the homogeneous system. (a) 3x + 6y = 18 (b) x + y = 1 (c) x1 + x3 = 4 x + 2y = 6 x − y = −1 x1 − x2 + 2x3 = 5 4x1 − x2 + 5x3 = 17 (d) 2a + b − c = 2 (e) x + 2y − z =3 (f ) x +z+w=4 2a +c=3 2x + y +w=4 2x + y −w=2 a−b =0 x− y+z+w=1 3x + y + z =7 3.16 Solve each system, giving the solution set in vector notation. Identify the particular solution and the solution of the homogeneous system. (a) 2x + y − z = 1 (b) x − z =1 (c) x − y + z =0 4x − y =3 y + 2z − w = 3 y +w=0 x + 2y + 3z − w = 7 3x − 2y + 3z + w = 0 −y −w=0 (d) a + 2b + 3c + d − e = 1 3a − b + c + d + e = 3 3.17 For the system 2x − y − w= 3 y + z + 2w = 2 x − 2y − z = −1

30

Chapter One. Linear Systems
which of these can be used as the particular solution part of some general solution?       −1 2 0 −4 1 −3 (c)   (b)   (a)   8 1 5 −1 0 0 3.18 Lemma 3.8 says that any particular solution may be used for p. Find, if possible, a general solution to this system x− y +w=4 2x + 3y − z =0 y+z+w=4 that uses the given vector its particular solution. as      0 −5 2 0 1 −1 (a)   (b)   (c)   0 −7 1 4 10 1 3.19 One of these is nonsingular while the other is singular. Which is which? 1 3 1 3 (b) 4 −12 4 12 3.20 Singular or nonsingular? 1 2 1 2 1 2 1 (a) (b) (c) (Careful!) 1 3 −3 −6 1 3 1 1 2 1 2 2 1 1 0 5 (d) 1 1 3 (e) 3 4 7 −1 1 4 3.21 Is the given vector in the set generated by the given set? 2 1 1 (a) ,{ , } 3 4 5 −1 2 1 0 ,{ 1 , 0 } (b) 1 0 1 1 1 2 3 4 (c) 3 , { 0 , 1 , 3 , 2 } 0 4 5 0 1       1 2 3 0 1 0 (d)   , {  ,  } 1 0 0 1 1 2 3.22 Prove that any linear system with a nonsingular matrix of coeﬃcients has a solution, and that the solution is unique. 3.23 To tell the whole truth, there is another tricky point to the proof of Lemma 3.7. What happens if there are no non-‘0 = 0’ equations? (There aren’t any more tricky points after this one.) 3.24 Prove that if s and t satisfy a homogeneous system then so do these vectors. (a) s + t (b) 3s (c) ks + mt for k, m ∈ R What’s wrong with: “These three show that if a homogeneous system has one solution then it has many solutions — any multiple of a solution is another solution, (a)

Section I. Solving Linear Systems

31

and any sum of solutions is a solution also — so there are no homogeneous systems with exactly one solution.”? 3.25 Prove that if a system with only rational coeﬃcients and constants has a solution then it has at least one all-rational solution. Must it have inﬁnitely many?

32

Chapter One. Linear Systems

II

Linear Geometry of n-Space

For readers who have seen the elements of vectors before, in calculus or physics, this section is an optional review. However, later work will refer to this material so it is not optional if it is not a review. In the ﬁrst section, we had to do a bit of work to show that there are only three types of solution sets — singleton, empty, and inﬁnite. But in the special case of systems with two equations and two unknowns this is easy to see. Draw each two-unknowns equation as a line in the plane and then the two lines could have a unique intersection, be parallel, or be the same line.
Unique solution No solutions Inﬁnitely many solutions

3x + 2y = 7 x − y = −1

3x + 2y = 7 3x + 2y = 4

3x + 2y = 7 6x + 4y = 14

These pictures don’t prove the results from the prior section, which apply to any number of linear equations and any number of unknowns, but nonetheless they do help us to understand those results. This section develops the ideas that we need to express our results from the prior section, and from some future sections, geometrically. In particular, while the two-dimensional case is familiar, to extend to systems with more than two unknowns we shall need some higherdimensional geometry.

II.1 Vectors in Space
“Higher-dimensional geometry” sounds exotic. It is exotic — interesting and eye-opening. But it isn’t distant or unreachable. We begin by deﬁning one-dimensional space to be the set R1 . To see that deﬁnition is reasonable, draw a one-dimensional space

and make the usual correspondence with R: pick a point to label 0 and another to label 1.
0 1

Now, with a scale and a direction, ﬁnding the point corresponding to, say +2.17, is easy — start at 0 and head in the direction of 1 (i.e., the positive direction), but don’t stop there, go 2.17 times as far.

Section II. Linear Geometry of n-Space

33

The basic idea here, combining magnitude with direction, is the key to extending to higher dimensions. An object comprised of a magnitude and a direction is a vector (we will use the same word as in the previous section because we shall show below how to describe such an object with a column vector). We can draw a vector as having some length, and pointing somewhere.

There is a subtlety here — these vectors

are equal, even though they start in diﬀerent places, because they have equal lengths and equal directions. Again: those vectors are not just alike, they are equal. How can things that are in diﬀerent places be equal? Think of a vector as representing a displacement (‘vector’ is Latin for “carrier” or “traveler”). These squares undergo the same displacement, despite that those displacements start in diﬀerent places.

Sometimes, to emphasize this property vectors have of not being anchored, they are referred to as free vectors. Thus, these free vectors are equal as each is a displacement of one over and two up.

More generally, vectors in the plane are the same if and only if they have the same change in ﬁrst components and the same change in second components: the vector extending from (a1 , a2 ) to (b1 , b2 ) equals the vector from (c1 , c2 ) to (d1 , d2 ) if and only if b1 − a1 = d1 − c1 and b2 − a2 = d2 − c2 . An expression like ‘the vector that, were it to start at (a1 , a2 ), would extend to (b1 , b2 )’ is awkward. We instead describe such a vector as b1 − a1 b2 − a2 so that, for instance, the ‘one over and two up’ arrows shown above picture this vector. 1 2

34

Chapter One. Linear Systems

We often draw the arrow as starting at the origin, and we then say it is in the canonical position (or natural position). When the vector b1 − a1 b2 − a2 is in its canonical position then it extends to the endpoint (b1 − a1 , b2 − a2 ). We typically just refer to “the point 1 ” 2 rather than “the endpoint of the canonical position of” that vector. Thus, we will call both of these sets R2 . {(x1 , x2 ) x1 , x2 ∈ R} { x1 x2 x1 , x2 ∈ R}

In the prior section we deﬁned vectors and vector operations with an algebraic motivation; r· v1 v2 = rv1 rv2 v1 v2 + w1 w2 = v 1 + w1 v 2 + w2

we can now interpret those operations geometrically. For instance, if v represents a displacement then 3v represents a displacement in the same direction but three times as far, and −1v represents a displacement of the same distance as v but in the opposite direction.
v 3v −v

And, where v and w represent displacements, v + w represents those displacements combined.
v+w w

v

The long arrow is the combined displacement in this sense: if, in one minute, a ship’s motion gives it the displacement relative to the earth of v and a passenger’s motion gives a displacement relative to the ship’s deck of w, then v + w is the displacement of the passenger relative to the earth. Another way to understand the vector sum is with the parallelogram rule. Draw the parallelogram formed by the vectors v1 , v2 and then the sum v1 + v2 extends along the diagonal to the far corner.

Section II. Linear Geometry of n-Space
v+w w

35

v

The above drawings show how vectors and vector operations behave in R2 . We can extend to R3 , or to even higher-dimensional spaces where we have no pictures, with the obvious generalization: the free vector that, if it starts at (a1 , . . . , an ), ends at (b1 , . . . , bn ), is represented by this column   b1 − a1  .   .  . bn − an (vectors are equal if they have the same representation), we aren’t too careful to distinguish between a point and the vector whose canonical representation ends at that point,   v1 . n R = { .  v1 , . . . , vn ∈ R} . vn and addition and scalar multiplication are component-wise. Having considered points, we now turn to the lines. In R2 , the line through (1, 2) and (3, 1) is comprised of (the endpoints of) the vectors in this set { 1 2 +t· 2 −1 t ∈ R}

That description expresses this picture.
2 −1 = 3 1 − 1 2

The vector associated with the parameter t has its whole body in the line — it is a direction vector for the line. Note that points on the line to the left of x = 1 are described using negative values of t. In R3 , the line through (1, 2, 1) and (2, 3, 2) is the set of (endpoints of) vectors of this form
{ 1 2 1 +t· 1 1 1 t ∈ R}

36

Chapter One. Linear Systems

and lines in even higher-dimensional spaces work in the same way. If a line uses one parameter, so that there is freedom to move back and forth in one dimension, then a plane must involve two. For example, the plane through the points (1, 0, 5), (2, 1, −3), and (−2, 4, 0.5) consists of (endpoints of) the vectors in       1 1 −3 {0 + t ·  1  + s ·  4  t, s ∈ R} 5 −8 −4.5 (the column vectors associated with the parameters             1 2 1 −3 −2 1  1  =  1  − 0  4  =  4  − 0 −8 −3 5 −4.5 0.5 5 are two vectors whose whole bodies lie in the plane). As with the line, note that some points in this plane are described with negative t’s or negative s’s or both. A description of planes that is often encountered in algebra and calculus uses a single equation as the condition that describes the relationship among the ﬁrst, second, and third coordinates of points in a plane.

P ={

x y z

2x + y + z = 4}

The translation from such a description to the vector description that we favor in this book is to think of the condition as a one-equation linear system and parametrize x = (1/2)(4 − y − z).

P ={

2 0 0

+

−0.5 1 0

y+

−0.5 0 1

z

y, z ∈ R}

Generalizing from lines and planes, we deﬁne a k-dimensional linear surface (or k-ﬂat) in Rn to be {p + t1 v1 + t2 v2 + · · · + tk vk t1 , . . . , tk ∈ R} where v1 , . . . , vk ∈ Rn . For example, in R4 ,     2 1  π  0    {  3  + t 0 t ∈ R} −0.5 0

Section II. Linear Geometry of n-Space is a line,       0 1 2 0 1 0 {  + t   + s   t, s ∈ R} 0 0 1 0 −1 0 is a plane, and        2 1 0 3 0 0 0 1 {  + r   + s   + t   r, s, t ∈ R} 1 1 0 −2 0 0 −1 0.5 

37

is a three-dimensional linear surface. Again, the intuition is that a line permits motion in one direction, a plane permits motion in combinations of two directions, etc. A linear surface description can be misleading about the dimension — this       2 1 1 2 1 0 L = {  + t   + s   t, s ∈ R} 0 0 −1 −2 −1 −2 is a degenerate plane because it is actually a line.     1 1 1 0 L = {  + r   r ∈ R} 0 −1 −1 −2 We shall see in the Linear Independence section of Chapter Two what relationships among vectors causes the linear surface they generate to be degenerate. We ﬁnish this subsection by restating our conclusions from the ﬁrst section in geometric terms. First, the solution set of a linear system with n unknowns is a linear surface in Rn . Speciﬁcally, it is a k-dimensional linear surface, where k is the number of free variables in an echelon form version of the system. Second, the solution set of a homogeneous linear system is a linear surface passing through the origin. Finally, we can view the general solution set of any linear system as being the solution set of its associated homogeneous system oﬀset from the origin by a vector, namely by any particular solution. Exercises
1.1 Find the canonical name for each vector. (a) the vector from (2, 1) to (4, 2) in R2 (b) the vector from (3, 3) to (2, 5) in R2 (c) the vector from (1, 0, 6) to (5, 0, 3) in R3 (d) the vector from (6, 8, 8) to (6, 8, 8) in R3 1.2 Decide if the two vectors are equal. (a) the vector from (5, 3) to (6, 2) and the vector from (1, −2) to (1, 1)

38

Chapter One. Linear Systems
(b) the vector from (2, 1, 1) to (3, 0, 4) and the vector from (5, 1, 4) to (6, 0, 7) 1.3 Does (1, 0, 2, 1) lie on the line through (−2, 1, 1, 0) and (5, 10, −1, 4)? 1.4 (a) Describe the plane through (1, 1, 5, −1), (2, 2, 2, 0), and (3, 1, 0, 4). (b) Is the origin in that plane? 1.5 Describe the plane that contains this point and line. 2 0 3 1.6 Intersect these planes. 1 { 1 1 t+ 0 1 3 s t, s ∈ R} 1 { 1 0 + 0 3 0 k+ 2 0 4 m k, m ∈ R} { −1 0 −4 + 1 1 2 t t ∈ R}

1.7 Intersect each pair, if possible. 1 0 1 0 (a) { 1 + t 1 t ∈ R}, { 3 +s 1 s ∈ R} 2 1 −2 2 2 1 0 0 (b) { 0 + t 1 t ∈ R}, {s 1 + w 4 s, w ∈ R} 1 −1 2 1 1.8 When a plane does not pass through the origin, performing operations on vectors whose bodies lie in it is more complicated than when the plane passes through the origin. Consider the picture in this subsection of the plane 2 { 0 0 + −0.5 1 0 y+ −0.5 0 1 z y, z ∈ R}

and the three vectors it shows, with endpoints (2, 0, 0), (1.5, 1, 0), and (1.5, 0, 1). (a) Redraw the picture, including the vector in the plane that is twice as long as the one with endpoint (1.5, 1, 0). The endpoint of your vector is not (3, 2, 0); what is it? (b) Redraw the picture, including the parallelogram in the plane that shows the sum of the vectors ending at (1.5, 0, 1) and (1.5, 1, 0). The endpoint of the sum, on the diagonal, is not (3, 1, 1); what is it? 1.9 Show that the line segments (a1 , a2 )(b1 , b2 ) and (c1 , c2 )(d1 , d2 ) have the same lengths and slopes if b1 − a1 = d1 − c1 and b2 − a2 = d2 − c2 . Is that only if? 1.10 How should R0 be deﬁned? ? 1.11 A person traveling eastward at a rate of 3 miles per hour ﬁnds that the wind appears to blow directly from the north. On doubling his speed it appears to come from the north east. What was the wind’s velocity? [Math. Mag., Jan. 1957] 1.12 Euclid describes a plane as “a surface which lies evenly with the straight lines on itself”. Commentators (e.g., Heron) have interpreted this to mean “(A plane surface is) such that, if a straight line pass through two points on it, the line coincides wholly with it at every spot, all ways”. (Translations from [Heath], pp. 171-172.) Do planes, as described in this section, have that property? Does this description adequately deﬁne planes?

Section II. Linear Geometry of n-Space

39

II.2 Length and Angle Measures
We’ve translated the ﬁrst section’s results about solution sets into geometric terms for insight into how those sets look. But we must watch out not to be mislead by our own terms; labeling subsets of Rk of the forms {p + tv t ∈ R} and {p + tv + sw t, s ∈ R} as “lines” and “planes” doesn’t make them act like the lines and planes of our prior experience. Rather, we must ensure that the names suit the sets. While we can’t prove that the sets satisfy our intuition — we can’t prove anything about intuition — in this subsection we’ll observe that a result familiar from R2 and R3 , when generalized to arbitrary Rk , supports the idea that a line is straight and a plane is ﬂat. Speciﬁcally, we’ll see how to do Euclidean geometry in a “plane” by giving a deﬁnition of the angle between two Rn vectors in the plane that they generate. 2.1 Deﬁnition The length of a vector v ∈ Rn is this. v =
2 2 v1 + · · · + vn

2.2 Remark This is a natural generalization of the Pythagorean Theorem. A classic discussion is in [Polya]. We can use that deﬁnition to derive a formula for the angle between two vectors. For a model of what to do, consider two vectors in R3 .

v u

Put them in canonical position and, in the plane that they determine, consider the triangle formed by u, v, and u − v.

Apply the Law of Cosines, u − v 2 = u 2 + v 2 − 2 u is the angle between the vectors. Expand both sides (u1 − v1 )2 + (u2 − v2 )2 + (u3 − v3 )2

v cos θ, where θ

2 2 2 = (u2 + u2 + u2 ) + (v1 + v2 + v3 ) − 2 u 1 2 3

v cos θ

and simplify. θ = arccos(

u1 v1 + u2 v2 + u3 v3 ) u v

40

Chapter One. Linear Systems

In higher dimensions no picture suﬃces but we can make the same argument analytically. First, the form of the numerator is clear — it comes from the middle terms of the squares (u1 − v1 )2 , (u2 − v2 )2 , etc. 2.3 Deﬁnition The dot product (or inner product, or scalar product) of two n-component real vectors is the linear combination of their components. u v = u1 v1 + u2 v2 + · · · + un vn Note that the dot product of two vectors is a real number, not a vector, and that the dot product of a vector from Rn with a vector from Rm is deﬁned only when n equals m. Note also this relationship between dot product and length: dotting a vector with itself gives its length squared u u = u1 u1 + · · · + un un = u 2 . 2.4 Remark The wording in that deﬁnition allows one or both of the two to be a row vector instead of a column vector. Some books require that the ﬁrst vector be a row vector and that the second vector be a column vector. We shall not be that strict. Still reasoning with letters, but guided by the pictures, we use the next theorem to argue that the triangle formed by u, v, and u − v in Rn lies in the planar subset of Rn generated by u and v. 2.5 Theorem (Triangle Inequality) For any u, v ∈ Rn , u+v ≤ u + v with equality if and only if one of the vectors is a nonnegative scalar multiple of the other one. This inequality is the source of the familiar saying, “The shortest distance between two points is in a straight line.”
ﬁnish u+v v

start

u

Proof. (We’ll use some algebraic properties of dot product that we have not

yet checked, for instance that u (a + b) = u a + u b and that u v = v u. See Exercise 17.) The desired inequality holds if and only if its square holds. u+v
2

≤ ( u + v )2
2

(u + v) (u + v) ≤ u

+2 u v

v + v v +v v

2

u u+u v+v u+v v ≤u u+2 u 2u v ≤ 2 u

Section II. Linear Geometry of n-Space

41

That, in turn, holds if and only if the relationship obtained by multiplying both sides by the nonnegative numbers u and v 2( v u) ( u v) ≤ 2 u and rewriting 0≤ u is true. But factoring 0 ≤ ( u v − v u) ( u v − v u) shows that this certainly is true since it only says that the square of the length of the vector u v − v u is not negative. As for equality, it holds when, and only when, u v − v u is 0. The check that u v = v u if and only if one vector is a nonnegative real scalar multiple of the other is easy. QED This result supports the intuition that even in higher-dimensional spaces, lines are straight and planes are ﬂat. For any two points in a linear surface, the line segment connecting them is contained in that surface (this is easily checked from the deﬁnition). But if the surface has a bend then that would allow for a shortcut (shown here grayed, while the segment from P to Q that is contained in the surface is solid).
Q 2 2

v

2

v

2

− 2( v u) ( u v) + u

2

v

2

P

Because the Triangle Inequality says that in any Rn , the shortest cut between two endpoints is simply the line segment connecting them, linear surfaces have no such bends. Back to the deﬁnition of angle measure. The heart of the Triangle Inequality’s proof is the ‘u v ≤ u v ’ line. At ﬁrst glance, a reader might wonder if some pairs of vectors satisfy the inequality in this way: while u v is a large number, with absolute value bigger than the right-hand side, it is a negative large number. The next result says that no such pair of vectors exists. 2.6 Corollary (Cauchy-Schwartz Inequality) For any u, v ∈ Rn , |u v| ≤ u v

with equality if and only if one vector is a scalar multiple of the other.
Proof. The Triangle Inequality’s proof shows that u v ≤ u v so if u v is positive or zero then we are done. If u v is negative then this holds.

| u v | = −( u v ) = (−u ) v ≤ − u The equality condition is Exercise 18.

v = u

v
QED

42

Chapter One. Linear Systems

The Cauchy-Schwartz inequality assures us that the next deﬁnition makes sense because the fraction has absolute value less than or equal to one. 2.7 Deﬁnition The angle between two nonzero vectors u, v ∈ Rn is θ = arccos( u v ) u v

(the angle between the zero vector and any other vector is deﬁned to be a right angle). Thus vectors from Rn are orthogonal if and only if their dot product is zero. 2.8 Example These vectors are orthogonal. 1 −1 1 1

=0

The arrows are shown away from canonical position but nevertheless the vectors are orthogonal. 2.9 Example The R3 angle formula given at the start of this subsection is a special case of the deﬁnition. Between these two

0 3 2

1 1 0

the angle is arccos( √ (1)(0) + (1)(3) + (0)(2) 3 √ ) = arccos( √ √ ) 2 + 1 2 + 0 2 02 + 3 2 + 2 2 2 13 1

approximately 0.94 radians. Notice that these vectors are not orthogonal. Although the yz-plane may appear to be perpendicular to the xy-plane, in fact the two planes are that way only in the weak sense that there are vectors in each orthogonal to all vectors in the other. Not every vector in each is orthogonal to all vectors in the other. Exercises
2.10 Find the length of each vector.

Section II. Linear Geometry of n-Space
1 −1 (e)   1 0

43  

(a)

3 1

(b)

−1 2

(c)

4 1 1

(d)

0 0 0

2.11 Find the angle between each two, if it is deﬁned. 1 0 1 1 1 1 (c) , 4 (a) , (b) 2 , 4 2 2 4 −1 1 0 2.12 During maneuvers preceding the Battle of Jutland, the British battle cruiser Lion moved as follows (in nautical miles): 1.2 miles north, 6.1 miles 38 degrees east of south, 4.0 miles at 89 degrees east of north, and 6.5 miles at 31 degrees east of north. Find the distance between starting and ending positions. [Ohanian] 2.13 Find k so that these two vectors are perpendicular. k 1 4 3

2.14 Describe the set of vectors in R3 orthogonal to this one. 1 3 −1 2.15 (a) Find the angle between the diagonal of the unit square in R2 and one of the axes. (b) Find the angle between the diagonal of the unit cube in R3 and one of the axes. (c) Find the angle between the diagonal of the unit cube in Rn and one of the axes. (d) What is the limit, as n goes to ∞, of the angle between the diagonal of the unit cube in Rn and one of the axes? 2.16 Is any vector perpendicular to itself? 2.17 Describe the algebraic properties of dot product. (a) Is it right-distributive over addition: (u + v) w = u w + v w? (b) Is is left-distributive (over addition)? (c) Does it commute? (d) Associate? (e) How does it interact with scalar multiplication? As always, any assertion must be backed by either a proof or an example. 2.18 Verify the equality condition in Corollary 2.6, the Cauchy-Schwartz Inequality. (a) Show that if u is a negative scalar multiple of v then u v and v u are less than or equal to zero. (b) Show that |u v| = u v if and only if one vector is a scalar multiple of the other. 2.19 Suppose that u v = u w and u = 0. Must v = w? 2.20 Does any vector have length zero except a zero vector? (If “yes”, produce an example. If “no”, prove it.) 2.21 Find the midpoint of the line segment connecting (x1 , y1 ) with (x2 , y2 ) in R2 . Generalize to Rn . 2.22 Show that if v = 0 then v/ v has length one. What if v = 0? 2.23 Show that if r ≥ 0 then rv is r times as long as v. What if r < 0?

44

Chapter One. Linear Systems

2.24 A vector v ∈ Rn of length one is a unit vector. Show that the dot product of two unit vectors has absolute value less than or equal to one. Can ‘less than’ happen? Can ‘equal to’ ? 2.25 Prove that u + v 2 + u − v 2 = 2 u 2 + 2 v 2 . 2.26 Show that if x y = 0 for every y then x = 0. 2.27 Is u1 + · · · + un ≤ u1 + · · · + un ? If it is true then it would generalize the Triangle Inequality. 2.28 What is the ratio between the sides in the Cauchy-Schwartz inequality? 2.29 Why is the zero vector deﬁned to be perpendicular to every vector? 2.30 Describe the angle between two vectors in R1 . 2.31 Give a simple necessary and suﬃcient condition to determine whether the angle between two vectors is acute, right, or obtuse. 2.32 Generalize to Rn the converse of the Pythagorean Theorem, that if u and v are perpendicular then u + v 2 = u 2 + v 2 . 2.33 Show that u = v if and only if u + v and u − v are perpendicular. Give an example in R2 . 2.34 Show that if a vector is perpendicular to each of two others then it is perpendicular to each vector in the plane they generate. (Remark. They could generate a degenerate plane — a line or a point — but the statement remains true.) 2.35 Prove that, where u, v ∈ Rn are nonzero vectors, the vector u v + u v bisects the angle between them. Illustrate in R2 . 2.36 Verify that the deﬁnition of angle is dimensionally correct: (1) if k > 0 then the cosine of the angle between ku and v equals the cosine of the angle between u and v, and (2) if k < 0 then the cosine of the angle between ku and v is the negative of the cosine of the angle between u and v. 2.37 Show that the inner product operation is linear : for u, v, w ∈ Rn and k, m ∈ R, u (kv + mw) = k(u v) + m(u w). √ 2.38 The geometric mean of two positive reals x, y is xy. It is analogous to the arithmetic mean (x + y)/2. Use the Cauchy-Schwartz inequality to show that the geometric mean of any x, y ∈ R is less than or equal to the arithmetic mean. ? 2.39 A ship is sailing with speed and direction v1 ; the wind blows apparently (judging by the vane on the mast) in the direction of a vector a; on changing the direction and speed of the ship from v1 to v2 the apparent wind is in the direction of a vector b. Find the vector velocity of the wind. [Am. Math. Mon., Feb. 1933] 2.40 Verify the Cauchy-Schwartz inequality by ﬁrst proving Lagrange’s identity:
2

aj bj
1≤j≤n

=
1≤j≤n

a2 j
1≤j≤n

b2 j

1≤k<j≤n

(ak bj − aj bk )2

and then noting that the ﬁnal term is positive. (Recall the meaning aj bj = a1 b1 + a2 b2 + · · · + an bn
1≤j≤n

and aj 2 = a1 2 + a2 2 + · · · + an 2
1≤j≤n

Section II. Linear Geometry of n-Space

45

of the Σ notation.) This result is an improvement over Cauchy-Schwartz because it gives a formula for the diﬀerence between the two sides. Interpret that diﬀerence in R2 .

46

Chapter One. Linear Systems

III

Reduced Echelon Form

After developing the mechanics of Gauss’ method, we observed that it can be done in more than one way. One example is that we sometimes have to swap rows and there can be more than one row to choose from. Another example is that from this matrix 2 2 4 3 Gauss’ method could derive any of these echelon form matrices. 2 0 2 −1 1 0 1 −1 2 0 0 −1

The ﬁrst results from −2ρ1 + ρ2 . The second comes from following (1/2)ρ1 with −4ρ1 + ρ2 . The third comes from −2ρ1 + ρ2 followed by 2ρ2 + ρ1 (after the ﬁrst pivot the matrix is already in echelon form so the second one is extra work but it is nonetheless a legal row operation). The fact that the echelon form outcome of Gauss’ method is not unique leaves us with some questions. Will any two echelon form versions of a system have the same number of free variables? Will they in fact have exactly the same variables free? In this section we will answer both questions “yes”. We will do more than answer the questions. We will give a way to decide if one linear system can be derived from another by row operations. The answers to the two questions will follow from this larger result.

III.1 Gauss-Jordan Reduction
Gaussian elimination coupled with back-substitution solves linear systems, but it’s not the only method possible. Here is an extension of Gauss’ method that has some advantages. 1.1 Example To solve x + y − 2z = −2 y + 3z = 7 x − z = −1 we can start by going to  1 −ρ1 +ρ3 −→ 0 0 echelon form as usual.   1 1 −2 1 −2 −2 ρ2 +ρ3 1 3 7  −→ 0 1 3 −1 1 1 0 0 4

 −2 7 8

We can keep going to a second stage  1 (1/4)ρ3 −→ 0 0

by making the leading entries into ones  1 −2 −2 1 3 7 0 1 2

Section III. Reduced Echelon Form and then to a third stage that uses the leading entries to other entries in each column by pivoting upwards.    1 1 0 2 1 0 0 −3ρ3 +ρ2 −ρ2 +ρ1 −→ 0 1 0 1 −→ 0 1 0 2ρ3 +ρ1 0 0 1 2 0 0 1 The answer is x = 1, y = 1, and z = 2.

47 eliminate all of the  1 1 2

Note that the pivot operations in the ﬁrst stage proceed from column one to column three while the pivot operations in the third stage proceed from column three to column one. 1.2 Example We often combine the operations of the middle stage into a single step, even though they are operations on diﬀerent rows. 2 4 1 −2 7 6
−2ρ1 +ρ2

−→

2 0

1 −4

7 −8 7/2 2 5/2 2

(1/2)ρ1

−→

(−1/4)ρ2 −(1/2)ρ2 +ρ1

1 1/2 0 1 1 0 0 1

−→

48

Chapter One. Linear Systems that is shown

of the reduced echelon form. Consider, for example, this system brought to echelon form and then to reduced echelon form.     2 6 1 2 5 2 6 1 2 5 −ρ2 +ρ3 0 3 1 4 1 −→ 0 3 1 4 1 0 3 1 2 5 0 0 0 −2 4  1 0 −1/2 (1/2)ρ1 (4/3)ρ3 +ρ2 −3ρ2 +ρ1 −→ 0 1 1/3 −→ −→ −ρ3 +ρ1 (1/3)ρ2 0 0 0 −(1/2)ρ3

0 0 1

 −9/2 3  −2

Starting with the middle matrix, the echelon form version, back substitution produces −2x4 = 4 so that x4 = −2, then another back substitution gives 3x2 + x3 + 4(−2) = 1 implying that x2 = 3 − (1/3)x3 , and then the ﬁnal back substitution gives 2x1 + 6(3 − (1/3)x3 ) + x3 + 2(−2) = 5 implying that x1 = −(9/2) + (1/2)x3 . Thus the solution set is this.       x1 1/2 −9/2 x   3  −1/3    S = { 2  =  x3   0  +  1  x3 x3 ∈ R} 0 x4 −2 Now, considering the ﬁnal matrix, the reduced echelon form version, note that adjusting the parametrization by moving the x3 terms to the other side does indeed give the description of this inﬁnite solution set. Part of the reason that this works is straightforward. While a set can have many parametrizations that describe it, e.g., both of these also describe the above set S (take t to be x3 /6 and s to be x3 − 1)         1/2 −4 3 −9/2 8/3 −1/3  3  −2     {  +  {  1   1  s s ∈ R}  0  +  6  t t ∈ R} 0 −2 0 −2 nonetheless we have in this book stuck to a convention of parametrizing using the unmodiﬁed free variables (that is, x3 = x3 instead of x3 = 6t). We can easily see that a reduced echelon form version of a system is equivalent to a parametrization in terms of unmodiﬁed free variables. For instance,   1 0 2 4 x1 = 4 − 2x3 ⇐⇒ 0 1 1 3 x2 = 3 − x3 0 0 0 0 (to move from left to right we also need to know how many equations are in the system). So, the convention of parametrizing with the free variables by solving each equation for its leading variable and then eliminating that leading variable from every other equation is exactly equivalent to the reduced echelon form conditions that each leading entry must be a one and must be the only nonzero entry in its column.

Section III. Reduced Echelon Form

49

Not as straightforward is the other part of the reason that the reduced echelon form version allows us to read oﬀ the parametrization that we would have gotten had we stopped at echelon form and then done back substitution. The prior paragraph shows that reduced echelon form corresponds to some parametrization, but why the same parametrization? A solution set can be parametrized in many ways, and Gauss’ method or the Gauss-Jordan method can be done in many ways, so a ﬁrst guess might be that we could derive many diﬀerent reduced echelon form versions of the same starting system and many diﬀerent parametrizations. But we never do. Experience shows that starting with the same system and proceeding with row operations in many diﬀerent ways always yields the same reduced echelon form and the same parametrization (using the unmodiﬁed free variables). In the rest of this section we will show that the reduced echelon form version of a matrix is unique. It follows that the parametrization of a linear system in terms of its unmodiﬁed free variables is unique because two diﬀerent ones would give two diﬀerent reduced echelon forms. We shall use this result, and the ones that lead up to it, in the rest of the book but perhaps a restatement in a way that makes it seem more immediately useful may be encouraging. Imagine that we solve a linear system, parametrize, and check in the back of the book for the answer. But the parametrization there appears diﬀerent. Have we made a mistake, or could these be diﬀerent-looking descriptions of the same set, as with the three descriptions above of S? The prior paragraph notes that we will show here that diﬀerent-looking parametrizations (using the unmodiﬁed free variables) describe genuinely diﬀerent sets. Here is an informal argument that the reduced echelon form version of a matrix is unique. Consider again the example that started this section of a matrix that reduces to three diﬀerent echelon form matrices. The ﬁrst matrix of the three is the natural echelon form version. The second matrix is the same as the ﬁrst except that a row has been halved. The third matrix, too, is just a cosmetic variant of the ﬁrst. The deﬁnition of reduced echelon form outlaws this kind of fooling around. In reduced echelon form, halving a row is not possible because that would change the row’s leading entry away from one, and neither is combining rows possible, because then a leading entry would no longer be alone in its column. This informal justiﬁcation is not a proof; we have argued that no two diﬀerent reduced echelon form matrices are related by a single row operation step, but we have not ruled out the possibility that multiple steps might do. Before we go to that proof, we ﬁnish this subsection by rephrasing our work in a terminology that will be enlightening. Many diﬀerent matrices yield the same reduced echelon form matrix. The three echelon form matrices from the start of this section, and the matrix they were derived from, all give this reduced echelon form matrix. 1 0 0 1

We think of these matrices as related to each other. The next result speaks to

50 this relationship.

Chapter One. Linear Systems

1.4 Lemma Elementary row operations are reversible.
Proof. For any matrix A, the eﬀect of swapping rows is reversed by swapping

them back, multiplying a row by a nonzero k is undone by multiplying by 1/k, and adding a multiple of row i to row j (with i = j) is undone by subtracting the same multiple of row i from row j. A −→ −→ A
ρi ↔ρj ρj ↔ρi

A −→ −→ A

kρi (1/k)ρi

A −→

kρi +ρj −kρi +ρj

−→

A
QED

(The i = j conditions is needed. See Exercise 13.)

This lemma suggests that ‘reduces to’ is misleading — where A −→ B, we shouldn’t think of B as “after” A or “simpler than” A. Instead we should think of them as interreducible or interrelated. Below is a picture of the idea. The matrices from the start of this section and their reduced echelon form version are shown in a cluster. They are all interreducible; these relationships are shown also.
2 0 0 −1 1 0 1 −1

2 4

2 3 1 0 0 1

2 0

2 −1

We say that matrices that reduce to each other are ‘equivalent with respect to the relationship of row reducibility’. The next result veriﬁes this statement using the deﬁnition of an equivalence.∗ 1.5 Lemma Between matrices, ‘reduces to’ is an equivalence relation.
Proof. We must check the conditions (i) reﬂexivity, that any matrix reduces to

itself, (ii) symmetry, that if A reduces to B then B reduces to A, and (iii) transitivity, that if A reduces to B and B reduces to C then A reduces to C. Reﬂexivity is easy; any matrix reduces to itself in zero row operations. That the relationship is symmetric is Lemma 1.4 — if A reduces to B by some row operations then also B reduces to A by reversing those operations. For transitivity, suppose that A reduces to B and that B reduces to C. Linking the reduction steps from A → · · · → B with those from B → · · · → C gives a reduction from A to C. QED 1.6 Deﬁnition Two matrices that are interreducible by the elementary row operations are row equivalent.

Section III. Reduced Echelon Form

51

The diagram below shows the collection of all matrices as a box. Inside that box, each matrix lies in some class. Matrices are in the same class if and only if they are interreducible. The classes are disjoint — no matrix is in two distinct classes. The collection of matrices has been partitioned into row equivalence classes.∗
A B

...

One of the classes in this partition is the cluster of matrices shown above, expanded to include all of the nonsingular 2×2 matrices. The next subsection proves that the reduced echelon form of a matrix is unique; that every matrix reduces to one and only one reduced echelon form matrix. Rephrased in terms of the row-equivalence relationship, we shall prove that every matrix is row equivalent to one and only one reduced echelon form matrix. In terms of the partition what we shall prove is: every equivalence class contains one and only one reduced echelon form matrix. So each reduced echelon form matrix serves as a representative of its class. After that proof we shall, as mentioned in the introduction to this section, have a way to decide if one matrix can be derived from another by row reduction. We just apply the Gauss-Jordan procedure to both and see whether or not they come to the same reduced echelon form. Exercises
1.7 Use Gauss-Jordan reduction to solve each system. (a) x + y = 2 (b) x −z=4 (c) 3x − 2y = 1 x−y=0 2x + 2y =1 6x + y = 1/2 (d) 2x − y = −1 x + 3y − z = 5 y + 2z = 5 1.8 Find the reduced echelon form of each matrix. 1 3 1 1 0 3 1 2 2 1 2 0 4 (a) (b) (c) 1 4 2 1 5 1 3 −1 −3 −3 3 4 8 1 2 0 1 3 2 (d) 0 0 5 6 1 5 1 5 1.9 Find each solution set by using Gauss-Jordan reduction, then reading oﬀ the parametrization. (a) 2x + y − z = 1 (b) x − z =1 (c) x − y + z =0 4x − y =3 y + 2z − w = 3 y +w=0 x + 2y + 3z − w = 7 3x − 2y + 3z + w = 0 −y −w=0 (d) a + 2b + 3c + d − e = 1 3a − b + c + d + e = 3

52
1.10 Give two distinct echelon form versions 2 1 1 6 4 1 1 5 1

Chapter One. Linear Systems
of this matrix. 3 2 5

1.11 List the reduced echelon forms possible for each size. (a) 2×2 (b) 2×3 (c) 3×2 (d) 3×3 1.12 What results from applying Gauss-Jordan reduction to a nonsingular matrix? 1.13 The proof of Lemma 1.4 contains a reference to the i = j condition on the row pivoting operation. (a) The deﬁnition of row operations has an i = j condition on the swap operation ρi ↔ ρj . Show that in A −→
ρi ↔ρj ρi ↔ρj

−→ A this condition is not needed.

(b) Write down a 2×2 matrix with nonzero entries, and show that the −1·ρ1 +ρ1 operation is not reversed by 1 · ρ1 + ρ1 . (c) Expand the proof of that lemma to make explicit exactly where the i = j condition on pivoting is used.

III.2 Row Equivalence
We will close this section and this chapter by proving that every matrix is row equivalent to one and only one reduced echelon form matrix. The ideas that appear here will reappear, and be further developed, in the next chapter. The underlying theme here is that one way to understand a mathematical situation is by being able to classify the cases that can happen. We have met this theme several times already. We have classiﬁed solution sets of linear systems into the no-elements, one-element, and inﬁnitely-many elements cases. We have also classiﬁed linear systems with the same number of equations as unknowns into the nonsingular and singular cases. We adopted these classiﬁcations because they give us a way to understand the situations that we were investigating. Here, where we are investigating row equivalence, we know that the set of all matrices breaks into the row equivalence classes. When we ﬁnish the proof here, we will have a way to understand each of those classes — its matrices can be thought of as derived by row operations from the unique reduced echelon form matrix in that class. To understand how row operations act to transform one matrix into another, we consider the eﬀect that they have on the parts of a matrix. The crucial observation is that row operations combine the rows linearly. 2.1 Deﬁnition A linear combination of x1 , . . . , xm is an expression of the form c1 x1 + c2 x2 + · · · + cm xm where the c’s are scalars. (We have already used the phrase ‘linear combination’ in this book. The meaning is unchanged, but the next result’s statement makes a more formal deﬁnition in order.)

Section III. Reduced Echelon Form

53

2.2 Lemma (Linear Combination Lemma) A linear combination of linear combinations is a linear combination.
Proof. Given the linear combinations c1,1 x1 + · · · + c1,n xn through cm,1 x1 +

· · · + cm,n xn , consider a combination of those d1 (c1,1 x1 + · · · + c1,n xn ) + · · · + dm (cm,1 x1 + · · · + cm,n xn ) where the d’s are scalars along with the c’s. Distributing those d’s and regrouping gives = (d1 c1,1 + · · · + dm cm,1 )x1 + · · · + (d1 c1,n + · · · + dm cm,n )xn which is a linear combination of the x’s.
QED

In this subsection we will use the convention that, where a matrix is named with an upper case roman letter, the matching lower-case greek letter names the rows.     · · · α1 · · · · · · β1 · · ·  ··· α ···   ··· β ···  2 2        A= B= . .    . .     . . · · · αm · · · · · · βm · · · 2.3 Corollary Where one matrix reduces to another, each row of the second is a linear combination of the rows of the ﬁrst. The proof below uses induction on the number of row operations used to reduce one matrix to the other. Before we proceed, here is an outline of the argument (readers unfamiliar with induction may want to compare this argument with the one used in the ‘General = Particular + Homogeneous’ proof).∗ First, for the base step of the argument, we will verify that the proposition is true when reduction can be done in zero row operations. Second, for the inductive step, we will argue that if being able to reduce the ﬁrst matrix to the second in some number t ≥ 0 of operations implies that each row of the second is a linear combination of the rows of the ﬁrst, then being able to reduce the ﬁrst to the second in t + 1 operations implies the same thing. Together, this base step and induction step prove this result because by the base step the proposition is true in the zero operations case, and by the inductive step the fact that it is true in the zero operations case implies that it is true in the one operation case, and the inductive step applied again gives that it is therefore true in the two operations case, etc.
Proof. We proceed by induction on the minimum number of row operations

that take a ﬁrst matrix A to a second one B.

54

Chapter One. Linear Systems

In the base step, that zero reduction operations suﬃce, the two matrices are equal and each row of B is obviously a combination of A’s rows: βi = 0 · α1 + · · · + 1 · αi + · · · + 0 · αm . For the inductive step, assume the inductive hypothesis: with t ≥ 0, if a matrix can be derived from A in t or fewer operations then its rows are linear combinations of the A’s rows. Consider a B that takes t+1 operations. Because there are more than zero operations, there must be a next-to-last matrix G so that A −→ · · · −→ G −→ B. This G is only t operations away from A and so the inductive hypothesis applies to it, that is, each row of G is a linear combination of the rows of A. If the last operation, the one from G to B, is a row swap then the rows of B are just the rows of G reordered and thus each row of B is also a linear combination of the rows of A. The other two possibilities for this last operation, that it multiplies a row by a scalar and that it adds a multiple of one row to another, both result in the rows of B being linear combinations of the rows of G. But therefore, by the Linear Combination Lemma, each row of B is a linear combination of the rows of A. With that, we have both the base step and the inductive step, and so the proposition follows. QED 2.4 Example In the reduction 0 1 2 1
ρ1 ↔ρ2

−→

1 0

1 2

(1/2)ρ2

−→

1 0

1 1

−ρ2 +ρ1

−→

1 0

0 1

call the matrices A, D, G, and B. The methods of the proof show that there are three sets of linear relationships. δ1 = 0 · α1 + 1 · α2 δ2 = 1 · α1 + 0 · α2 γ1 = 0 · α1 + 1 · α2 γ2 = (1/2)α1 + 0 · α2 β1 = (−1/2)α1 + 1 · α2 β2 = (1/2)α1 + 0 · α2

The prior result gives us the insight that Gauss’ method works by taking linear combinations of the rows. But to what end; why do we go to echelon form as a particularly simple, or basic, version of a linear system? The answer, of course, is that echelon form is suitable for back substitution, because we have isolated the variables. For instance, in this matrix   2 3 7 8 0 0  0 0 1 5 1 1  R=  0 0 0 3 3 0 0 0 0 0 2 1 x1 has been removed from x5 ’s equation. That is, Gauss’ method has made x5 ’s row independent of x1 ’s row. Independence of a collection of row vectors, or of any kind of vectors, will be precisely deﬁned and explored in the next chapter. But a ﬁrst take on it is that we can show that, say, the third row above is not comprised of the other

Section III. Reduced Echelon Form

55

rows, that ρ3 = c1 ρ1 + c2 ρ2 + c4 ρ4 . For, suppose that there are scalars c1 , c2 , and c4 such that this relationship holds. 0 0 0 3 3 0 = c1 2 + c2 0 + c4 0 3 0 0 7 1 0 8 5 0 0 1 2 0 1 1

The ﬁrst row’s leading entry is in the ﬁrst column and narrowing our consideration of the above relationship to consideration only of the entries from the ﬁrst column 0 = 2c1 +0c2 +0c4 gives that c1 = 0. The second row’s leading entry is in the third column and the equation of entries in that column 0 = 7c1 + 1c2 + 0c4 , along with the knowledge that c1 = 0, gives that c2 = 0. Now, to ﬁnish, the third row’s leading entry is in the fourth column and the equation of entries in that column 3 = 8c1 + 5c2 + 0c4 , along with c1 = 0 and c2 = 0, gives an impossibility. The following result shows that this eﬀect always holds. It shows that what Gauss’ linear elimination method eliminates is linear relationships among the rows. 2.5 Lemma In an echelon form matrix, no nonzero row is a linear combination of the other rows.
Proof. Let R be in echelon form. Suppose, to obtain a contradiction, that some nonzero row is a linear combination of the others.

ρi = c1 ρ1 + . . . + ci−1 ρi−1 + ci+1 ρi+1 + . . . + cm ρm We will ﬁrst use induction to show that the coeﬃcients c1 , . . . , ci−1 associated with rows above ρi are all zero. The contradiction will come from consideration of ρi and the rows below it. The base step of the induction argument is to show that the ﬁrst coeﬃcient c1 is zero. Let the ﬁrst row’s leading entry be in column number 1 and consider the equation of entries in that column. ρi,
1

= c1 ρ1, 1 + . . . + ci−1 ρi−1, 1 + ci+1 ρi+1, 1 + . . . + cm ρm,

1

The matrix is in echelon form so the entries ρ2, 1 , . . . , ρm, 1 , including ρi, 1 , are all zero. 0 = c1 ρ1, 1 + · · · + ci−1 · 0 + ci+1 · 0 + · · · + cm · 0 Because the entry ρ1, 1 is nonzero as it leads its row, the coeﬃcient c1 must be zero. The inductive step is to show that for each row index k between 1 and i − 2, if the coeﬃcient c1 and the coeﬃcients c2 , . . . , ck are all zero then ck+1 is also zero. That argument, and the contradiction that ﬁnishes this proof, is saved for Exercise 21. QED

56

Chapter One. Linear Systems

We can now prove that each matrix is row equivalent to one and only one reduced echelon form matrix. We will ﬁnd it convenient to break the ﬁrst half of the argument oﬀ as a preliminary lemma. For one thing, it holds for any echelon form whatever, not just reduced echelon form. 2.6 Lemma If two echelon form matrices are row equivalent then the leading entries in their ﬁrst rows lie in the same column. The same is true of all the nonzero rows — the leading entries in their second rows lie in the same column, etc. For the proof we rephrase the result in more technical terms. Deﬁne the form of an m×n matrix to be the sequence 1 , 2 , . . . , m where i is the column number of the leading entry in row i and i = ∞ if there is no leading entry in that row. The lemma says that if two echelon form matrices are row equivalent then their forms are equal sequences.
Proof. Let B and D be echelon form matrices that are row equivalent. Because

they are row equivalent they must be the same size, say m×n. Let the column number of the leading entry in row i of B be i and let the column number of the leading entry in row j of D be kj . We will show that 1 = k1 , that 2 = k2 , etc., by induction. This induction argument relies on the fact that the matrices are row equivalent, because the Linear Combination Lemma and its corollary therefore give that each row of B is a linear combination of the rows of D and vice versa: βi = si,1 δ1 + si,2 δ2 + · · · + si,m δm and δj = tj,1 β1 + tj,2 β2 + · · · + tj,m βm

where the s’s and t’s are scalars. The base step of the induction is to verify the lemma for the ﬁrst rows of the matrices, that is, to verify that 1 = k1 . If either row is a zero row then the entire matrix is a zero matrix since it is in echelon form, and therefore both matrices are zero matrices (by Corollary 2.3), and so both 1 and k1 are ∞. For the case where neither β1 nor δ1 is a zero row, consider the i = 1 instance of the linear relationship above. β1 = s1,1 δ1 + s1,2 δ2 + · · · + s1,m δm 0 ··· b1,
1

···

= s1,1 0 + s1,2 0 . . .

··· ···

d1,k1 0 0

··· ··· ···

+ s1,m 0

···

First, note that 1 < k1 is impossible: in the columns of D to the left of column k1 the entries are all zeroes (as d1,k1 leads the ﬁrst row) and so if 1 < k1 then the equation of entries from column 1 would be b1, 1 = s1,1 · 0 + · · · + s1,m · 0, but b1, 1 isn’t zero since it leads its row and so this is an impossibility. Next, a symmetric argument shows that k1 < 1 also is impossible. Thus the 1 = k1 base case holds.

Section III. Reduced Echelon Form

57

The inductive step is to show that if 1 = k1 , and 2 = k2 , . . . , and r = kr , then also r+1 = kr+1 (for r in the interval 1 .. m − 1). This argument is saved for Exercise 22. QED That lemma answers two of the questions that we have posed: (i) any two echelon form versions of a matrix have the same free variables, and consequently, and (ii) any two echelon form versions have the same number of free variables. There is no linear system and no combination of row operations such that, say, we could solve the system one way and get y and z free but solve it another way and get y and w free, or solve it one way and get two free variables while solving it another way yields three. We ﬁnish now by specializing to the case of reduced echelon form matrices. 2.7 Theorem Each matrix is row equivalent to a unique reduced echelon form matrix.
Proof. Clearly any matrix is row equivalent to at least one reduced echelon

form matrix, via Gauss-Jordan reduction. For the other half, that any matrix is equivalent to at most one reduced echelon form matrix, we will show that if a matrix Gauss-Jordan reduces to each of two others then those two are equal. Suppose that a matrix is row equivalent to two reduced echelon form matrices B and D, which are therefore row equivalent to each other. The Linear Combination Lemma and its corollary allow us to write the rows of one, say B, as a linear combination of the rows of the other βi = ci,1 δ1 + · · · + ci,m δm . The preliminary result, Lemma 2.6, says that in the two matrices, the same collection of rows are nonzero. Thus, if β1 through βr are the nonzero rows of B then the nonzero rows of D are δ1 through δr . Zero rows don’t contribute to the sum so we can rewrite the relationship to include just the nonzero rows. βi = ci,1 δ1 + · · · + ci,r δr (∗)

The preliminary result also says that for each row j between 1 and r, the leading entries of the j-th row of B and D appear in the same column, denoted j . Rewriting the above relationship to focus on the entries in the j -th column ··· bi,
j

···

= ci,1 + ci,2 . . . + ci,r

··· ··· ···

d1,

j j

··· ··· ···

d2, dr,

j

gives this set of equations for i = 1 up to i = r. b1, bj, br,
j

= c1,1 d1, j + · · · + c1,j dj, j + · · · + c1,r dr, . . . = cj,1 d1, j + · · · + cj,j dj, j + · · · + cj,r dr, . . . = cr,1 d1, j + · · · + cr,j dj, j + · · · + cr,r dr,

j

j

j

j

j

58

Chapter One. Linear Systems

Since D is in reduced echelon form, all of the d’s in column j are zero except for dj, j , which is 1. Thus each equation above simpliﬁes to bi, j = ci,j dj, j = ci,j · 1. But B is also in reduced echelon form and so all of the b’s in column j are zero except for bj, j , which is 1. Therefore, each ci,j is zero, except that c1,1 = 1, and c2,2 = 1, . . . , and cr,r = 1. We have shown that the only nonzero coeﬃcient in the linear combination labelled (∗) is cj,j , which is 1. Therefore βj = δj . Because this holds for all nonzero rows, B = D. QED We end with a recap. In Gauss’ method we start with a matrix and then derive a sequence of other matrices. We deﬁned two matrices to be related if one can be derived from the other. That relation is an equivalence relation, called row equivalence, and so partitions the set of all matrices into row equivalence classes.

13 27 13 01

...

(There are inﬁnitely many matrices in the pictured class, but we’ve only got room to show two.) We have proved there is one and only one reduced echelon form matrix in each row equivalence class. So the reduced echelon form is a canonical form∗ for row equivalence: the reduced echelon form matrices are representatives of the classes.

10 01

...

We can answer questions about the classes by translating them into questions about the representatives. 2.8 Example We can decide if matrices are interreducible by seeing if GaussJordan reduction produces the same reduced echelon form result. Thus, these are not row equivalent 1 −2

−3 6

1 −2

−3 5

Section III. Reduced Echelon Form because their reduced echelon forms are not equal. 1 0 −3 0 1 0 0 1

59

2.9 Example Any nonsingular 3×3 matrix Gauss-Jordan reduces to this.  1 0 0 0 1 0  0 0 1

2.10 Example We can describe the classes by listing all possible reduced echelon form matrices. Any 2×2 matrix lies in one of these: the class of matrices row equivalent to this, 0 0 0 0 the inﬁnitely many classes of matrices row equivalent to one of this type 1 0 a 0

where a ∈ R (including a = 0), the class of matrices row equivalent to this, 0 0 1 0

and the class of matrices row equivalent to this 1 0 0 1

(this is the class of nonsingular 2×2 matrices). Exercises
2.11 Decide if the matrices are row equivalent. 1 0 2 1 2 0 1 (a) , (b) 3 −1 1 , 4 8 1 2 5 −1 5 2 1 −1 1 0 2 1 0 , (c) 1 1 (d) 0 2 10 −1 4 3 −1 1 1 1 0 1 2 (e) , 0 0 3 1 −1 1 1 0 2 1 2 0 2 0 2 10 4 3 2 −1 5

1 0 , 2 2

2.12 Describe the matrices in each of the classes represented in Example 2.10. 2.13 Describe all matrices in the row equivalence class of these.

60

Chapter One. Linear Systems
1 0 1 2 1 1 (b) (c) 0 0 2 4 1 3 2.14 How many row equivalence classes are there? 2.15 Can row equivalence classes contain diﬀerent-sized matrices? 2.16 How big are the row equivalence classes? (a) Show that the class of any zero matrix is ﬁnite. (b) Do any other classes contain only ﬁnitely many members? 2.17 Give two reduced echelon form matrices that have their leading entries in the same columns, but that are not row equivalent. 2.18 Show that any two n × n nonsingular matrices are row equivalent. Are any two singular matrices row equivalent? 2.19 Describe all of the row equivalence classes containing these. (a) 2 × 2 matrices (b) 2 × 3 matrices (c) 3 × 2 matrices (d) 3×3 matrices 2.20 (a) Show that a vector β0 is a linear combination of members of the set {β1 , . . . , βn } if and only if there is a linear relationship 0 = c0 β0 + · · · + cn βn where c0 is not zero. (Hint. Watch out for the β0 = 0 case.) (b) Use that to simplify the proof of Lemma 2.5. 2.21 Finish the proof of Lemma 2.5. (a) First illustrate the inductive step by showing that c2 = 0. (b) Do the full inductive step: where 1 ≤ n < i − 1, assume that ck = 0 for 1 < k < n and deduce that cn+1 = 0 also. (c) Find the contradiction. 2.22 Finish the induction argument in Lemma 2.6. (a) State the inductive hypothesis, Also state what must be shown to follow from that hypothesis. (b) Check that the inductive hypothesis implies that in the relationship βr+1 = sr+1,1 δ1 + sr+2,2 δ2 + · · · + sr+1,m δm the coeﬃcients sr+1,1 , . . . , sr+1,r are each zero. (c) Finish the inductive step by arguing, as in the base case, that r+1 < kr+1 and kr+1 < r+1 are impossible. 2.23 Why, in the proof of Theorem 2.7, do we bother to restrict to the nonzero rows? Why not just stick to the relationship that we began with, βi = ci,1 δ1 +· · ·+ci,m δm , with m instead of r, and argue using it that the only nonzero coeﬃcient is ci,i , which is 1? 2.24 Three truck drivers went into a roadside cafe. One truck driver purchased four sandwiches, a cup of coﬀee, and ten doughnuts for \$8.45. Another driver purchased three sandwiches, a cup of coﬀee, and seven doughnuts for \$6.30. What did the third truck driver pay for a sandwich, a cup of coﬀee, and a doughnut? [Trono] 2.25 The fact that Gaussian reduction disallows multiplication of a row by zero is needed for the proof of uniqueness of reduced echelon form, or else every matrix would be row equivalent to a matrix of all zeros. Where is it used? 2.26 The Linear Combination Lemma says which equations can be gotten from Gaussian reduction from a given linear system. (a) (1) Produce an equation not implied by this system. 3x + 4y = 8 2x + y = 3

Section III. Reduced Echelon Form
(2) Can any equation be derived from an inconsistent system?

61

2.27 Extend the deﬁnition of row equivalence to linear systems. Under your deﬁnition, do equivalent systems have the same solution set? [Hoﬀman & Kunze] 2.28 In this matrix 1 2 3 3 0 3 1 4 5 the ﬁrst and second columns add to the third. (a) Show that remains true under any row operation. (b) Make a conjecture. (c) Prove that it holds.

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Chapter One. Linear Systems

Topic: Computer Algebra Systems
The linear systems in this chapter are small enough that their solution by hand is easy. But large systems are easiest, and safest, to do on a computer. There are special purpose programs such as LINPACK for this job. Another popular tool is a general purpose computer algebra system, including both commercial packages such as Maple, Mathematica, or MATLAB, or free packages such as SciLab,, MuPAD, or Octave. For example, in the Topic on Networks, we need to solve this. i0 − i1 − i2 i1 − i2 = 0 − i5 = 0 − i4 + i5 = 0 i3 + i4 − i6 = 0 5i1 + 10i3 = 10 2i2 + 4i4 = 10 5i1 − 2i2 + 50i5 = 0 i3

It can be done by hand, but it would take a while and be error-prone. Using a computer is better. We illustrate by solving that system under Maple (for another system, a user’s manual would obviously detail the exact syntax needed). The array of coeﬃcients can be entered in this way
> A:=array( [[1,-1,-1,0,0,0,0], [0,1,0,-1,0,-1,0], [0,0,1,0,-1,1,0], [0,0,0,1,1,0,-1], [0,5,0,10,0,0,0], [0,0,2,0,4,0,0], [0,5,-1,0,0,10,0]] );

(putting the rows on separate lines is not necessary, but is done for clarity). The vector of constants is entered similarly.
> u:=array( [0,0,0,0,10,10,0] );

Then the system is solved, like magic.
> linsolve(A,u); 7 2 5 2 5 7 [ -, -, -, -, -, 0, - ] 3 3 3 3 3 3

Systems with inﬁnitely many solutions are solved in the same way — the computer simply returns a parametrization. Exercises
Answers for this Topic use Maple as the computer algebra system. In particular, all of these were tested on Maple V running under MS-DOS NT version 4.0. (On all of them, the preliminary command to load the linear algebra package along with Maple’s responses to the Enter key, have been omitted.) Other systems have similar commands.

Topic: Computer Algebra Systems
1 Use the computer to solve the two problems that opened this chapter. (a) This is the Statics problem. 40h + 15c = 100 25c = 50 + 50h (b) This is the Chemistry problem. 7h = 7j 8h + 1i = 5j + 2k 1i = 3j 3i = 6j + 1k

63

2 Use the computer to solve these systems from the ﬁrst subsection, or conclude ‘many solutions’ or ‘no solutions’. (a) 2x + 2y = 5 (b) −x + y = 1 (c) x − 3y + z = 1 x − 4y = 0 x+y=2 x + y + 2z = 14 (d) −x − y = 1 (e) 4y + z = 20 (f ) 2x + z+w= 5 −3x − 3y = 2 2x − 2y + z = 0 y − w = −1 x +z= 5 3x − z−w= 0 x + y − z = 10 4x + y + 2z + w = 9 3 Use the computer to solve these systems from the second subsection. (a) 3x + 6y = 18 (b) x + y = 1 (c) x1 + x3 = 4 x + 2y = 6 x − y = −1 x1 − x2 + 2x3 = 5 4x1 − x2 + 5x3 = 17 (d) 2a + b − c = 2 (e) x + 2y − z =3 (f ) x +z+w=4 2a +c=3 2x + y +w=4 2x + y −w=2 a−b =0 x− y+z+w=1 3x + y + z =7 4 What does the computer give for the solution of the general 2×2 system? ax + cy = p bx + dy = q

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Chapter One. Linear Systems

Topic: Input-Output Analysis
An economy is an immensely complicated network of interdependences. Changes in one part can ripple out to aﬀect other parts. Economists have struggled to be able to describe, and to make predictions about, such a complicated object. Mathematical models using systems of linear equations have emerged as a key tool. One is Input-Output Analysis, pioneered by W. Leontief, who won the 1973 Nobel Prize in Economics. Consider an economy with many parts, two of which are the steel industry and the auto industry. As they work to meet the demand for their product from other parts of the economy, that is, from users external to the steel and auto sectors, these two interact tightly. For instance, should the external demand for autos go up, that would lead to an increase in the auto industry’s usage of steel. Or, should the external demand for steel fall, then it would lead to a fall in steel’s purchase of autos. The type of Input-Output model we will consider takes in the external demands and then predicts how the two interact to meet those demands. We start with a listing of production and consumption statistics. (These numbers, giving dollar values in millions, are excerpted from [Leontief 1965], describing the 1958 U.S. economy. Today’s statistics would be quite diﬀerent, both because of inﬂation and because of technical changes in the industries.) used by steel value of steel value of auto 5 395 48 used by auto 2 664 9 030 used by others total 25 448 30 346

For instance, the dollar value of steel used by the auto industry in this year is 2, 664 million. Note that industries may consume some of their own output. We can ﬁll in the blanks for the external demand. This year’s value of the steel used by others this year is 17, 389 and this year’s value of the auto used by others is 21, 268. With that, we have a complete description of the external demands and of how auto and steel interact, this year, to meet them. Now, imagine that the external demand for steel has recently been going up by 200 per year and so we estimate that next year it will be 17, 589. Imagine also that for similar reasons we estimate that next year’s external demand for autos will be down 25 to 21, 243. We wish to predict next year’s total outputs. That prediction isn’t as simple as adding 200 to this year’s steel total and subtracting 25 from this year’s auto total. For one thing, a rise in steel will cause that industry to have an increased demand for autos, which will mitigate, to some extent, the loss in external demand for autos. On the other hand, the drop in external demand for autos will cause the auto industry to use less steel, and so lessen somewhat the upswing in steel’s business. In short, these two industries form a system, and we need to predict the totals at which the system as a whole will settle.

Topic: Input-Output Analysis

65

For that prediction, let s be next years total production of steel and let a be next year’s total output of autos. We form these equations. next year’s production of steel = next year’s use of steel by steel + next year’s use of steel by auto + next year’s use of steel by others next year’s production of autos = next year’s use of autos by steel + next year’s use of autos by auto + next year’s use of autos by others On the left side of those equations go the unknowns s and a. At the ends of the right sides go our external demand estimates for next year 17, 589 and 21, 243. For the remaining four terms, we look to the table of this year’s information about how the industries interact. For instance, for next year’s use of steel by steel, we note that this year the steel industry used 5395 units of steel input to produce 25, 448 units of steel output. So next year, when the steel industry will produce s units out, we expect that doing so will take s · (5395)/(25 448) units of steel input — this is simply the assumption that input is proportional to output. (We are assuming that the ratio of input to output remains constant over time; in practice, models may try to take account of trends of change in the ratios.) Next year’s use of steel by the auto industry is similar. This year, the auto industry uses 2664 units of steel input to produce 30346 units of auto output. So next year, when the auto industry’s total output is a, we expect it to consume a · (2664)/(30346) units of steel. Filling in the other equation in the same way, we get this system of linear equation. 5 395 2 664 ·s+ · a + 17 589 = s 25 448 30 346 48 9 030 ·s+ · a + 21 243 = a 25 448 30 346 Gauss’ method on this system. (20 053/25 448)s − (2 664/30 346)a = 17 589 −(48/25 448)s + (21 316/30 346)a = 21 243 gives s = 25 698 and a = 30 311. Looking back, recall that above we described why the prediction of next year’s totals isn’t as simple as adding 200 to last year’s steel total and subtracting 25 from last year’s auto total. In fact, comparing these totals for next year to the ones given at the start for the current year shows that, despite the drop in external demand, the total production of the auto industry is predicted to rise. The increase in internal demand for autos caused by steel’s sharp rise in business more than makes up for the loss in external demand for autos. One of the advantages of having a mathematical model is that we can ask “What if . . . ?” questions. For instance, we can ask “What if the estimates for

66

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next year’s external demands are somewhat oﬀ?” To try to understand how much the model’s predictions change in reaction to changes in our estimates, we can try revising our estimate of next year’s external steel demand from 17, 589 down to 17, 489, while keeping the assumption of next year’s external demand for autos ﬁxed at 21, 243. The resulting system (20 053/25 448)s − (2 664/30 346)a = 17 489 −(48/25 448)s + (21 316/30 346)a = 21 243 when solved gives s = 25 571 and a = 30 311. This kind of exploration of the model is sensitivity analysis. We are seeing how sensitive the predictions of our model are to the accuracy of the assumptions. Obviously, we can consider larger models that detail the interactions among more sectors of an economy. These models are typically solved on a computer, using the techniques of matrix algebra that we will develop in Chapter Three. Some examples are given in the exercises. Obviously also, a single model does not suit every case; expert judgment is needed to see if the assumptions underlying the model are reasonable for a particular case. With those caveats, however, this model has proven in practice to be a useful and accurate tool for economic analysis. For further reading, try [Leontief 1951] and [Leontief 1965]. Exercises
Hint: these systems are easiest to solve on a computer. 1 With the steel-auto system given above, estimate next year’s total productions in these cases. (a) Next year’s external demands are: up 200 from this year for steel, and unchanged for autos. (b) Next year’s external demands are: up 100 for steel, and up 200 for autos. (c) Next year’s external demands are: up 200 for steel, and up 200 for autos. 2 In the steel-auto system, the ratio for the use of steel by the auto industry is 2 664/30 346, about 0.0878. Imagine that a new process for making autos reduces this ratio to .0500. (a) How will the predictions for next year’s total productions change compared to the ﬁrst example discussed above (i.e., taking next year’s external demands to be 17, 589 for steel and 21, 243 for autos)? (b) Predict next year’s totals if, in addition, the external demand for autos rises to be 21, 500 because the new cars are cheaper. 3 This table gives the numbers for the auto-steel system from a diﬀerent year, 1947 (see [Leontief 1951]). The units here are billions of 1947 dollars. used by used by used by steel auto others total value of 6.90 1.28 18.69 steel value of 0 4.40 14.27 autos (a) Solve for total output if next year’s external demands are: steel’s demand up 10% and auto’s demand up 15%. (b) How do the ratios compare to those given above in the discussion for the 1958 economy?

Topic: Input-Output Analysis

67

(c) Solve the 1947 equations with the 1958 external demands (note the diﬀerence in units; a 1947 dollar buys about what \$1.30 in 1958 dollars buys). How far oﬀ are the predictions for total output? 4 Predict next year’s total productions of each of the three sectors of the hypothetical economy shown below used by used by used by used by farm rail shipping others total value of 25 50 100 800 farm value of 25 50 50 300 rail value of 15 10 0 500 shipping if next year’s external demands are as stated. (a) 625 for farm, 200 for rail, 475 for shipping (b) 650 for farm, 150 for rail, 450 for shipping 5 This table gives the interrelationships among three segments of an economy (see [Clark & Coupe]). used by used by used by used by total food wholesale retail others value of 0 2 318 4 679 11 869 food value of wholesale 393 1 089 22 459 122 242 value of retail 3 53 75 116 041 We will do an Input-Output analysis on this system. (a) Fill in the numbers for this year’s external demands. (b) Set up the linear system, leaving next year’s external demands blank. (c) Solve the system where next year’s external demands are calculated by taking this year’s external demands and inﬂating them 10%. Do all three sectors increase their total business by 10%? Do they all even increase at the same rate? (d) Solve the system where next year’s external demands are calculated by taking this year’s external demands and reducing them 7%. (The study from which these numbers are taken concluded that because of the closing of a local military facility, overall personal income in the area would fall 7%, so this might be a ﬁrst guess at what would actually happen.)

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Chapter One. Linear Systems

Topic: Accuracy of Computations
Gauss’ method lends itself nicely to computerization. The code below illustrates. It operates on an n×n matrix a, pivoting with the ﬁrst row, then with the second row, etc.
for(pivot_row=1;pivot_row<=n-1;pivot_row++){ for(row_below=pivot_row+1;row_below<=n;row_below++){ multiplier=a[row_below,pivot_row]/a[pivot_row,pivot_row]; for(col=pivot_row;col<=n;col++){ a[row_below,col]-=multiplier*a[pivot_row,col]; } } }

(This code is in the C language. Here is a brief translation. The loop construct for(pivot row=1;pivot row<=n-1;pivot row++){· · · } sets pivot row to 1 and then iterates while pivot row is less than or equal to n − 1, each time through incrementing pivot row by one with the ‘++’ operation. The other non-obvious construct is that the ‘-=’ in the innermost loop amounts to the a[row below,col] = −multiplier ∗ a[pivot row,col] + a[row below,col] operation.) While this code provides a quick take on how Gauss’ method can be mechanized, it is not ready to use. It is naive in many ways. The most glaring way is that it assumes that a nonzero number is always found in the pivot row, pivot row position for use as the pivot entry. To make it practical, one way in which this code needs to be reworked is to cover the case where ﬁnding a zero in that location leads to a row swap, or to the conclusion that the matrix is singular. Adding some if · · · statements to cover those cases is not hard, but we will instead consider some more subtle ways in which the code is naive. There are pitfalls arising from the computer’s reliance on ﬁnite-precision ﬂoating point arithmetic. For example, we have seen above that we must handle as a separate case a system that is singular. But systems that are nearly singular also require care. Consider this one. x + 2y = 3 1.000 000 01x + 2y = 3.000 000 01 By eye we get the solution x = 1 and y = 1. But a computer has more trouble. A computer that represents real numbers to eight signiﬁcant places (as is common, usually called single precision) will represent the second equation internally as 1.000 000 0x + 2y = 3.000 000 0, losing the digits in the ninth place. Instead of reporting the correct solution, this computer will report something that is not even close — this computer thinks that the system is singular because the two equations are represented internally as equal. For some intuition about how the computer could come up with something that far oﬀ, we can graph the system.

Topic: Accuracy of Computations

69

(1, 1)

At the scale of this graph, the two lines cannot be resolved apart. This system is nearly singular in the sense that the two lines are nearly the same line. Nearsingularity gives this system the property that a small change in the system can cause a large change in its solution; for instance, changing the 3.000 000 01 to 3.000 000 03 changes the intersection point from (1, 1) to (3, 0). This system changes radically depending on a ninth digit, which explains why the eightplace computer has trouble. A problem that is very sensitive to inaccuracy or uncertainties in the input values is ill-conditioned. The above example gives one way in which a system can be diﬃcult to solve on a computer. It has the advantage that the picture of nearly-equal lines gives a memorable insight into one way that numerical diﬃculties can arise. Unfortunately this insight isn’t very useful when we wish to solve some large system. We cannot, typically, hope to understand the geometry of an arbitrary large system. In addition, there are ways that a computer’s results may be unreliable other than that the angle between some of the linear surfaces is quite small. For an example, consider the system below, from [Hamming]. 0.001x + y = 1 x−y=0 (∗)

The second equation gives x = y, so x = y = 1/1.001 and thus both variables have values that are just less than 1. A computer using two digits represents the system internally in this way (we will do this example in two-digit ﬂoating point arithmetic, but a similar one with eight digits is easy to invent). (1.0 × 10−2 )x + (1.0 × 100 )y = 1.0 × 100 (1.0 × 100 )x − (1.0 × 100 )y = 0.0 × 100 The computer’s row reduction step −1000ρ1 + ρ2 produces a second equation −1001y = −999, which the computer rounds to two places as (−1.0 × 103 )y = −1.0 × 103 . Then the computer decides from the second equation that y = 1 and from the ﬁrst equation that x = 0. This y value is fairly good, but the x is quite bad. Thus, another cause of unreliable output is a mixture of ﬂoating point arithmetic and a reliance on pivots that are small. An experienced programmer may respond that we should go to double precision where sixteen signiﬁcant digits are retained. This will indeed solve many problems. However, there are some diﬃculties with it as a general approach. For one thing, double precision takes longer than single precision (on a ’486

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Chapter One. Linear Systems

chip, multiplication takes eleven ticks in single precision but fourteen in double precision [Programmer’s Ref.]) and has twice the memory requirements. So attempting to do all calculations in double precision is just not practical. And besides, the above systems can obviously be tweaked to give the same trouble in the seventeenth digit, so double precision won’t ﬁx all problems. What we need is a strategy to minimize the numerical trouble arising from solving systems on a computer, and some guidance as to how far the reported solutions can be trusted. Mathematicians have made a careful study of how to get the most reliable results. A basic improvement on the naive code above is to not simply take the entry in the pivot row , pivot row position for the pivot, but rather to look at all of the entries in the pivot row column below the pivot row row, and take the one that is most likely to give reliable results (e.g., take one that is not too small). This strategy is partial pivoting. For example, to solve the troublesome system (∗) above, we start by looking at both equations for a best ﬁrst pivot, and taking the 1 in the second equation as more likely to give good results. Then, the pivot step of −.001ρ2 + ρ1 gives a ﬁrst equation of 1.001y = 1, which the computer will represent as (1.0×100 )y = 1.0×100 , leading to the conclusion that y = 1 and, after back-substitution, x = 1, both of which are close to right. The code from above can be adapted to this purpose.
for(pivot_row=1;pivot_row<=n-1;pivot_row++){ /* find the largest pivot in this column (in row max) */ max=pivot_row; for(row_below=pivot_row+1;pivot_row<=n;row_below++){ if (abs(a[row_below,pivot_row]) > abs(a[max,row_below])) max=row_below; } /* swap rows to move that pivot entry up */ for(col=pivot_row;col<=n;col++){ temp=a[pivot_row,col]; a[pivot_row,col]=a[max,col]; a[max,col]=temp; } /* proceed as before */ for(row_below=pivot_row+1;row_below<=n;row_below++){ multiplier=a[row_below,pivot_row]/a[pivot_row,pivot_row]; for(col=pivot_row;col<=n;col++){ a[row_below,col]-=multiplier*a[pivot_row,col]; } } }

A full analysis of the best way to implement Gauss’ method is outside the scope of the book (see [Wilkinson 1965]), but the method recommended by most experts is a variation on the code above that ﬁrst ﬁnds the best pivot among the candidates, and then scales it to a number that is less likely to give trouble. This is scaled partial pivoting.

Topic: Accuracy of Computations

71

In addition to returning a result that is likely to be reliable, most well-done code will return a number, called the conditioning number that describes the factor by which uncertainties in the input numbers could be magniﬁed to become inaccuracies in the results returned (see [Rice]). The lesson of this discussion is that just because Gauss’ method always works in theory, and just because computer code correctly implements that method, and just because the answer appears on green-bar paper, doesn’t mean that the answer is reliable. In practice, always use a package where experts have worked hard to counter what can go wrong. Exercises
1 Using two decimal places, add 253 and 2/3. 2 This intersect-the-lines problem contrasts with the example discussed above.

(1, 1)

x + 2y = 3 3x − 2y = 1

Illustrate that in this system some small change in the numbers will produce only a small change in the solution by changing the constant in the bottom equation to 1.008 and solving. Compare it to the solution of the unchanged system. 3 Solve this system by hand ([Rice]). 0.000 3x + 1.556y = 1.569 0.345 4x − 2.346y = 1.018 (a) Solve it accurately, by hand. (b) Solve it by rounding at each step to four signiﬁcant digits. 4 Rounding inside the computer often has an eﬀect on the result. Assume that your machine has eight signiﬁcant digits. (a) Show that the machine will compute (2/3) + ((2/3) − (1/3)) as unequal to ((2/3) + (2/3)) − (1/3). Thus, computer arithmetic is not associative. (b) Compare the computer’s version of (1/3)x + y = 0 and (2/3)x + 2y = 0. Is twice the ﬁrst equation the same as the second? 5 Ill-conditioning is not only dependent on the matrix of coeﬃcients. This example [Hamming] shows that it can arise from an interaction between the left and right sides of the system. Let ε be a small real. 3x + 2y + z = 6 2x + 2εy + 2εz = 2 + 4ε x + 2εy − εz = 1 + ε (a) Solve the system by hand. Notice that the ε’s divide out only because there is an exact cancelation of the integer parts on the right side as well as on the left. (b) Solve the system by hand, rounding to two decimal places, and with ε = 0.001.

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Chapter One. Linear Systems

Topic: Analyzing Networks
The diagram below shows some of a car’s electrical network. The battery is on the left, drawn as stacked line segments. The wires are drawn as lines, shown straight and with sharp right angles for neatness. Each light is a circle enclosing a loop.
Brake Actuated Switch 12V Light Switch Oﬀ Dimmer Hi Door Actuated Switch Dome Light

Lo

L

R Brake Lights

L

R Parking Lights

L

R Rear Lights

L

R

L

R

The designer of such a network needs to answer questions like: How much electricity ﬂows when both the hi-beam headlights and the brake lights are on? Below, we will use linear systems to analyze simpler versions of electrical networks. For the analysis we need two facts about electricity and two facts about electrical networks. The ﬁrst fact about electricity is that a battery is like a pump: it provides a force impelling the electricity to ﬂow through the circuits connecting the battery’s ends, if there are any such circuits. We say that the battery provides a potential to ﬂow. Of course, this network accomplishes its function when, as the electricity ﬂows through a circuit, it goes through a light. For instance, when the driver steps on the brake then the switch makes contact and a circuit is formed on the left side of the diagram, and the electrical current ﬂowing through that circuit will make the brake lights go on, warning drivers behind. The second electrical fact is that in some kinds of network components the amount of ﬂow is proportional to the force provided by the battery. That is, for each such component there is a number, it’s resistance, such that the potential is equal to the ﬂow times the resistance. The units of measurement are: potential is described in volts, the rate of ﬂow is in amperes, and resistance to the ﬂow is in ohms. These units are deﬁned so that volts = amperes · ohms. Components with this property, that the voltage-amperage response curve is a line through the origin, are called resistors. (Light bulbs such as the ones shown above are not this kind of component, because their ohmage changes as they heat up.) For example, if a resistor measures 2 ohms then wiring it to a 12 volt battery results in a ﬂow of 6 amperes. Conversely, if we have ﬂow of electrical current of 2 amperes through it then there must be a 4 volt potential

Topic: Analyzing Networks

73

diﬀerence between it’s ends. This is the voltage drop across the resistor. One way to think of a electrical circuits like the one above is that the battery provides a voltage rise while the other components are voltage drops. The two facts that we need about networks are Kirchhoﬀ’s Laws. Current Law. For any point in a network, the ﬂow in equals the ﬂow out. Voltage Law. Around any circuit the total drop equals the total rise. In the above network there is only one voltage rise, at the battery, but some networks have more than one. For a start we can consider the network below. It has a battery that provides the potential to ﬂow and three resistors (resistors are drawn as zig-zags). When components are wired one after another, as here, they are said to be in series.
2 ohm resistance 20 volt potential 3 ohm resistance 5 ohm resistance

By Kirchhoﬀ’s Voltage Law, because the voltage rise is 20 volts, the total voltage drop must also be 20 volts. Since the resistance from start to ﬁnish is 10 ohms (the resistance of the wires is negligible), we get that the current is (20/10) = 2 amperes. Now, by Kirchhoﬀ’s Current Law, there are 2 amperes through each resistor. (And therefore the voltage drops are: 4 volts across the 2 oh m resistor, 10 volts across the 5 ohm resistor, and 6 volts across the 3 ohm resistor.) The prior network is so simple that we didn’t use a linear system, but the next network is more complicated. In this one, the resistors are in parallel. This network is more like the car lighting diagram shown earlier.

20 volt

12 ohm

8 ohm

We begin by labeling the branches, shown below. Let the current through the left branch of the parallel portion be i1 and that through the right branch be i2 , and also let the current through the battery be i0 . (We are following Kirchoﬀ’s Current Law; for instance, all points in the right branch have the same current, which we call i2 . Note that we don’t need to know the actual direction of ﬂow — if current ﬂows in the direction opposite to our arrow then we will simply get a negative number in the solution.)

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Chapter One. Linear Systems

↑ i0

i1 ↓

↓ i2

The Current Law, applied to the point in the upper right where the ﬂow i0 meets i1 and i2 , gives that i0 = i1 + i2 . Applied to the lower right it gives i1 + i2 = i0 . In the circuit that loops out of the top of the battery, down the left branch of the parallel portion, and back into the bottom of the battery, the voltage rise is 20 while the voltage drop is i1 · 12, so the Voltage Law gives that 12i1 = 20. Similarly, the circuit from the battery to the right branch and back to the battery gives that 8i2 = 20. And, in the circuit that simply loops around in the left and right branches of the parallel portion (arbitrarily taken clockwise), there is a voltage rise of 0 and a voltage drop of 8i2 − 12i1 so the Voltage Law gives that 8i2 − 12i1 = 0. i0 − −i0 + i1 − i2 = 0 i1 + i2 = 0 12i1 = 20 8i2 = 20 −12i1 + 8i2 = 0

The solution is i0 = 25/6, i1 = 5/3, and i2 = 5/2, all in amperes. (Incidentally, this illustrates that redundant equations do indeed arise in practice.) Kirchhoﬀ’s laws can be used to establish the electrical properties of networks of great complexity. The next diagram shows ﬁve resistors, wired in a seriesparallel way.
5 ohm 10 volt 10 ohm 50 ohm 2 ohm

4 ohm

This network is a Wheatstone bridge (see Exercise 4). To analyze it, we can place the arrows in this way.
i1 ↑ i0 i3 i5 → i4 i2

Topic: Analyzing Networks

75

Kirchoﬀ’s Current Law, applied to the top node, the left node, the right node, and the bottom node gives these. i0 = i1 + i2 i1 = i3 + i5 i2 + i5 = i4 i3 + i4 = i0 Kirchhoﬀ’s Voltage Law, applied to the inside loop (the i0 to i1 to i3 to i0 loop), the outside loop, and the upper loop not involving the battery, gives these. 5i1 + 10i3 = 10 2i2 + 4i4 = 10 5i1 + 50i5 − 2i2 = 0 Those suﬃce to determine the solution i0 = 7/3, i1 = 2/3, i2 = 5/3, i3 = 2/3, i4 = 5/3, and i5 = 0. Networks of other kinds, not just electrical ones, can also be analyzed in this way. For instance, networks of streets are given in the exercises. Exercises
Many of the systems for these problems are mostly easily solved on a computer. 1 Calculate the amperages in each part of each network. (a) This is a simple network. 3 ohm 9 volt 2 ohm (b) Compare this one with the parallel case discussed above. 3 ohm 9 volt 2 ohm 2 ohm (c) This is a reasonably complicated network. 3 ohm 9 volt 3 ohm 2 ohm 2 ohm 2 ohm 3 ohm 4 ohm 2 ohm 2 ohm

76

Chapter One. Linear Systems
2 In the ﬁrst network that we analyzed, with the three resistors in series, we just added to get that they acted together like a single resistor of 10 ohms. We can do a similar thing for parallel circuits. In the second circuit analyzed,

20 volt

12 ohm

8 ohm

the electric current through the battery is 25/6 amperes. Thus, the parallel portion is equivalent to a single resistor of 20/(25/6) = 4.8 ohms. (a) What is the equivalent resistance if we change the 12 ohm resistor to 5 ohms? (b) What is the equivalent resistance if the two are each 8 ohms? (c) Find the formula for the equivalent resistance if the two resistors in parallel are r1 ohms and r2 ohms. 3 For the car dashboard example that opens this Topic, solve for these amperages (assume that all resistances are 2 ohms). (a) If the driver is stepping on the brakes, so the brake lights are on, and no other circuit is closed. (b) If the hi-beam headlights and the brake lights are on. 4 Show that, in this Wheatstone Bridge, r1 rg r2 r4 r3

r2 /r1 equals r4 /r3 if and only if the current ﬂowing through rg is zero. (The way that this device is used in practice is that an unknown resistance at r4 is compared to the other three r1 , r2 , and r3 . At rg is placed a meter that shows the current. The three resistances r1 , r2 , and r3 are varied — typically they each have a calibrated knob — until the current in the middle reads 0, and then the above equation gives the value of r4 .) There are networks other than electrical ones, and we can ask how well Kirchoﬀ ’s laws apply to them. The remaining questions consider an extension to networks of streets. 5 Consider this traﬃc circle. North Avenue Main Street

Pier Boulevard

Topic: Analyzing Networks

77

This is the traﬃc volume, in units of cars per ﬁve minutes. North Pier Main into 100 150 25 out of 75 150 50 We can set up equations to model how the traﬃc ﬂows. (a) Adapt Kirchoﬀ’s Current Law to this circumstance. Is it a reasonable modelling assumption? (b) Label the three between-road arcs in the circle with a variable. Using the (adapted) Current Law, for each of the three in-out intersections state an equation describing the traﬃc ﬂow at that node. (c) Solve that system. (d) Interpret your solution. (e) Restate the Voltage Law for this circumstance. How reasonable is it? 6 This is a network of streets. Shelburne St Willow west Winooski Ave Jay Ln east

The hourly ﬂow of cars into this network’s entrances, and out of its exits can be observed. east Winooski west Winooski Willow Jay Shelburne into 80 50 65 – 40 30 5 70 55 75 out of (Note that to reach Jay a car must enter the network via some other road ﬁrst, which is why there is no ‘into Jay’ entry in the table. Note also that over a long period of time, the total in must approximately equal the total out, which is why both rows add to 235 cars.) Once inside the network, the traﬃc may ﬂow in diﬀerent ways, perhaps ﬁlling Willow and leaving Jay mostly empty, or perhaps ﬂowing in some other way. Kirchhoﬀ’s Laws give the limits on that freedom. (a) Determine the restrictions on the ﬂow inside this network of streets by setting up a variable for each block, establishing the equations, and solving them. Notice that some streets are one-way only. (Hint: this will not yield a unique solution, since traﬃc can ﬂow through this network in various ways; you should get at least one free variable.) (b) Suppose that some construction is proposed for Winooski Avenue East between Willow and Jay, so traﬃc on that block will be reduced. What is the least amount of traﬃc ﬂow that can be allowed on that block without disrupting the hourly ﬂow into and out of the network?

Chapter Two

Vector Spaces
The ﬁrst chapter began by introducing Gauss’ method and ﬁnished with a fair understanding, keyed on the Linear Combination Lemma, of how it ﬁnds the solution set of a linear system. Gauss’ method systematically takes linear combinations of the rows. With that insight, we now move to a general study of linear combinations. We need a setting for this study. At times in the ﬁrst chapter, we’ve combined vectors from R2 , at other times vectors from R3 , and at other times vectors from even higher-dimensional spaces. Thus, our ﬁrst impulse might be to work in Rn , leaving n unspeciﬁed. This would have the advantage that any of the results would hold for R2 and for R3 and for many other spaces, simultaneously. But, if having the results apply to many spaces at once is advantageous then sticking only to Rn ’s is overly restrictive. We’d like the results to also apply to combinations of row vectors, as in the ﬁnal section of the ﬁrst chapter. We’ve even seen some spaces that are not just a collection of all of the same-sized column vectors or row vectors. For instance, we’ve seen a solution set of a homogeneous system that is a plane, inside of R3 . This solution set is a closed system in the sense that a linear combination of these solutions is also a solution. But it is not just a collection of all of the three-tall column vectors; only some of them are in this solution set. We want the results about linear combinations to apply anywhere that linear combinations are sensible. We shall call any such set a vector space. Our results, instead of being phrased as “Whenever we have a collection in which we can sensibly take linear combinations . . . ”, will be stated as “In any vector space . . . ”. Such a statement describes at once what happens in many spaces. The step up in abstraction from studying a single space at a time to studying a class of spaces can be hard to make. To understand its advantages, consider this analogy. Imagine that the government made laws one person at a time: “Leslie Jones can’t jay walk.” That would be a bad idea; statements have the virtue of economy when they apply to many cases at once. Or, suppose that they ruled, “Kim Ke must stop when passing the scene of an accident.” Contrast that with, “Any doctor must stop when passing the scene of an accident.” More general statements, in some ways, are clearer. 79

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Chapter Two. Vector Spaces

I

Deﬁnition of Vector Space

We shall study structures with two operations, an addition and a scalar multiplication, that are subject to some simple conditions. We will reﬂect more on the conditions later, but on ﬁrst reading notice how reasonable they are. For instance, surely any operation that can be called an addition (e.g., column vector addition, row vector addition, or real number addition) will satisfy all the conditions in (1) below.

I.1 Deﬁnition and Examples

1.1 Deﬁnition A vector space (over R) consists of a set V along with two operations ‘+’ and ‘·’ subject to these conditions. Where v, w ∈ V , (1) their vector sum v+ w is an element of V . If u, v, w ∈ V then (2) v + w = w + v and (3) (v + w) + u = v + (w + u). (4) There is a zero vector 0 ∈ V such that v + 0 = v for all v ∈ V . (5) Each v ∈ V has an additive inverse w ∈ V such that w + v = 0. If r, s are scalars, members of R, and v, w ∈ V then (6) each scalar multiple r · v is in V . If r, s ∈ R and v, w ∈ V then (7) (r + s) · v = r · v + s · v, and (8) r · (v + w) = r · v + r · w, and (9) (rs) · v = r · (s · v), and (10) 1 · v = v. 1.2 Remark Because it involves two kinds of addition and two kinds of multiplication, that deﬁnition may seem confused. For instance, in condition (7) ‘(r + s) · v = r · v + s · v ’, the ﬁrst ‘+’ is the real number addition operator while the ‘+’ to the right of the equals sign represents vector addition in the structure V . These expressions aren’t ambiguous because, e.g., r and s are real numbers so ‘r + s’ can only mean real number addition. The best way to go through the examples below is to check all ten conditions in the deﬁnition. That check is written out at length in the ﬁrst example. Use it as a model for the others. Especially important are the ﬁrst condition ‘v + w is in V ’ and the sixth condition ‘r · v is in V ’. These are the closure conditions. They specify that the addition and scalar multiplication operations are always sensible — they are deﬁned for every pair of vectors, and every scalar and vector, and the result of the operation is a member of the set (see Example 1.4). 1.3 Example The set R2 is a vector space if the operations ‘+’ and ‘·’ have their usual meaning. x1 x2 + y1 y2 = x1 + y1 x2 + y2 r· x1 x2 = rx1 rx2

We shall check all of the conditions.

Section I. Deﬁnition of Vector Space

81

There are ﬁve conditions in item (1). For (1), closure of addition, note that for any v1 , v2 , w1 , w2 ∈ R the result of the sum v1 v2 + w1 w2 = v1 + w1 v2 + w2

is a column array with two real entries, and so is in R2 . For (2), that addition of vectors commutes, take all entries to be real numbers and compute v1 v2 + w1 w2 = v1 + w1 v2 + w2 = w1 + v1 w2 + v2 = w1 w2 + v1 v2

(the second equality follows from the fact that the components of the vectors are real numbers, and the addition of real numbers is commutative). Condition (3), associativity of vector addition, is similar. ( v1 v2 + w1 u1 )+ w2 u2 = = = (v1 + w1 ) + u1 (v2 + w2 ) + u2 v1 + (w1 + u1 ) v2 + (w2 + u2 ) v1 v2 +( w1 w2 + u1 ) u2

For the fourth condition we must produce a zero element — the vector of zeroes is it. v1 0 v + = 1 v2 0 v2 For (5), to produce an additive inverse, note that for any v1 , v2 ∈ R we have −v1 −v2 + v1 v2 = 0 0

so the ﬁrst vector is the desired additive inverse of the second. The checks for the ﬁve conditions having to do with scalar multiplication are just as routine. For (6), closure under scalar multiplication, where r, v1 , v2 ∈ R, r· v1 v2 = rv1 rv2

is a column array with two real entries, and so is in R2 . Next, this checks (7). (r + s) · v1 v2 = (r + s)v1 (r + s)v2 = rv1 + sv1 rv2 + sv2 =r· v1 v2 +s· v1 v2

For (8), that scalar multiplication distributes from the left over vector addition, we have this. r·( v1 v2 + w1 )= w2 r(v1 + w1 ) r(v2 + w2 ) = rv1 + rw1 rv2 + rw2 =r· v1 v2 +r· w1 w2

82 The ninth (rs) · v1 v2 = (rs)v1 (rs)v2 = r(sv1 ) r(sv2 )

Chapter Two. Vector Spaces

= r · (s ·

v1 ) v2

and tenth conditions are also straightforward. 1· v1 v2 = 1v1 1v2 = v1 v2

In a similar way, each Rn is a vector space with the usual operations of vector addition and scalar multiplication. (In R1 , we usually do not write the members as column vectors, i.e., we usually do not write ‘(π)’. Instead we just write ‘π’.) 1.4 Example This subset of R3 that is a plane through the origin   x P = {y  x + y + z = 0} z is a vector space if ‘+’ and ‘·’ are interpreted in this way.           x1 x2 x1 + x2 rx x  y1  +  y2  =  y1 + y2  r · y  = ry  z1 rz z2 z z1 + z2 The addition and scalar multiplication operations here are just the ones of R3 , reused on its subset P . We say that P inherits these operations from R3 . This example of an addition in P       −1 0 1  1 + 0 = 1  −2 1 −1 illustrates that P is closed under addition. We’ve added two vectors from P — that is, with the property that the sum of their three entries is zero — and the result is a vector also in P . Of course, this example of closure is not a proof of closure. To prove that P is closed under addition, take two elements of P     x1 x2  y1   y2  z1 z2 (membership in P means that x1 + y1 + z1 = 0 and x2 + y2 + z2 = 0), and observe that their sum   x1 + x2  y1 + y2  z1 + z2

Section I. Deﬁnition of Vector Space

83

is also in P since its entries add (x1 + x2 ) + (y1 + y2 ) + (z1 + z2 ) = (x1 + y1 + z1 ) + (x2 + y2 + z2 ) to 0. To show that P is closed under scalar multiplication, start with a vector from P   x y  z (so that x + y + z = 0) and then for r ∈ R observe that the scalar multiple     x rx r · y  = ry  z rz satisﬁes that rx + ry + rz = r(x + y + z) = 0. Thus the two closure conditions are satisﬁed. Veriﬁcation of the other conditions in the deﬁnition of a vector space are just as straightforward. 1.5 Example Example 1.3 shows that the set of all two-tall vectors with real entries is a vector space. Example 1.4 gives a subset of an Rn that is also a vector space. In contrast with those two, consider the set of two-tall columns with entries that are integers (under the obvious operations). This is a subset of a vector space, but it is not itself a vector space. The reason is that this set is not closed under scalar multiplication, that is, it does not satisfy condition (6). Here is a column with integer entries, and a scalar, such that the outcome of the operation 4 2 0.5 · = 3 1.5 is not a member of the set, since its entries are not all integers. 1.6 Example The singleton set   0 0 { } 0 0 is a vector space under the operations       0 0 0 0 0 0  + =  0 0 0 0 0 0 that it inherits from R4 . A vector space must have at least one element, its zero vector. Thus a one-element vector space is the smallest one possible. 1.7 Deﬁnition A one-element vector space is a trivial space.

    0 0 0 0 r· =  0 0 0 0

84

Chapter Two. Vector Spaces

Warning! The examples so far involve sets of column vectors with the usual operations. But vector spaces need not be collections of column vectors, or even of row vectors. Below are some other types of vector spaces. The term ‘vector space’ does not mean ‘collection of columns of reals’. It means something more like ‘collection in which any linear combination is sensible’. 1.8 Example Consider P3 = {a0 + a1 x + a2 x2 + a3 x3 a0 , . . . , a3 ∈ R}, the set of polynomials of degree three or less (in this book, we’ll take constant polynomials, including the zero polynomial, to be of degree zero). It is a vector space under the operations (a0 + a1 x + a2 x2 + a3 x3 ) + (b0 + b1 x + b2 x2 + b3 x3 ) = (a0 + b0 ) + (a1 + b1 )x + (a2 + b2 )x2 + (a3 + b3 )x3 and r · (a0 + a1 x + a2 x2 + a3 x3 ) = (ra0 ) + (ra1 )x + (ra2 )x2 + (ra3 )x3 (the veriﬁcation is easy). This vector space is worthy of attention because these are the polynomial operations familiar from high school algebra. For instance, 3 · (1 − 2x + 3x2 − 4x3 ) − 2 · (2 − 3x + x2 − (1/2)x3 ) = −1 + 7x2 − 11x3 . Although this space is not a subset of any Rn , there is a sense in which we can think of P3 as “the same” as R4 . If we identify these two spaces’s elements in this way   a0 a1  a0 + a1 x + a2 x2 + a3 x3 corresponds to   a2  a3 then the operations also correspond. Here is an example of corresponding additions.       3 2 1 1 − 2x + 0x2 + 1x3 −2  3   1  2 3 + 2 + 3x + 7x − 4x corresponds to   +   =   0 7 7 3 + 1x + 7x2 − 3x3 −3 −4 1 Things we are thinking of as “the same” add to “the same” sum. Chapter Three makes precise this idea of vector space correspondence. For now we shall just leave it as an intuition. 1.9 Example The set M2×2 of 2 × 2 matrices with real number entries is a vector space under the natural entry-by-entry operations. a c b w + d y x z = a+w c+y b+x d+z r· a c b d = ra rc rb rd

As in the prior example, we can think of this space as “the same” as R4 .

Section I. Deﬁnition of Vector Space

85

1.10 Example The set {f f : N → R} of all real-valued functions of one natural number variable is a vector space under the operations (f1 + f2 ) (n) = f1 (n) + f2 (n) (r · f ) (n) = r f (n)

so that if, for example, f1 (n) = n2 + 2 sin(n) and f2 (n) = − sin(n) + 0.5 then (f1 + 2f2 ) (n) = n2 + 1. We can view this space as a generalization of Example 1.3 — instead of 2-tall vectors, these functions are like inﬁnitely-tall vectors. n 0 1 2 3 . . . f (n) = n2 + 1 1 2 5 10 . . .  1 2   5   10   . . .  corresponds to

Addition and scalar multiplication are component-wise, as in Example 1.3. (We can formalize “inﬁnitely-tall” by saying that it means an inﬁnite sequence, or that it means a function from N to R.) 1.11 Example The set of polynomials with real coeﬃcients {a0 + a1 x + · · · + an xn n ∈ N and a0 , . . . , an ∈ R} makes a vector space when given the natural ‘+’ (a0 + a1 x + · · · + an xn ) + (b0 + b1 x + · · · + bn xn ) = (a0 + b0 ) + (a1 + b1 )x + · · · + (an + bn )xn and ‘·’. r · (a0 + a1 x + . . . an xn ) = (ra0 ) + (ra1 )x + . . . (ran )xn This space diﬀers from the space P3 of Example 1.8. This space contains not just degree three polynomials, but degree thirty polynomials and degree three hundred polynomials, too. Each individual polynomial of course is of a ﬁnite degree, but the set has no single bound on the degree of all of its members. This example, like the prior one, can be thought of in terms of inﬁnite-tuples. For instance, we can think of 1 + 3x + 5x2 as corresponding to (1, 3, 5, 0, 0, . . .). However, don’t confuse this space with the one from Example 1.10. Each member of this set has a bounded degree, so under our correspondence there are no elements from this space matching (1, 2, 5, 10, . . . ). The vectors in this space correspond to inﬁnite-tuples that end in zeroes. 1.12 Example The set {f f : R → R} of all real-valued functions of one real variable is a vector space under these. (f1 + f2 ) (x) = f1 (x) + f2 (x) (r · f ) (x) = r f (x)

The diﬀerence between this and Example 1.10 is the domain of the functions.

86

Chapter Two. Vector Spaces

1.13 Example The set F = {a cos θ+b sin θ a, b ∈ R} of real-valued functions of the real variable θ is a vector space under the operations (a1 cos θ + b1 sin θ) + (a2 cos θ + b2 sin θ) = (a1 + a2 ) cos θ + (b1 + b2 ) sin θ and r · (a cos θ + b sin θ) = (ra) cos θ + (rb) sin θ inherited from the space in the prior example. (We can think of F as “the same” as R2 in that a cos θ + b sin θ corresponds to the vector with components a and b.) 1.14 Example The set {f : R → R d2 f + f = 0} dx2

is a vector space under the, by now natural, interpretation. (f + g) (x) = f (x) + g(x) (r · f ) (x) = r f (x)

In particular, notice that closure is a consequence: d2 (f + g) d2 f d2 g + (f + g) = ( 2 + f ) + ( 2 + g) dx2 dx dx d2 (rf ) d2 f + (rf ) = r( 2 + f ) dx2 dx of basic Calculus. This turns out to equal the space from the prior example — functions satisfying this diﬀerential equation have the form a cos θ + b sin θ — but this description suggests an extension to solutions sets of other diﬀerential equations. 1.15 Example The set of solutions of a homogeneous linear system in n variables is a vector space under the operations inherited from Rn . For closure under addition, if     v1 w1 .  .  v= .  w= .  . . vn wn both satisfy the condition that their entries add to 0 then v + w also satisﬁes that condition: c1 (v1 + w1 ) + · · · + cn (vn + wn ) = (c1 v1 + · · · + cn vn ) + (c1 w1 + · · · + cn wn ) = 0. The checks of the other conditions are just as routine. As we’ve done in those equations, we often omit the multiplication symbol ‘·’. We can distinguish the multiplication in ‘c1 v1 ’ from that in ‘rv ’ since if both multiplicands are real numbers then real-real multiplication must be meant, while if one is a vector then scalar-vector multiplication must be meant. The prior example has brought us full circle since it is one of our motivating examples. and

Section I. Deﬁnition of Vector Space

87

1.16 Remark Now, with some feel for the kinds of structures that satisfy the deﬁnition of a vector space, we can reﬂect on that deﬁnition. For example, why specify in the deﬁnition the condition that 1 · v = v but not a condition that 0 · v = 0? One answer is that this is just a deﬁnition — it gives the rules of the game from here on, and if you don’t like it, put the book down and walk away. Another answer is perhaps more satisfying. People in this area have worked hard to develop the right balance of power and generality. This deﬁnition has been shaped so that it contains the conditions needed to prove all of the interesting and important properties of spaces of linear combinations. As we proceed, we shall derive all of the properties natural to collections of linear combinations from the conditions given in the deﬁnition. The next result is an example. We do not need to include these properties in the deﬁnition of vector space because they follow from the properties already listed there. 1.17 Lemma In any vector space V , for any v ∈ V and r ∈ R, we have (1) 0 · v = 0, and (2) (−1 · v) + v = 0, and (3) r · 0 = 0.
Proof. For (1), note that v = (1 + 0) · v = v + (0 · v). Add to both sides the additive inverse of v, the vector w such that w + v = 0.

w+v =w+v+0·v 0=0+0·v 0=0·v The second item is easy: (−1 · v) + v = (−1 + 1) · v = 0 · v = 0 shows that we can write ‘−v ’ for the additive inverse of v without worrying about possible confusion with (−1) · v. For (3), this r · 0 = r · (0 · 0) = (r · 0) · 0 = 0 will do. QED We ﬁnish with a recap. Our study in Chapter One of Gaussian reduction led us to consider collections of linear combinations. So in this chapter we have deﬁned a vector space to be a structure in which we can form such combinations, expressions of the form c1 · v1 + · · · + cn · vn (subject to simple conditions on the addition and scalar multiplication operations). In a phrase: vector spaces are the right context in which to study linearity. Finally, a comment. From the fact that it forms a whole chapter, and especially because that chapter is the ﬁrst one, a reader could come to think that the study of linear systems is our purpose. The truth is, we will not so much use vector spaces in the study of linear systems as we will instead have linear systems start us on the study of vector spaces. The wide variety of examples from this subsection shows that the study of vector spaces is interesting and important in its own right, aside from how it helps us understand linear systems. Linear systems won’t go away. But from now on our primary objects of study will be vector spaces.

88 Exercises

Chapter Two. Vector Spaces

1.18 Name the zero vector for each of these vector spaces. (a) The space of degree three polynomials under the natural operations (b) The space of 2×4 matrices (c) The space {f : [0..1] → R f is continuous} (d) The space of real-valued functions of one natural number variable 1.19 Find the additive inverse, in the vector space, of the vector. (a) In P3 , the vector −3 − 2x + x2 . (b) In the space 2×2, 1 −1 . 0 3 (c) In {aex + be−x a, b ∈ R}, the space of functions of the real variable x under the natural operations, the vector 3ex − 2e−x . 1.20 Show that each of these is a vector space. (a) The set of linear polynomials P1 = {a0 + a1 x a0 , a1 ∈ R} under the usual polynomial addition and scalar multiplication operations. (b) The set of 2×2 matrices with real entries under the usual matrix operations. (c) The set of three-component row vectors with their usual operations. (d) The set   x y  L = {  ∈ R4 x + y − z + w = 0} z w under the operations inherited from R4 . 1.21 Show that each of these is not a vector space. (Hint. Start by listing two members of each set.) (a) Under the operations inherited from R3 , this set x { y z x { y z ∈ R3 x + y + z = 1}

(b) Under the operations inherited from R3 , this set ∈ R3 x2 + y 2 + z 2 = 1}

(c) Under the usual matrix operations, { a b 1 c a, b, c ∈ R}

(d) Under the usual polynomial operations, {a0 + a1 x + a2 x2 a0 , a1 , a2 ∈ R+ } where R+ is the set of reals greater than zero (e) Under the inherited operations, { x y ∈ R2 x + 3y = 4 and 2x − y = 3 and 6x + 4y = 10}

1.22 Deﬁne addition and scalar multiplication operations to make the complex numbers a vector space over R.

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1.23 Is the set of rational numbers a vector space over R under the usual addition and scalar multiplication operations? 1.24 Show that the set of linear combinations of the variables x, y, z is a vector space under the natural addition and scalar multiplication operations. 1.25 Prove that this is not a vector space: the set of two-tall column vectors with real entries subject to these operations. x rx x1 − x2 x2 x1 r· = = + y ry y1 − y2 y2 y1 1.26 Prove or disprove that R3 is a vector space under these operations. rx x 0 x2 x1 and r y = ry (a) y1 + y2 = 0 rz z 0 z2 z1 x1 x2 0 x 0 (b) y1 + y2 = 0 and r y = 0 z1 z2 0 z 0 1.27 For each, decide if it is a vector space; the intended operations are the natural ones. (a) The diagonal 2×2 matrices a 0 { a, b ∈ R} 0 b (b) This set of 2×2 matrices x x+y { x, y ∈ R} x+y y (c) This set   x y  {  ∈ R4 x + y + w = 1} z w (d) The set of functions {f : R → R df /dx + 2f = 0} (e) The set of functions {f : R → R df /dx + 2f = 1} 1.28 Prove or disprove that this is a vector space: the real-valued functions f of one real variable such that f (7) = 0. 1.29 Show that the set R+ of positive reals is a vector space when ‘x + y’ is interpreted to mean the product of x and y (so that 2 + 3 is 6), and ‘r · x’ is interpreted as the r-th power of x. 1.30 Is {(x, y) x, y ∈ R} a vector space under these operations? (a) (x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 ) and r · (x, y) = (rx, y) (b) (x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 ) and r · (x, y) = (rx, 0) 1.31 Prove or disprove that this is a vector space: the set of polynomials of degree greater than or equal to two, along with the zero polynomial. 1.32 At this point “the same” is only an intuition, but nonetheless for each vector space identify the k for which the space is “the same” as Rk . (a) The 2×3 matrices under the usual operations (b) The n×m matrices (under their usual operations) (c) This set of 2×2 matrices a 0 { a, b, c ∈ R} b c

90
(d) This set of 2×2 matrices { a b 0 c

Chapter Two. Vector Spaces

a + b + c = 0}

1.33 Using + to represent vector addition and · for scalar multiplication, restate the deﬁnition of vector space. 1.34 Prove these. (a) Any vector is the additive inverse of the additive inverse of itself. (b) Vector addition left-cancels: if v, s, t ∈ V then v + s = v + t implies that s = t. 1.35 The deﬁnition of vector spaces does not explicitly say that 0+v = v (it instead says that v + 0 = v). Show that it must nonetheless hold in any vector space. 1.36 Prove or disprove that this is a vector space: the set of all matrices, under the usual operations. 1.37 In a vector space every element has an additive inverse. Can some elements have two or more? 1.38 (a) Prove that every point, line, or plane thru the origin in R3 is a vector space under the inherited operations. (b) What if it doesn’t contain the origin? 1.39 Using the idea of a vector space we can easily reprove that the solution set of a homogeneous linear system has either one element or inﬁnitely many elements. Assume that v ∈ V is not 0. (a) Prove that r · v = 0 if and only if r = 0. (b) Prove that r1 · v = r2 · v if and only if r1 = r2 . (c) Prove that any nontrivial vector space is inﬁnite. (d) Use the fact that a nonempty solution set of a homogeneous linear system is a vector space to draw the conclusion. 1.40 Is this a vector space under the natural operations: the real-valued functions of one real variable that are diﬀerentiable? 1.41 A vector space over the complex numbers C has the same deﬁnition as a vector space over the reals except that scalars are drawn from C instead of from R. Show that each of these is a vector space over the complex numbers. (Recall how complex numbers add and multiply: (a0 + a1 i) + (b0 + b1 i) = (a0 + b0 ) + (a1 + b1 )i and (a0 + a1 i)(b0 + b1 i) = (a0 b0 − a1 b1 ) + (a0 b1 + a1 b0 )i.) (a) The set of degree two polynomials with complex coeﬃcients (b) This set 0 a { a, b ∈ C and a + b = 0 + 0i} b 0 1.42 Name a property shared by all of the Rn ’s but not listed as a requirement for a vector space. 1.43 (a) Prove that a sum of four vectors v1 , . . . , v4 ∈ V can be associated in any way without changing the result. ((v1 + v2 ) + v3 ) + v4 = (v1 + (v2 + v3 )) + v4 = (v1 + v2 ) + (v3 + v4 ) = v1 + ((v2 + v3 ) + v4 ) = v1 + (v2 + (v3 + v4 )) This allows us to simply write ‘v1 + v2 + v3 + v4 ’ without ambiguity.

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(b) Prove that any two ways of associating a sum of any number of vectors give the same sum. (Hint. Use induction on the number of vectors.) 1.44 For any vector space, a subset that is itself a vector space under the inherited operations (e.g., a plane through the origin inside of R3 ) is a subspace. (a) Show that {a0 + a1 x + a2 x2 a0 + a1 + a2 = 0} is a subspace of the vector space of degree two polynomials. (b) Show that this is a subspace of the 2×2 matrices. { a c b 0 a + b = 0}

(c) Show that a nonempty subset S of a real vector space is a subspace if and only if it is closed under linear combinations of pairs of vectors: whenever c1 , c2 ∈ R and s1 , s2 ∈ S then the combination c1 v1 + c2 v2 is in S.

I.2 Subspaces and Spanning Sets
One of the examples that led us to introduce the idea of a vector space was the solution set of a homogeneous system. For instance, we’ve seen in Example 1.4 such a space that is a planar subset of R3 . There, the vector space R3 contains inside it another vector space, the plane. 2.1 Deﬁnition For any vector space, a subspace is a subset that is itself a vector space, under the inherited operations. 2.2 Example The plane from the prior subsection,   x P = {y  x + y + z = 0} z is a subspace of R3 . As speciﬁed in the deﬁnition, the operations are the ones that are inherited from the larger space, that is, vectors add in P as they add in R3       x1 x2 x1 + x2  y1  +  y2  =  y1 + y2  z1 z2 z1 + z2 and scalar multiplication is also the same as it is in R3 . To show that P is a subspace, we need only note that it is a subset and then verify that it is a space. Checking that P satisﬁes the conditions in the deﬁnition of a vector space is routine. For instance, for closure under addition, just note that if the summands satisfy that x1 + y1 + z1 = 0 and x2 + y2 + z2 = 0 then the sum satisﬁes that (x1 + x2 ) + (y1 + y2 ) + (z1 + z2 ) = (x1 + y1 + z1 ) + (x2 + y2 + z2 ) = 0.

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2.3 Example The x-axis in R2 is a subspace where the addition and scalar multiplication operations are the inherited ones. x1 0 + x2 0 = x1 + x2 0 r· x 0 = rx 0

As above, to verify that this is a subspace, we simply note that it is a subset and then check that it satisﬁes the conditions in deﬁnition of a vector space. For instance, the two closure conditions are satisﬁed: (1) adding two vectors with a second component of zero results in a vector with a second component of zero, and (2) multiplying a scalar times a vector with a second component of zero results in a vector with a second component of zero. 2.4 Example Another subspace of R2 is { its trivial subspace. Any vector space has a trivial subspace {0 }. At the opposite extreme, any vector space has itself for a subspace. These two are the improper subspaces. Other subspaces are proper . 2.5 Example The condition in the deﬁnition requiring that the addition and scalar multiplication operations must be the ones inherited from the larger space is important. Consider the subset {1} of the vector space R1 . Under the operations 1+1 = 1 and r ·1 = 1 that set is a vector space, speciﬁcally, a trivial space. But it is not a subspace of R1 because those aren’t the inherited operations, since of course R1 has 1 + 1 = 2. 2.6 Example All kinds of vector spaces, not just Rn ’s, have subspaces. The vector space of cubic polynomials {a + bx + cx2 + dx3 a, b, c, d ∈ R} has a subspace comprised of all linear polynomials {m + nx m, n ∈ R}. 2.7 Example Another example of a subspace not taken from an Rn is one from the examples following the deﬁnition of a vector space. The space of all real-valued functions of one real variable f : R → R has a subspace of functions satisfying the restriction (d2 f /dx2 ) + f = 0. 2.8 Example Being vector spaces themselves, subspaces must satisfy the closure conditions. The set R+ is not a subspace of the vector space R1 because with the inherited operations it is not closed under scalar multiplication: if v = 1 then −1 · v ∈ R+ . The next result says that Example 2.8 is prototypical. The only way that a subset can fail to be a subspace (if it is nonempty and the inherited operations are used) is if it isn’t closed. 0 } 0

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2.9 Lemma For a nonempty subset S of a vector space, under the inherited operations, the following are equivalent statements.∗ (1) S is a subspace of that vector space (2) S is closed under linear combinations of pairs of vectors: for any vectors s1 , s2 ∈ S and scalars r1 , r2 the vector r1 s1 + r2 s2 is in S (3) S is closed under linear combinations of any number of vectors: for any vectors s1 , . . . , sn ∈ S and scalars r1 , . . . , rn the vector r1 s1 + · · · + rn sn is in S. Brieﬂy, the way that a subset gets to be a subspace is by being closed under linear combinations.
Proof. ‘The following are equivalent’ means that each pair of statements are

equivalent. (1) ⇐⇒ (2) (2) ⇐⇒ (3) (3) ⇐⇒ (1) We will show this equivalence by establishing that (1) =⇒ (3) =⇒ (2) =⇒ (1). This strategy is suggested by noticing that (1) =⇒ (3) and (3) =⇒ (2) are easy and so we need only argue the single implication (2) =⇒ (1). For that argument, assume that S is a nonempty subset of a vector space V and that S is closed under combinations of pairs of vectors. We will show that S is a vector space by checking the conditions. The ﬁrst item in the vector space deﬁnition has ﬁve conditions. First, for closure under addition, if s1 , s2 ∈ S then s1 + s2 ∈ S, as s1 + s2 = 1 · s1 + 1 · s2 . Second, for any s1 , s2 ∈ S, because addition is inherited from V , the sum s1 + s2 in S equals the sum s1 + s2 in V , and that equals the sum s2 + s1 in V (because V is a vector space, its addition is commutative), and that in turn equals the sum s2 + s1 in S. The argument for the third condition is similar to that for the second. For the fourth, consider the zero vector of V and note that closure of S under linear combinations of pairs of vectors gives that (where s is any member of the nonempty set S) 0 · s + 0 · s = 0 is in S; showing that 0 acts under the inherited operations as the additive identity of S is easy. The ﬁfth condition is satisﬁed because for any s ∈ S, closure under linear combinations shows that the vector 0 · 0 + (−1) · s is in S; showing that it is the additive inverse of s under the inherited operations is routine. The checks for item (2) are similar and are saved for Exercise 32. QED We usually show that a subset is a subspace with (2) =⇒ (1). 2.10 Remark At the start of this chapter we introduced vector spaces as collections in which linear combinations are “sensible”. The above result speaks to this. The vector space deﬁnition has ten conditions but eight of them — the conditions not about closure — simply ensure that referring to the operations as an ‘addition’ and a ‘scalar multiplication’ is sensible. The proof above checks that these eight are inherited from the surrounding vector space provided that the

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nonempty set S satisﬁes Theorem 2.9’s statement (2) (e.g., commutativity of addition in S follows right from commutativity of addition in V ). So, in this context, this meaning of “sensible” is automatically satisﬁed. In assuring us that this ﬁrst meaning of the word is met, the result draws our attention to the second meaning of “sensible”. It has to do with the two remaining conditions, the closure conditions. Above, the two separate closure conditions inherent in statement (1) are combined in statement (2) into the single condition of closure under all linear combinations of two vectors, which is then extended in statement (3) to closure under combinations of any number of vectors. The latter two statements say that we can always make sense of an expression like r1 s1 + r2 s2 , without restrictions on the r’s — such expressions are “sensible” in that the vector described is deﬁned and is in the set S. This second meaning suggests that a good way to think of a vector space is as a collection of unrestricted linear combinations. The next two examples take some spaces and describe them in this way. That is, in these examples we parametrize, just as we did in Chapter One to describe the solution set of a homogeneous linear system. 2.11 Example This subset of R3   x S = {y  x − 2y + z = 0} z is a subspace under the usual addition and scalar multiplication operations of column vectors (the check that it is nonempty and closed under linear combinations of two vectors is just like the one in Example 2.2). To parametrize, we can take x − 2y + z = 0 to be a one-equation linear system and expressing the leading variable in terms of the free variables x = 2y − z.       −1 2 2y − z S = { y  y, z ∈ R} = {y 1 + z  0  y, z ∈ R} 1 0 z Now the subspace is described as the collection of unrestricted linear combinations of those two vectors. Of course, in either description, this is a plane through the origin. 2.12 Example This is a subspace of the 2×2 matrices L={ a 0 b c a + b + c = 0}

(checking that it is nonempty and closed under linear combinations is easy). To parametrize, express the condition as a = −b − c. L={ −b − c 0 b c b, c ∈ R} = {b −1 1 0 −1 +c 0 0 0 1 b, c ∈ R}

As above, we’ve described the subspace as a collection of unrestricted linear combinations (by coincidence, also of two elements).

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Parametrization is an easy technique, but it is important. We shall use it often. 2.13 Deﬁnition The span (or linear closure) of a nonempty subset S of a vector space is the set of all linear combinations of vectors from S. [S] = {c1 s1 + · · · + cn sn c1 , . . . , cn ∈ R and s1 , . . . , sn ∈ S} The span of the empty subset of a vector space is the trivial subspace. No notation for the span is completely standard. The square brackets used here are common, but so are ‘span(S)’ and ‘sp(S)’. 2.14 Remark In Chapter One, after we showed that the solution set of a homogeneous linear system can be written as {c1 β1 + · · · + ck βk c1 , . . . , ck ∈ R}, we described that as the set ‘generated’ by the β’s. We now have the technical term; we call that the ‘span’ of the set {β1 , . . . , βk }. Recall also the discussion of the “tricky point” in that proof. The span of the empty set is deﬁned to be the set {0} because we follow the convention that a linear combination of no vectors sums to 0. Besides, deﬁning the empty set’s span to be the trivial subspace is a convienence in that it keeps results like the next one from having annoying exceptional cases. 2.15 Lemma In a vector space, the span of any subset is a subspace.
Proof. Call the subset S. If S is empty then by deﬁnition its span is the trivial subspace. If S is not empty then by Lemma 2.9 we need only check that the span [S] is closed under linear combinations. For a pair of vectors from that span, v = c1 s1 +· · ·+cn sn and w = cn+1 sn+1 +· · ·+cm sm , a linear combination

p · (c1 s1 + · · · + cn sn ) + r · (cn+1 sn+1 + · · · + cm sm ) = pc1 s1 + · · · + pcn sn + rcn+1 sn+1 + · · · + rcm sm (p, r scalars) is a linear combination of elements of S and so is in [S] (possibly some of the si ’s forming v equal some of the sj ’s from w, but it does not matter). QED The converse of the lemma holds: any subspace is the span of some set, because a subspace is obviously the span of the set of its members. Thus a subset of a vector space is a subspace if and only if it is a span. This ﬁts the intuition that a good way to think of a vector space is as a collection in which linear combinations are sensible. Taken together, Lemma 2.9 and Lemma 2.15 show that the span of a subset S of a vector space is the smallest subspace containing all the members of S. 2.16 Example In any vector space V , for any vector v, the set {r · v r ∈ R} is a subspace of V . For instance, for any vector v ∈ R3 , the line through the origin containing that vector, {kv k ∈ R} is a subspace of R3 . This is true even when v is the zero vector, in which case the subspace is the degenerate line, the trivial subspace.

96 2.17 Example The span of this set is all of R2 . { 1 1 , } 1 −1

Chapter Two. Vector Spaces

To check this we must show that any member of R2 is a linear combination of these two vectors. So we ask: for which vectors (with real components x and y) are there scalars c1 and c2 such that this holds? c1 Gauss’ method c1 + c2 = x c1 − c2 = y
−ρ1 +ρ2

1 1 + c2 1 −1

=

x y

−→

c1 +

c2 = x −2c2 = −x + y

with back substitution gives c2 = (x − y)/2 and c1 = (x + y)/2. These two equations show that for any x and y that we start with, there are appropriate coeﬃcients c1 and c2 making the above vector equation true. For instance, for x = 1 and y = 2 the coeﬃcients c2 = −1/2 and c1 = 3/2 will do. That is, any vector in R2 can be written as a linear combination of the two given vectors. Since spans are subspaces, and we know that a good way to understand a subspace is to parametrize its description, we can try to understand a set’s span in that way. 2.18 Example Consider, in P2 , the span of the set {3x − x2 , 2x}. By the deﬁnition of span, it is the set of unrestricted linear combinations of the two {c1 (3x − x2 ) + c2 (2x) c1 , c2 ∈ R}. Clearly polynomials in this span must have a constant term of zero. Is that necessary condition also suﬃcient? We are asking: for which members a2 x2 + a1 x + a0 of P2 are there c1 and c2 such that a2 x2 + a1 x + a0 = c1 (3x − x2 ) + c2 (2x)? Since polynomials are equal if and only if their coeﬃcients are equal, we are looking for conditions on a2 , a1 , and a0 satisfying these. −c1 = a2 3c1 + 2c2 = a1 0 = a0 Gauss’ method gives that c1 = −a2 , c2 = (3/2)a2 + (1/2)a1 , and 0 = a0 . Thus the only condition on polynomials in the span is the condition that we knew of — as long as a0 = 0, we can give appropriate coeﬃcients c1 and c2 to describe the polynomial a0 + a1 x + a2 x2 as in the span. For instance, for the polynomial 0 − 4x + 3x2 , the coeﬃcients c1 = −3 and c2 = 5/2 will do. So the span of the given set is {a1 x + a2 x2 a1 , a2 ∈ R}. This shows, incidentally, that the set {x, x2 } also spans this subspace. A space can have more than one spanning set. Two other sets spanning this subspace are {x, x2 , −x + 2x2 } and {x, x + x2 , x + 2x2 , . . . }. (Naturally, we usually prefer to work with spanning sets that have only a few members.)

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2.19 Example These are the subspaces of R3 that we now know of, the trivial subspace, the lines through the origin, the planes through the origin, and the whole space (of course, the picture shows only a few of the inﬁnitely many subspaces). In the next section we will prove that R3 has no other type of subspaces, so in fact this picture shows them all.
{x 1 0 0 +y 0 1 0 +z 0 0 1 }

{x

1 0 0

+y

0 1 0

\$\$\$   \$\$\$ \$    \$\$\$
} {x 1 0 0 +z 0 0 1 } {x 1 1 0 +z 0 0 1 }

...

{x

1 0 0

r  £ e rr  £  e r
} 0 1 0

2 1 0 1 1 1

... {y } {y } {y }     rrr d   r d r   { 0 0
0

}

The subsets are described as spans of sets, using a minimal number of members, and are shown connected to their supersets. Note that these subspaces fall naturally into levels — planes on one level, lines on another, etc. — according to how many vectors are in a minimal-sized spanning set. So far in this chapter we have seen that to study the properties of linear combinations, the right setting is a collection that is closed under these combinations. In the ﬁrst subsection we introduced such collections, vector spaces, and we saw a great variety of examples. In this subsection we saw still more spaces, ones that happen to be subspaces of others. In all of the variety we’ve seen a commonality. Example 2.19 above brings it out: vector spaces and subspaces are best understood as a span, and especially as a span of a small number of vectors. The next section studies spanning sets that are minimal. Exercises
2.20 Which of these subsets of the vector space of 2 × 2 matrices are subspaces under the inherited operations? For each one that is a subspace, parametrize its description. For each that is not, give a condition that fails. a 0 (a) { a, b ∈ R} 0 b (b) { (c) { (d) { a 0 a 0 a 0 0 b 0 b c b a + b = 0} a + b = 5} a + b = 0, c ∈ R}

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2.21 Is this a subspace of P2 : {a0 + a1 x + a2 x2 a0 + 2a1 + a2 = 4}? If it is then parametrize its description. 2.22 Decide if the vector lies in the span of the set, inside of the space. 0 1 2 (a) 0 , { 0 , 0 }, in R3 1 0 1 (b) x − x3 , {x2 , 2x + x2 , x + x3 }, in P3 0 1 1 0 2 0 (c) ,{ , }, in M2×2 4 2 1 1 2 3 2.23 Which of these are members of the span [{cos2 x, sin2 x}] in the vector space of real-valued functions of one real variable? (a) f (x) = 1 (b) f (x) = 3 + x2 (c) f (x) = sin x (d) f (x) = cos(2x) 2.24 Which of these sets spans R3 ? That is, which of these sets has the property that any three-tall vector can be expressed as a suitable linear combination of the set’s elements? 1 0 0 2 1 0 1 3 (a) { 0 , 2 , 0 } (b) { 0 , 1 , 0 } (c) { 1 , 0 } 0 0 3 1 0 1 0 0 1 3 −1 2 2 3 5 6 (e) { 1 , 0 , 1 , 0 } (d) { 0 , 1 , 0 , 1 } 1 0 0 5 1 1 2 2 2.25 Parametrize each subspace’s description. Then express each subspace as a span. (a) The subset { a b c a − c = 0} of the three-wide row vectors (b) This subset of M2×2 { (c) This subset of M2×2 { a c b d 2a − c − d = 0 and a + 3b = 0} a c b d a + d = 0}

(d) The subset {a + bx + cx3 a − 2b + c = 0} of P3 (e) The subset of P2 of quadratic polynomials p such that p(7) = 0 2.26 Find a set to span the given subspace of the given space. (Hint. Parametrize each.) (a) the xz-plane in R3 x (b) { y 3x + 2y + z = 0} in R3 z   x y  (c) {  2x + y + w = 0 and y + 2z = 0} in R4 z w (d) {a0 + a1 x + a2 x2 + a3 x3 a0 + a1 = 0 and a2 − a3 = 0} in P3 (e) The set P4 in the space P4 (f ) M2×2 in M2×2 2.27 Is R2 a subspace of R3 ?

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2.28 Decide if each is a subspace of the vector space of real-valued functions of one real variable. (a) The even functions {f : R → R f (−x) = f (x) for all x}. For example, two members of this set are f1 (x) = x2 and f2 (x) = cos(x). (b) The odd functions {f : R → R f (−x) = −f (x) for all x}. Two members are f3 (x) = x3 and f4 (x) = sin(x). 2.29 Example 2.16 says that for any vector v that is an element of a vector space V , the set {r · v r ∈ R} is a subspace of V . (This is of course, simply the span of the singleton set {v}.) Must any such subspace be a proper subspace, or can it be improper? 2.30 An example following the deﬁnition of a vector space shows that the solution set of a homogeneous linear system is a vector space. In the terminology of this subsection, it is a subspace of Rn where the system has n variables. What about a non-homogeneous linear system; do its solutions form a subspace (under the inherited operations)? 2.31 Example 2.19 shows that R3 has inﬁnitely many subspaces. Does every nontrivial space have inﬁnitely many subspaces? 2.32 Finish the proof of Lemma 2.9. 2.33 Show that each vector space has only one trivial subspace. 2.34 Show that for any subset S of a vector space, the span of the span equals the span [[S]] = [S]. (Hint. Members of [S] are linear combinations of members of S. Members of [[S]] are linear combinations of linear combinations of members of S.) 2.35 All of the subspaces that we’ve seen use zero in their description in some way. For example, the subspace in Example 2.3 consists of all the vectors from R2 with a second component of zero. In contrast, the collection of vectors from R2 with a second component of one does not form a subspace (it is not closed under scalar multiplication). Another example is Example 2.2, where the condition on the vectors is that the three components add to zero. If the condition were that the three components add to one then it would not be a subspace (again, it would fail to be closed). This exercise shows that a reliance on zero is not strictly necessary. Consider the set x { y x + y + z = 1} z under these operations. x1 y1 z1 + x2 y2 z2 = x1 + x2 − 1 y1 + y2 z1 + z2 r x y z = rx − r + 1 ry rz

(a) Show that it is not a subspace of R3 . (Hint. See Example 2.5). (b) Show that it is a vector space. Note that by the prior item, Lemma 2.9 can not apply. (c) Show that any subspace of R3 must pass through the origin, and so any subspace of R3 must involve zero in its description. Does the converse hold? Does any subset of R3 that contains the origin become a subspace when given the inherited operations? 2.36 We can give a justiﬁcation for the convention that the sum of zero-many vectors equals the zero vector. Consider this sum of three vectors v1 + v2 +

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v3 . (a) What is the diﬀerence between this sum of three vectors and the sum of the ﬁrst two of these three? (b) What is the diﬀerence between the prior sum and the sum of just the ﬁrst one vector? (c) What should be the diﬀerence between the prior sum of one vector and the sum of no vectors? (d) So what should be the deﬁnition of the sum of no vectors? 2.37 Is a space determined by its subspaces? That is, if two vector spaces have the same subspaces, must the two be equal? 2.38 (a) Give a set that is closed under scalar multiplication but not addition. (b) Give a set closed under addition but not scalar multiplication. (c) Give a set closed under neither. 2.39 Show that the span of a set of vectors does not depend on the order in which the vectors are listed in that set. 2.40 Which trivial subspace is the span of the empty set? Is it 0 { 0 } ⊆ R3 , or {0 + 0x} ⊆ P1 , 0 or some other subspace? 2.41 Show that if a vector is in the span of a set then adding that vector to the set won’t make the span any bigger. Is that also ‘only if’ ? 2.42 Subspaces are subsets and so we naturally consider how ‘is a subspace of’ interacts with the usual set operations. (a) If A, B are subspaces of a vector space, must A ∩ B be a subspace? Always? Sometimes? Never? (b) Must A ∪ B be a subspace? (c) If A is a subspace, must its complement be a subspace? (Hint. Try some test subspaces from Example 2.19.) 2.43 Does the span of a set depend on the enclosing space? That is, if W is a subspace of V and S is a subset of W (and so also a subset of V ), might the span of S in W diﬀer from the span of S in V ? 2.44 Is the relation ‘is a subspace of’ transitive? That is, if V is a subspace of W and W is a subspace of X, must V be a subspace of X? 2.45 Because ‘span of’ is an operation on sets we naturally consider how it interacts with the usual set operations. (a) If S ⊆ T are subsets of a vector space, is [S] ⊆ [T ]? Always? Sometimes? Never? (b) If S, T are subsets of a vector space, is [S ∪ T ] = [S] ∪ [T ]? (c) If S, T are subsets of a vector space, is [S ∩ T ] = [S] ∩ [T ]? (d) Is the span of the complement equal to the complement of the span? 2.46 Reprove Lemma 2.15 without doing the empty set separately. 2.47 Find a structure that is closed under linear combinations, and yet is not a vector space. (Remark. This is a bit of a trick question.)

Section II. Linear Independence

101

II

Linear Independence

The prior section shows that a vector space can be understood as an unrestricted linear combination of some of its elements — that is, as a span. For example, the space of linear polynomials {a + bx a, b ∈ R} is spanned by the set {1, x}. The prior section also showed that a space can have many sets that span it. The space of linear polynomials is also spanned by {1, 2x} and {1, x, 2x}. At the end of that section we described some spanning sets as ‘minimal’, but we never precisely deﬁned that word. We could take ‘minimal’ to mean one of two things. We could mean that a spanning set is minimal if it contains the smallest number of members of any set with the same span. With this meaning {1, x, 2x} is not minimal because it has one member more than the other two. Or we could mean that a spanning set is minimal when it has no elements that can be removed without changing the span. Under this meaning {1, x, 2x} is not minimal because removing the 2x and getting {1, x} leaves the span unchanged. The ﬁrst sense of minimality appears to be a global requirement, in that to check if a spanning set is minimal we seemingly must look at all the spanning sets of a subspace and ﬁnd one with the least number of elements. The second sense of minimality is local in that we need to look only at the set under discussion and consider the span with and without various elements. For instance, using the second sense, we could compare the span of {1, x, 2x} with the span of {1, x} and note that the 2x is a “repeat” in that its removal doesn’t shrink the span. In this section we will use the second sense of ‘minimal spanning set’ because of this technical convenience. However, the most important result of this book is that the two senses coincide; we will prove that in the section after this one.

II.1 Deﬁnition and Examples
We ﬁrst characterize when a vector can be removed from a set without changing the span of that set. 1.1 Lemma Where S is a subset of a vector space V , [S] = [S ∪ {v}] for any v ∈ V .
Proof. The left to right implication is easy. If [S] = [S ∪ {v}] then, since

if and only if v ∈ [S]

v ∈ [S ∪ {v}], the equality of the two sets gives that v ∈ [S]. For the right to left implication assume that v ∈ [S] to show that [S] = [S ∪ {v}] by mutual inclusion. The inclusion [S] ⊆ [S ∪ {v}] is obvious. For the other inclusion [S] ⊇ [S ∪ {v}], write an element of [S ∪ {v}] as d0 v + d1 s1 + · · · + dm sm and substitute v’s expansion as a linear combination of members of the same set d0 (c0 t0 + · · · + ck tk ) + d1 s1 + · · · + dm sm . This is a linear combination of linear combinations and so distributing d0 results in a linear combination of vectors from S. Hence each member of [S ∪ {v}] is also a member of [S]. QED

102 1.2 Example In R3 , where   1 v1 = 0 0

Chapter Two. Vector Spaces

  0 v2 = 1 0

  2 v3 = 1 0

the spans [{v1 , v2 }] and [{v1 , v2 , v3 }] are equal since v3 is in the span [{v1 , v2 }]. The lemma says that if we have a spanning set then we can remove a v to get a new set S with the same span if and only if v is a linear combination of vectors from S. Thus, under the second sense described above, a spanning set is minimal if and only if it contains no vectors that are linear combinations of the others in that set. We have a term for this important property. 1.3 Deﬁnition A subset of a vector space is linearly independent if none of its elements is a linear combination of the others. Otherwise it is linearly dependent. Here is an important observation: although this way of writing one vector as a combination of the others s0 = c1 s1 + c2 s2 + · · · + cn sn visually sets s0 oﬀ from the other vectors, algebraically there is nothing special in that equation about s0 . For any si with a coeﬃcient ci that is nonzero, we can rewrite the relationship to set oﬀ si . si = (1/ci )s0 + (−c1 /ci )s1 + · · · + (−cn /ci )sn When we don’t want to single out any vector by writing it alone on one side of the equation we will instead say that s0 , s1 , . . . , sn are in a linear relationship and write the relationship with all of the vectors on the same side. The next result rephrases the linear independence deﬁnition in this style. It gives what is usually the easiest way to compute whether a ﬁnite set is dependent or independent. 1.4 Lemma A subset S of a vector space is linearly independent if and only if for any distinct s1 , . . . , sn ∈ S the only linear relationship among those vectors c1 s1 + · · · + cn sn = 0 is the trivial one: c1 = 0, . . . , cn = 0.
Proof. This is a direct consequence of the observation above.

c1 , . . . , cn ∈ R

If the set S is linearly independent then no vector si can be written as a linear combination of the other vectors from S so there is no linear relationship where some of the s ’s have nonzero coeﬃcients. If S is not linearly independent then some si is a linear combination si = c1 s1 +· · ·+ci−1 si−1 +ci+1 si+1 +· · ·+cn sn of other vectors from S, and subtracting si from both sides of that equation gives a linear relationship involving a nonzero coeﬃcient, namely the −1 in front of si . QED

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103

1.5 Example In the vector space of two-wide row vectors, the two-element set { 40 15 , −50 25 } is linearly independent. To check this, set c1 · 40 15 + c2 · −50 25 = 0 0

and solving the resulting system 40c1 − 50c2 = 0 15c1 + 25c2 = 0
−(15/40)ρ1 +ρ2

−→

40c1 −

50c2 = 0 (175/4)c2 = 0

shows that both c1 and c2 are zero. So the only linear relationship between the two given row vectors is the trivial relationship. In the same vector space, { 40 15 , 20 7.5 } is linearly dependent since we can satisfy c1 40 15 + c2 · 20 7.5 = 0 0 with c1 = 1 and c2 = −2. 1.6 Remark Recall the Statics example that began this book. We ﬁrst set the unknown-mass objects at 40 cm and 15 cm and got a balance, and then we set the objects at −50 cm and 25 cm and got a balance. With those two pieces of information we could compute values of the unknown masses. Had we instead ﬁrst set the unknown-mass objects at 40 cm and 15 cm, and then at 20 cm and 7.5 cm, we would not have been able to compute the values of the unknown masses (try it). Intuitively, the problem is that the 20 7.5 information is a “repeat” of the 40 15 information — that is, 20 7.5 is in the span of the set { 40 15 } — and so we would be trying to solve a two-unknowns problem with what is essentially one piece of information. 1.7 Example The set {1 + x, 1 − x} is linearly independent in P2 , the space of quadratic polynomials with real coeﬃcients, because 0 + 0x + 0x2 = c1 (1 + x) + c2 (1 − x) = (c1 + c2 ) + (c1 − c2 )x + 0x2 gives c1 + c2 = 0 c1 − c2 = 0
−ρ1 +ρ2

−→

c1 + c2 = 0 2c2 = 0

since polynomials are equal only if their coeﬃcients are equal. Thus, the only linear relationship between these two members of P2 is the trivial one. 1.8 Example In R3 , where   3 v1 = 4 5   2 v2 = 9 2  4 v3 = 18 4 

the set S = {v1 , v2 , v3 } is linearly dependent because this is a relationship 0 · v1 + 2 · v2 − 1 · v3 = 0 where not all of the scalars are zero (the fact that some of the scalars are zero doesn’t matter).

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1.9 Remark That example illustrates why, although Deﬁnition 1.3 is a clearer statement of what independence is, Lemma 1.4 is more useful for computations. Working straight from the deﬁnition, someone trying to compute whether S is linearly independent would start by setting v1 = c2 v2 + c3 v3 and concluding that there are no such c2 and c3 . But knowing that the ﬁrst vector is not dependent on the other two is not enough. This person would have to go on to try v2 = c1 v1 + c3 v3 to ﬁnd the dependence c1 = 0, c3 = 1/2. Lemma 1.4 gets the same conclusion with only one computation. 1.10 Example The empty subset of a vector space is linearly independent. There is no nontrivial linear relationship among its members as it has no members. 1.11 Example In any vector space, any subset containing the zero vector is linearly dependent. For example, in the space P2 of quadratic polynomials, consider the subset {1 + x, x + x2 , 0}. One way to see that this subset is linearly dependent is to use Lemma 1.4: we have 0 · v1 + 0 · v2 + 1 · 0 = 0, and this is a nontrivial relationship as not all of the coeﬃcients are zero. Another way to see that this subset is linearly dependent is to go straight to Deﬁnition 1.3: we can express the third member of the subset as a linear combination of the ﬁrst two, namely, c1 v1 + c2 v2 = 0 is satisﬁed by taking c1 = 0 and c2 = 0 (in contrast to the lemma, the deﬁnition allows all of the coeﬃcients to be zero). (There is still another way to see that this subset is dependent that is subtler. The zero vector is equal to the trivial sum, that is, it is the sum of no vectors. So in a set containing the zero vector, there is an element that can be written as a combination of a collection of other vectors from the set, speciﬁcally, the zero vector can be written as a combination of the empty collection.) The above examples, especially Example 1.5, underline the discussion that begins this section. The next result says that given a ﬁnite set, we can produce a linearly independent subset by discarding what Remark 1.6 calls “repeats”. 1.12 Theorem In a vector space, any ﬁnite subset has a linearly independent subset with the same span.
Proof. If the set S = {s1 , . . . , sn } is linearly independent then S itself satisﬁes

the statement, so assume that it is linearly dependent. By the deﬁnition of dependence, there is a vector si that is a linear combination of the others. Call that vector v1 . Discard it — deﬁne the set S1 = S −{v1 }. By Lemma 1.1, the span does not shrink [S1 ] = [S]. Now, if S1 is linearly independent then we are ﬁnished. Otherwise iterate the prior paragraph: take a vector v2 that is a linear combination of other members of S1 and discard it to derive S2 = S1 − {v2 } such that [S2 ] = [S1 ]. Repeat this until a linearly independent set Sj appears; one must appear eventually because S is ﬁnite and the empty set is linearly independent. (Formally, this argument uses induction on n, the number of elements in the starting set. Exercise 37 asks for the details.) QED

Section II. Linear Independence 1.13 Example This set spans R3 .           1 0 1 0 3 S = {0 , 2 , 2 , −1 , 3} 0 0 0 1 0 Looking for a linear relationship             1 0 1 0 3 0 c1 0 + c2 2 + c3 2 + c4 −1 + c5 3 = 0 0 0 0 1 0 0

105

gives a three equations/ﬁve unknowns linear system whose solution set can be parametrized in this way.       c1 −3 −1 c2  −3/2 −1       c3  = c3  1  + c5  0  c3 , c5 ∈ R} {      c4   0  0 1 0 c5 So S is linearly dependent. Setting c3 = 0 and c5 = 1 shows that the ﬁfth vector is a linear combination of the ﬁrst two. Thus, Lemma 1.1 says that discarding the ﬁfth vector         0 1 0 1 S1 = {0 , 2 , 2 , −1} 0 0 1 0 leaves the span unchanged [S1 ] = [S]. Now, the third vector of S1 is a linear combination of the ﬁrst two and we get       0 0 1 S2 = {0 , 2 , −1} 0 0 1 with the same span as S1 , and therefore the same span as S, but with one diﬀerence. The set S2 is linearly independent (this is easily checked), and so discarding any of its elements will shrink the span. Theorem 1.12 describes producing a linearly independent set by shrinking, that is, by taking subsets. We ﬁnish this subsection by considering how linear independence and dependence, which are properties of sets, interact with the subset relation between sets. 1.14 Lemma Any subset of a linearly independent set is also linearly independent. Any superset of a linearly dependent set is also linearly dependent.
Proof. This is clear. QED

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Restated, independence is preserved by subset and dependence is preserved by superset. Those are two of the four possible cases of interaction that we can consider. The third case, whether linear dependence is preserved by the subset operation, is covered by Example 1.13, which gives a linearly dependent set S with a subset S1 that is linearly dependent and another subset S2 that is linearly independent. That leaves one case, whether linear independence is preserved by superset. The next example shows what can happen. 1.15 Example In each of these three paragraphs the subset S is linearly independent. For the set   1 S = {0} 0 the span [S] is the x axis. Here are two supersets of S, one linearly dependent and the other linearly independent.         1 −3 1 0 dependent: {0 ,  0 } independent: {0 , 1} 0 0 0 0 Checking the dependence or independence of these sets is easy. For     0 1 S = {0 , 1} 0 0 the span [S] is the xy plane. These are two supersets.             0 0 0 3 1 1 independent: {0 , 1 , 0} dependent: {0 , 1 , −2} 0 0 0 0 1 0 If       1 0 0 S = {0 , 1 , 0} 0 0 1

then [S] = R3 . A linearly dependent superset is         1 0 0 2 dependent: {0 , 1 , 0 , −1} 0 0 1 3 but there are no linearly independent supersets of S. The reason is that for any vector that we would add to make a superset, the linear dependence equation         x 1 0 0 y  = c1 0 + c2 1 + c3 0 z 0 0 1 has a solution c1 = x, c2 = y, and c3 = z.

Section II. Linear Independence

107

So, in general, a linearly independent set may have a superset that is dependent. And, in general, a linearly independent set may have a superset that is independent. We can characterize when the superset is one and when it is the other. 1.16 Lemma Where S is a linearly independent subset of a vector space V , S ∪ {v} is linearly dependent for any v ∈ V with v ∈ S.
Proof. One implication is clear: if v ∈ [S] then v = c1 s1 + c2 s2 + · · · + cn sn

if and only if

v ∈ [S]

where each si ∈ S and ci ∈ R, and so 0 = c1 s1 + c2 s2 + · · · + cn sn + (−1)v is a nontrivial linear relationship among elements of S ∪ {v}. The other implication requires the assumption that S is linearly independent. With S ∪ {v} linearly dependent, there is a nontrivial linear relationship c0 v + c1 s1 + c2 s2 + · · · + cn sn = 0 and independence of S then implies that c0 = 0, or else that would be a nontrivial relationship among members of S. Now rewriting this equation as v = −(c1 /c0 )s1 − · · · − (cn /c0 )sn shows that v ∈ [S]. QED (Compare this result with Lemma 1.1. Both say, roughly, that v is a “repeat” if it is in the span of S. However, note the additional hypothesis here of linear independence.) 1.17 Corollary A subset S = {s1 , . . . , sn } of a vector space is linearly dependent if and only if some si is a linear combination of the vectors s1 , . . . , si−1 listed before it.
Proof. Consider S0 = {}, S1 = {s1 }, S2 = {s1 , s2 }, etc. Some index i ≥ 1 is the ﬁrst one with Si−1 ∪ {si } linearly dependent, and there si ∈ [Si−1 ]. QED

Lemma 1.16 can be restated in terms of independence instead of dependence: if S is linearly independent and v ∈ S then the set S ∪ {v} is also linearly independent if and only if v ∈ [S]. Applying Lemma 1.1, we conclude that if S is linearly independent and v ∈ S then S ∪ {v} is also linearly independent if and only if [S ∪ {v}] = [S]. Brieﬂy, when passing from S to a superset S1 , to preserve linear independence we must expand the span [S1 ] ⊃ [S]. Example 1.15 shows that some linearly independent sets are maximal — have as many elements as possible — in that they have no supersets that are linearly independent. By the prior paragraph, a linearly independent sets is maximal if and only if it spans the entire space, because then no vector exists that is not already in the span. This table summarizes the interaction between the properties of independence and dependence and the relations of subset and superset. S1 ⊂ S S1 must be independent S1 may be either S1 ⊃ S S1 may be either S1 must be dependent

S independent S dependent

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Chapter Two. Vector Spaces

In developing this table we’ve uncovered an intimate relationship between linear independence and span. Complementing the fact that a spanning set is minimal if and only if it is linearly independent, a linearly independent set is maximal if and only if it spans the space. In summary, we have introduced the deﬁnition of linear independence to formalize the idea of the minimality of a spanning set. We have developed some properties of this idea. The most important is Lemma 1.16, which tells us that a linearly independent set is maximal when it spans the space. Exercises
1.18 Decide whether each subset of R3 is linearly dependent or linearly independent. 1 2 4 (a) { −3 , 2 , −4 } 5 4 14 1 2 3 (b) { 7 , 7 , 7 } 7 7 7 0 1 (c) { 0 , 0 } −1 4 9 2 3 12 (d) { 9 , 0 , 5 , 12 } 0 1 −4 −1 1.19 Which of these subsets of P3 are linearly dependent and which are independent? (a) {3 − x + 9x2 , 5 − 6x + 3x2 , 1 + 1x − 5x2 } (b) {−x2 , 1 + 4x2 } (c) {2 + x + 7x2 , 3 − x + 2x2 , 4 − 3x2 } (d) {8 + 3x + 3x2 , x + 2x2 , 2 + 2x + 2x2 , 8 − 2x + 5x2 } 1.20 Prove that each set {f, g} is linearly independent in the vector space of all functions from R+ to R. (a) f (x) = x and g(x) = 1/x (b) f (x) = cos(x) and g(x) = sin(x) (c) f (x) = ex and g(x) = ln(x) 1.21 Which of these subsets of the space of real-valued functions of one real variable is linearly dependent and which is linearly independent? (Note that we have abbreviated some constant functions; e.g., in the ﬁrst item, the ‘2’ stands for the constant function f (x) = 2.) (a) {2, 4 sin2 (x), cos2 (x)} (b) {1, sin(x), sin(2x)} (c) {x, cos(x)} (d) {(1 + x)2 , x2 + 2x, 3} (e) {cos(2x), sin2 (x), cos2 (x)} (f ) {0, x, x2 } 2 2 2 1.22 Does the equation sin (x)/ cos (x) = tan (x) show that this set of functions {sin2 (x), cos2 (x), tan2 (x)} is a linearly dependent subset of the set of all real-valued functions with domain the interval (−π/2..π/2) of real numbers between −π/2 and π/2)? 1.23 Why does Lemma 1.4 say “distinct”? 1.24 Show that the nonzero rows of an echelon form matrix form a linearly independent set.

Section II. Linear Independence

109

1.25 (a) Show that if the set {u, v, w} is linearly independent set then so is the set {u, u + v, u + v + w}. (b) What is the relationship between the linear independence or dependence of the set {u, v, w} and the independence or dependence of {u − v, v − w, w − u}? 1.26 Example 1.10 shows that the empty set is linearly independent. (a) When is a one-element set linearly independent? (b) How about a set with two elements? 1.27 In any vector space V , the empty set is linearly independent. What about all of V ? 1.28 Show that if {x, y, z} is linearly independent then so are all of its proper subsets: {x, y}, {x, z}, {y, z}, {x},{y}, {z}, and {}. Is that ‘only if’ also? 1.29 (a) Show that this 1 −1 S={ 1 , 2 } 0 0 is a linearly independent subset of R3 . (b) Show that 3 2 0 is in the span of S by ﬁnding c1 and c2 giving a linear relationship. c1 1 1 0 + c2 −1 2 0 = 3 2 0

Show that the pair c1 , c2 is unique. (c) Assume that S is a subset of a vector space and that v is in [S], so that v is a linear combination of vectors from S. Prove that if S is linearly independent then a linear combination of vectors from S adding to v is unique (that is, unique up to reordering and adding or taking away terms of the form 0 · s). Thus S as a spanning set is minimal in this strong sense: each vector in [S] is “hit” a minimum number of times — only once. (d) Prove that it can happen when S is not linearly independent that distinct linear combinations sum to the same vector. 1.30 Prove that a polynomial gives rise to the zero function if and only if it is the zero polynomial. (Comment. This question is not a Linear Algebra matter, but we often use the result. A polynomial gives rise to a function in the obvious way: x → cn xn + · · · + c1 x + c0 .) 1.31 Return to Section 1.2 and redeﬁne point, line, plane, and other linear surfaces to avoid degenerate cases. 1.32 (a) Show that any set of four vectors in R2 is linearly dependent. (b) Is this true for any set of ﬁve? Any set of three? (c) What is the most number of elements that a linearly independent subset of R2 can have? 1.33 Is there a set of four vectors in R3 , any three of which form a linearly independent set? 1.34 Must every linearly dependent set have a subset that is dependent and a subset that is independent?

110

Chapter Two. Vector Spaces

1.35 In R4 , what is the biggest linearly independent set you can ﬁnd? The smallest? The biggest linearly dependent set? The smallest? (‘Biggest’ and ‘smallest’ mean that there are no supersets or subsets with the same property.) 1.36 Linear independence and linear dependence are properties of sets. We can thus naturally ask how those properties act with respect to the familiar elementary set relations and operations. In this body of this subsection we have covered the subset and superset relations. We can also consider the operations of intersection, complementation, and union. (a) How does linear independence relate to intersection: can an intersection of linearly independent sets be independent? Must it be? (b) How does linear independence relate to complementation? (c) Show that the union of two linearly independent sets need not be linearly independent. (d) Characterize when the union of two linearly independent sets is linearly independent, in terms of the intersection of the span of each. 1.37 For Theorem 1.12, (a) ﬁll in the induction for the proof; (b) give an alternate proof that starts with the empty set and builds a sequence of linearly independent subsets of the given ﬁnite set until one appears with the same span as the given set. 1.38 With a little calculation we can get formulas to determine whether or not a set of vectors is linearly independent. (a) Show that this subset of R2 a b { , } c d is linearly independent if and only if ad − bc = 0. (b) Show that this subset of R3 a b c { d , e , f } g h i is linearly independent iﬀ aei + bf g + cdh − hf a − idb − gec = 0. (c) When is this subset of R3 a b { d , e } g h linearly independent? (d) This is an opinion question: for a set of four vectors from R4 , must there be a formula involving the sixteen entries that determines independence of the set? (You needn’t produce such a formula, just decide if one exists.) 1.39 (a) Prove that a set of two perpendicular nonzero vectors from Rn is linearly independent when n > 1. (b) What if n = 1? n = 0? (c) Generalize to more than two vectors. 1.40 Consider the set of functions from the open interval (−1..1) to R. (a) Show that this set is a vector space under the usual operations. (b) Recall the formula for the sum of an inﬁnite geometric series: 1+x+x2 +· · · = 1/(1−x) for all x ∈ (−1..1). Why does this not express a dependence inside of the set {g(x) = 1/(1 − x), f0 (x) = 1, f1 (x) = x, f2 (x) = x2 , . . .} (in the vector space that we are considering)? (Hint. Review the deﬁnition of linear combination.)

Section II. Linear Independence

111

(c) Show that the set in the prior item is linearly independent. This shows that some vector spaces exist with linearly independent subsets that are inﬁnite. 1.41 Show that, where S is a subspace of V , if a subset T of S is linearly independent in S then T is also linearly independent in V . Is that ‘only if’ ?

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Chapter Two. Vector Spaces

III

Basis and Dimension

The prior section ends with the statement that a spanning set is minimal when it is linearly independent and a linearly independent set is maximal when it spans the space. So the notions of minimal spanning set and maximal independent set coincide. In this section we will name this idea and study its properties.

III.1 Basis
1.1 Deﬁnition A basis for a vector space is a sequence of vectors that form a set that is linearly independent and that spans the space. We denote a basis with angle brackets β1 , β2 , . . . to signify that this collection is a sequence∗ — the order of the elements is signiﬁcant. (The requirement that a basis be ordered will be needed, for instance, in Deﬁnition 1.13.) 1.2 Example This is a basis for R2 . 2 1 , 4 1 It is linearly independent c1 2 1 + c2 4 1 = 0 0 =⇒ 2c1 + 1c2 = 0 4c1 + 1c2 = 0 =⇒ c1 = c2 = 0

and it spans R2 . 2c1 + 1c2 = x 4c1 + 1c2 = y =⇒ c2 = 2x − y and c1 = (y − x)/2

1.3 Example This basis for R2 1 2 , 1 4 diﬀers from the prior one because the vectors are in a diﬀerent order. The veriﬁcation that it is a basis is just as in the prior example. 1.4 Example The space R2 has many bases. Another one is this. 1 0 , 0 1 The veriﬁcation is easy.

Section III. Basis and Dimension 1.5 Deﬁnition For any Rn ,     0 1 0 1     En =  .  ,  .  , . . . , . . . . 0 0   0 0   . . . 1

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is the standard (or natural ) basis. We denote these vectors by e1 , . . . , en . (Calculus books refer to R2 ’s standard basis vectors ı and  instead of e1 and e2 , and they refer to R3 ’s standard basis vectors ı, , and k instead of e1 , e2 , and e3 .) Note that the symbol ‘e1 ’ means something diﬀerent in a discussion of R3 than it means in a discussion of R2 . 1.6 Example Consider the space {a · cos θ + b · sin θ a, b ∈ R} of functions of the real variable θ. This is a natural basis. 1 · cos θ + 0 · sin θ, 0 · cos θ + 1 · sin θ = cos θ, sin θ Another, more generic, basis is cos θ − sin θ, 2 cos θ + 3 sin θ . Verﬁcation that these two are bases is Exercise 22. 1.7 Example A natural basis for the vector space of cubic polynomials P3 is 1, x, x2 , x3 . Two other bases for this space are x3 , 3x2 , 6x, 6 and 1, 1 + x, 1 + x + x2 , 1 + x + x2 + x3 . Checking that these are linearly independent and span the space is easy. 1.8 Example The trivial space {0} has only one basis, the empty one .

1.9 Example The space of ﬁnite-degree polynomials has a basis with inﬁnitely many elements 1, x, x2 , . . . . 1.10 Example We have seen bases before. In the ﬁrst chapter we described the solution set of homogeneous systems such as this one x+y −w=0 z+w=0

by parametrizing.     −1 1 1 0 {  y +   w y, w ∈ R} 0 −1 0 1 That is, we described the vector space of solutions as the span of a two-element set. We can easily check that this two-vector set is also linearly independent. Thus the solution set is a subspace of R4 with a two-element basis.

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1.11 Example Parameterization helps ﬁnd bases for other vector spaces, not just for solution sets of homogeneous systems. To ﬁnd a basis for this subspace of M2×2 a b { a + b − 2c = 0} c 0 we rewrite the condition as a = −b + 2c. { −b + 2c b c 0 b, c ∈ R} = {b −1 0 1 2 +c 0 1 0 0 b, c ∈ R}

Thus, this is a natural candidate for a basis. −1 0 1 2 , 0 1 0 0

The above work shows that it spans the space. To show that it is linearly independent is routine. Consider again Example 1.2. It involves two veriﬁcations. In the ﬁrst, to check that the set is linearly independent we looked at linear combinations of the set’s members that total to the zero vector c1 β1 +c2 β2 = 0 . 0 The resulting calculation shows that such a combination is unique, that c1 must be 0 and c2 must be 0. The second veriﬁcation, that the set spans the space, looks at linear combinations that total to any member of the space c1 β1 +c2 β2 = x . In Example 1.2 y we noted only that the resulting calculation shows that such a combination exists, that for each x, y there is a c1 , c2 . However, in fact the calculation also shows that the combination is unique: c1 must be (y − x)/2 and c2 must be 2x − y. That is, the ﬁrst calculation is a special case of the second. The next result says that this holds in general for a spanning set: the combination totaling to the zero vector is unique if and only if the combination totaling to any vector is unique. 1.12 Theorem In any vector space, a subset is a basis if and only if each vector in the space can be expressed as a linear combination of elements of the subset in a unique way. We consider combinations to be the same if they diﬀer only in the order of summands or in the addition or deletion of terms of the form ‘0 · β’.
Proof. By deﬁnition, a sequence is a basis if and only if its vectors form both

a spanning set and a linearly independent set. A subset is a spanning set if and only if each vector in the space is a linear combination of elements of that subset in at least one way. Thus, to ﬁnish we need only show that a subset is linearly independent if and only if every vector in the space is a linear combination of elements from the subset in at most one way. Consider two expressions of a vector as a linear

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combination of the members of the basis. We can rearrange the two sums, and if necessary add some 0βi terms, so that the two sums combine the same β’s in the same order: v = c1 β1 + c2 β2 + · · · + cn βn and v = d1 β1 + d2 β2 + · · · + dn βn . Now c1 β1 + c2 β2 + · · · + cn βn = d1 β1 + d2 β2 + · · · + dn βn holds if and only if (c1 − d1 )β1 + · · · + (cn − dn )βn = 0 holds, and so asserting that each coeﬃcient in the lower equation is zero is the same thing as asserting that ci = di for each i. QED 1.13 Deﬁnition In a vector space with basis B the representation of v with respect to B is the column vector of the coeﬃcients used to express v as a linear combination of the basis vectors:   c1  c2    RepB (v) =  .  . . cn where B = β1 , . . . , βn and v = c1 β1 + c2 β2 + · · · + cn βn . The c’s are the coordinates of v with respect to B. We will later do representations in contexts that involve more than one basis. To help with the bookkeeping, we shall often attach a subscript B to the column vector. 1.14 Example In P3 , with respect to the basis B = 1, 2x, 2x2 , 2x3 , the representation of x + x2 is   0 1/2  RepB (x + x2 ) =  1/2 0 B (note that the coordinates are scalars, not vectors). With respect to a diﬀerent basis D = 1 + x, 1 − x, x + x2 , x + x3 , the representation   0 0 2 RepD (x + x ) =   1 0 D is diﬀerent.

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1.15 Remark This use of column notation and the term ‘coordinates’ has both a down side and an up side. The down side is that representations look like vectors from Rn , which can be confusing when the vector space we are working with is Rn , especially since we sometimes omit the subscript base. We must then infer the intent from the context. For example, the phrase ‘in R2 , where v = 3 ’ refers to the plane 2 vector that, when in canonical position, ends at (3, 2). To ﬁnd the coordinates of that vector with respect to the basis B= we solve c1 1 0 + c2 1 2 = 3 2 1 0 , 1 2

to get that c1 = 3 and c2 = 1/2. Then we have this. RepB (v) = 3 −1/2

Here, although we’ve ommited the subscript B from the column, the fact that the right side is a representation is clear from the context. The up side of the notation and the term ‘coordinates’ is that they generalize the use that we are familiar with: in Rn and with respect to the standard basis En , the vector starting at the origin and ending at (v1 , . . . , vn ) has this representation.     v1 v1 . . RepEn ( . ) =  .  . . vn vn
En

Our main use of representations will come in the third chapter. The deﬁnition appears here because the fact that every vector is a linear combination of basis vectors in a unique way is a crucial property of bases, and also to help make two points. First, we ﬁx an order for the elements of a basis so that coordinates can be stated in that order. Second, for calculation of coordinates, among other things, we shall restrict our attention to spaces with bases having only ﬁnitely many elements. We will see that in the next subsection. Exercises
1.16 Decide if each is a basis for R3 . 1 3 0 1 2 , 2 , 0 2 (a) (b) 3 1 1 3 0 1 1 2 , 1 , 3 (d) −1 1 0 , 3 2 1 (c) 0 2 −1 , 1 1 1 , 2 5 0

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1.17 Represent the vector with respect to the basis. 1 1 −1 (a) ,B= , ⊆ R2 2 1 1 (b) x2 + x3 , D = 1, 1 + x, 1 + x + x2 , 1 + x + x2 + x3 ⊆ P3   0 −1 (c)  , E4 ⊆ R4 0 1 1.18 Find a basis for P2 , the space of all quadratic polynomials. Must any such basis contain a polynomial of each degree: degree zero, degree one, and degree two? 1.19 Find a basis for the solution set of this system. x1 − 4x2 + 3x3 − x4 = 0 2x1 − 8x2 + 6x3 − 2x4 = 0 1.20 Find a basis for M2×2 , the space of 2×2 matrices. 1.21 Find a basis for each. (a) The subspace {a2 x2 + a1 x + a0 a2 − 2a1 = a0 } of P2 (b) The space of three-wide row vectors whose ﬁrst and second components add to zero (c) This subspace of the 2×2 matrices { a 0 b c c − 2b = 0}

1.22 Check Example 1.6. 1.23 Find the span of each set and then ﬁnd a basis for that span. (a) {1 + x, 1 + 2x} in P2 (b) {2 − 2x, 3 + 4x2 } in P2 1.24 Find a basis for each of these subspaces of the space P3 of cubic polynomials. (a) The subspace of cubic polynomials p(x) such that p(7) = 0 (b) The subspace of polynomials p(x) such that p(7) = 0 and p(5) = 0 (c) The subspace of polynomials p(x) such that p(7) = 0, p(5) = 0, and p(3) = 0 (d) The space of polynomials p(x) such that p(7) = 0, p(5) = 0, p(3) = 0, and p(1) = 0 1.25 We’ve seen that it is possible for a basis to remain a basis when it is reordered. Must it remain a basis? 1.26 Can a basis contain a zero vector? 1.27 Let β1 , β2 , β3 be a basis for a vector space. (a) Show that c1 β1 , c2 β2 , c3 β3 is a basis when c1 , c2 , c3 = 0. What happens when at least one ci is 0? (b) Prove that α1 , α2 , α3 is a basis where αi = β1 + βi . 1.28 Find one vector v that will make each into a basis for the space. 1 0 1 1 , 1 , v in R3 (a) , v in R2 (b) (c) x, 1 + x2 , v in P2 1 0 0 1.29 Where β1 , . . . , βn is a basis, show that in this equation c1 β1 + · · · + ck βk = ck+1 βk+1 + · · · + cn βn each of the ci ’s is zero. Generalize. 1.30 A basis contains some of the vectors from a vector space; can it contain them all?

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1.31 Theorem 1.12 shows that, with respect to a basis, every linear combination is unique. If a subset is not a basis, can linear combinations be not unique? If so, must they be? 1.32 A square matrix is symmetric if for all indices i and j, entry i, j equals entry j, i. (a) Find a basis for the vector space of symmetric 2×2 matrices. (b) Find a basis for the space of symmetric 3×3 matrices. (c) Find a basis for the space of symmetric n×n matrices. 1.33 We can show that every basis for R3 contains the same number of vectors. (a) Show that no linearly independent subset of R3 contains more than three vectors. (b) Show that no spanning subset of R3 contains fewer than three vectors. Hint: recall how to calculate the span of a set and show that this method cannot yield all of R3 when it is applied to fewer than three vectors. 1.34 One of the exercises in the Subspaces subsection shows that the set x { y z x1 y1 z1 Find a basis. x2 y2 z2 x + y + z = 1}

is a vector space under these operations. + = x1 + x2 − 1 y1 + y2 z1 + z2 r x y z = rx − r + 1 ry rz

III.2 Dimension
In the prior subsection we deﬁned the basis of a vector space, and we saw that a space can have many diﬀerent bases. For example, following the deﬁnition of a basis, we saw three diﬀerent bases for R2 . So we cannot talk about “the” basis for a vector space. True, some vector spaces have bases that strike us as more natural than others, for instance, R2 ’s basis E2 or R3 ’s basis E3 or P2 ’s basis 1, x, x2 . But, for example in the space {a2 x2 + a1 x + a0 2a2 − a0 = a1 }, no particular basis leaps out at us as the most natural one. We cannot, in general, associate with a space any single basis that best describes that space. We can, however, ﬁnd something about the bases that is uniquely associated with the space. This subsection shows that any two bases for a space have the same number of elements. So, with each space we can associate a number, the number of vectors in any of its bases. This brings us back to when we considered the two things that could be meant by the term ‘minimal spanning set’. At that point we deﬁned ‘minimal’ as linearly independent, but we noted that another reasonable interpretation of the term is that a spanning set is ‘minimal’ when it has the fewest number of elements of any set with the same span. At the end of this subsection, after we

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have shown that all bases have the same number of elements, then we will have shown that the two senses of ‘minimal’ are equivalent. Before we start, we ﬁrst limit our attention to spaces where at least one basis has only ﬁnitely many members. 2.1 Deﬁnition A vector space is ﬁnite-dimensional if it has a basis with only ﬁnitely many vectors. (One reason for sticking to ﬁnite-dimensional spaces is so that the representation of a vector with respect to a basis is a ﬁnitely-tall vector, and so can be easily written.) From now on we study only ﬁnite-dimensional vector spaces. We shall take the term ‘vector space’ to mean ‘ﬁnite-dimensional vector space’. Other spaces are interesting and important, but they lie outside of our scope. To prove the main theorem we shall use a technical result. 2.2 Lemma (Exchange Lemma) Assume that B = β1 , . . . , βn is a basis for a vector space, and that for the vector v the relationship v = c1 β1 + c2 β2 + · · · + cn βn has ci = 0. Then exchanging βi for v yields another basis for the space. ˆ Proof. Call the outcome of the exchange B = β1 , . . . , βi−1 , v, βi+1 , . . . , βn . ˆ We ﬁrst show that B is linearly independent. Any relationship d1 β1 + · · · + ˆ di v + · · · + dn βn = 0 among the members of B, after substitution for v, d1 β1 + · · · + di · (c1 β1 + · · · + ci βi + · · · + cn βn ) + · · · + dn βn = 0 (∗)

gives a linear relationship among the members of B. The basis B is linearly independent, so the coeﬃcient di ci of βi is zero. Because ci is assumed to be nonzero, di = 0. Using this in equation (∗) above gives that all of the other d’s ˆ are also zero. Therefore B is linearly independent. ˆ We ﬁnish by showing that B has the same span as B. Half of this argument, ˆ ˆ that [B] ⊆ [B], is easy; any member d1 β1 + · · · + di v + · · · + dn βn of [B] can be written d1 β1 + · · · + di · (c1 β1 + · · · + cn βn ) + · · · + dn βn , which is a linear combination of linear combinations of members of B, and hence is in [B]. For ˆ the [B] ⊆ [B] half of the argument, recall that when v = c1 β1 + · · · + cn βn with ci = 0, then the equation can be rearranged to βi = (−c1 /ci )β1 + · · · + (1/ci )v + · · · + (−cn /ci )βn . Now, consider any member d1 β1 + · · · + di βi + · · · + dn βn of [B], substitute for βi its expression as a linear combination of the members ˆ of B, and recognize (as in the ﬁrst half of this argument) that the result is a ˆ linear combination of linear combinations, of members of B, and hence is in ˆ [B]. QED 2.3 Theorem In any ﬁnite-dimensional vector space, all of the bases have the same number of elements.
Proof. Fix a vector space with at least one ﬁnite basis. Choose, from among

all of this space’s bases, one B = β1 , . . . , βn of minimal size. We will show

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that any other basis D = δ1 , δ2 , . . . also has the same number of members, n. Because B has minimal size, D has no fewer than n vectors. We will argue that it cannot have more than n vectors. The basis B spans the space and δ1 is in the space, so δ1 is a nontrivial linear combination of elements of B. By the Exchange Lemma, δ1 can be swapped for a vector from B, resulting in a basis B1 , where one element is δ and all of the n − 1 other elements are β’s. The prior paragraph forms the basis step for an induction argument. The inductive step starts with a basis Bk (for 1 ≤ k < n) containing k members of D and n − k members of B. We know that D has at least n members so there is a δk+1 . Represent it as a linear combination of elements of Bk . The key point: in that representation, at least one of the nonzero scalars must be associated with a βi or else that representation would be a nontrivial linear relationship among elements of the linearly independent set D. Exchange δk+1 for βi to get a new basis Bk+1 with one δ more and one β fewer than the previous basis Bk . Repeat the inductive step until no β’s remain, so that Bn contains δ1 , . . . , δn . Now, D cannot have more than these n vectors because any δn+1 that remains would be in the span of Bn (since it is a basis) and hence would be a linear combination of the other δ’s, contradicting that D is linearly independent. QED 2.4 Deﬁnition The dimension of a vector space is the number of vectors in any of its bases. 2.5 Example Any basis for Rn has n vectors since the standard basis En has n vectors. Thus, this deﬁnition generalizes the most familiar use of term, that Rn is n-dimensional. 2.6 Example The space Pn of polynomials of degree at most n has dimension n+1. We can show this by exhibiting any basis — 1, x, . . . , xn comes to mind — and counting its members. 2.7 Example A trivial space is zero-dimensional since its basis is empty. Again, although we sometimes say ‘ﬁnite-dimensional’ as a reminder, in the rest of this book all vector spaces are assumed to be ﬁnite-dimensional. An instance of this is that in the next result the word ‘space’ should be taken to mean ‘ﬁnite-dimensional vector space’. 2.8 Corollary No linearly independent set can have a size greater than the dimension of the enclosing space.
Proof. Inspection of the above proof shows that it never uses that D spans the space, only that D is linearly independent. QED

2.9 Example Recall the subspace diagram from the prior section showing the subspaces of R3 . Each subspace shown is described with a minimal spanning set, for which we now have the term ‘basis’. The whole space has a basis with three members, the plane subspaces have bases with two members, the line

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subspaces have bases with one member, and the trivial subspace has a basis with zero members. When we saw that diagram we could not show that these are the only subspaces that this space has. We can show it now. The prior corollary proves that the only subspaces of R3 are either three-, two-, one-, or zero-dimensional. Therefore, the diagram indicates all of the subspaces. There are no subspaces somehow, say, between lines and planes. 2.10 Corollary Any linearly independent set can be expanded to make a basis.
Proof. If a linearly independent set is not already a basis then it must not

span the space. Adding to it a vector that is not in the span preserves linear independence. Keep adding, until the resulting set does span the space, which the prior corollary shows will happen after only a ﬁnite number of steps. QED 2.11 Corollary Any spanning set can be shrunk to a basis.
Proof. Call the spanning set S. If S is empty then it is already a basis (the

space must be a trivial space). If S = {0} then it can be shrunk to the empty basis, thereby making it linearly independent, without changing its span. Otherwise, S contains a vector s1 with s1 = 0 and we can form a basis B1 = s1 . If [B1 ] = [S] then we are done. If not then there is a s2 ∈ [S] such that s2 ∈ [B1 ]. Let B2 = s1 , s2 ; if [B2 ] = [S] then we are done. We can repeat this process until the spans are equal, which must happen in at most ﬁnitely many steps. QED 2.12 Corollary In an n-dimensional space, a set of n vectors is linearly independent if and only if it spans the space.
Proof. First we will show that a subset with n vectors is linearly independent

if and only if it is a basis. ‘If’ is trivially true — bases are linearly independent. ‘Only if’ holds because a linearly independent set can be expanded to a basis, but a basis has n elements, so this expansion is actually the set that we began with. To ﬁnish, we will show that any subset with n vectors spans the space if and only if it is a basis. Again, ‘if’ is trivial. ‘Only if’ holds because any spanning set can be shrunk to a basis, but a basis has n elements and so this shrunken set is just the one we started with. QED The main result of this subsection, that all of the bases in a ﬁnite-dimensional vector space have the same number of elements, is the single most important result in this book because, as Example 2.9 shows, it describes what vector spaces and subspaces there can be. We will see more in the next chapter. 2.13 Remark The case of inﬁnite-dimensional vector spaces is somewhat controversial. The statement ‘any inﬁnite-dimensional vector space has a basis’ is known to be equivalent to a statement called the Axiom of Choice (see [Blass 1984]). Mathematicians diﬀer philosophically on whether to accept or reject this statement as an axiom on which to base mathematics (although, the

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great majority seem to accept it). Consequently the question about inﬁnitedimensional vector spaces is still somewhat up in the air. (A discussion of the Axiom of Choice can be found in the Frequently Asked Questions list for the Usenet group sci.math. Another accessible reference is [Rucker].) Exercises
Assume that all spaces are ﬁnite-dimensional unless otherwise stated. 2.14 Find a basis for, and the dimension of, P2 . 2.15 Find a basis for, and the dimension of, the solution set of this system. x1 − 4x2 + 3x3 − x4 = 0 2x1 − 8x2 + 6x3 − 2x4 = 0 2.16 Find a basis for, and the dimension of, M2×2 , the vector space of 2×2 matrices. 2.17 Find the dimension of the vector space of matrices a c subject to each condition. (a) a, b, c, d ∈ R (b) a − b + 2c = 0 and d ∈ R (c) a + b + c = 0, a + b − c = 0, and d ∈ R 2.18 Find the dimension of each. (a) The space of cubic polynomials p(x) such that p(7) = 0 (b) The space of cubic polynomials p(x) such that p(7) = 0 and p(5) = 0 (c) The space of cubic polynomials p(x) such that p(7) = 0, p(5) = 0, and p(3) = 0 (d) The space of cubic polynomials p(x) such that p(7) = 0, p(5) = 0, p(3) = 0, and p(1) = 0 2.19 What is the dimension of the span of the set {cos2 θ, sin2 θ, cos 2θ, sin 2θ}? This span is a subspace of the space of all real-valued functions of one real variable. 2.20 Find the dimension of C47 , the vector space of 47-tuples of complex numbers. 2.21 What is the dimension of the vector space M3×5 of 3×5 matrices? 2.22 Show that this is a basis for R4 .         1 1 1 1 0 1 1 1 0 , 0 , 1 , 1 0 0 0 1 (The results of this subsection can be used to simplify this job.) 2.23 Refer to Example 2.9. (a) Sketch a similar subspace diagram for P2 . (b) Sketch one for M2×2 . 2.24 Where S is a set, the functions f : S → R form a vector space under the natural operations: the sum f + g is the function given by f + g (s) = f (s) + g(s) and the scalar product is given by r · f (s) = r · f (s). What is the dimension of the space resulting for each domain? b d

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(a) S = {1} (b) S = {1, 2} (c) S = {1, . . . , n} 2.25 (See Exercise 24.) Prove that this is an inﬁnite-dimensional space: the set of all functions f : R → R under the natural operations. 2.26 (See Exercise 24.) What is the dimension of the vector space of functions f : S → R, under the natural operations, where the domain S is the empty set? 2.27 Show that any set of four vectors in R2 is linearly dependent. 2.28 Show that α1 , α2 , α3 ⊂ R3 is a basis if and only if there is no plane through the origin containing all three vectors. 2.29 (a) Prove that any subspace of a ﬁnite dimensional space has a basis. (b) Prove that any subspace of a ﬁnite dimensional space is ﬁnite dimensional. 2.30 Where is the ﬁniteness of B used in Theorem 2.3? 2.31 Prove that if U and W are both three-dimensional subspaces of R5 then U ∩W is non-trivial. Generalize. 2.32 A basis for a space consists of elements of that space. So we are naturally led to how the property ‘is a basis’ interacts with operations ⊆ and ∩ and ∪. (Of course, a basis is actually a sequence in that it is ordered, but there is a natural extension of these operations.) (a) Consider ﬁrst how bases might be related by ⊆. Assume that U, W are subspaces of some vector space and that U ⊆ W . Can there exist bases BU for U and BW for W such that BU ⊆ BW ? Must such bases exist? For any basis BU for U , must there be a basis BW for W such that BU ⊆ BW ? For any basis BW for W , must there be a basis BU for U such that BU ⊆ BW ? For any bases BU , BW for U and W , must BU be a subset of BW ? (b) Is the ∩ of bases a basis? For what space? (c) Is the ∪ of bases a basis? For what space? (d) What about the complement operation? (Hint. Test any conjectures against some subspaces of R3 .) 2.33 Consider how ‘dimension’ interacts with ‘subset’. Assume U and W are both subspaces of some vector space, and that U ⊆ W . (a) Prove that dim(U ) ≤ dim(W ). (b) Prove that equality of dimension holds if and only if U = W . (c) Show that the prior item does not hold if they are inﬁnite-dimensional. ? 2.34 For any vector v in Rn and any permutation σ of the numbers 1, 2, . . . , n (that is, σ is a rearrangement of those numbers into a new order), deﬁne σ(v) to be the vector whose components are vσ(1) , vσ(2) , . . . , and vσ(n) (where σ(1) is the ﬁrst number in the rearrangement, etc.). Now ﬁx v and let V be the span of {σ(v) σ permutes 1, . . . , n}. What are the possibilities for the dimension of V ? [Wohascum no. 47]

III.3 Vector Spaces and Linear Systems
We will now reconsider linear systems and Gauss’ method, aided by the tools and terms of this chapter. We will make three points. For the ﬁrst point, recall the ﬁrst chapter’s Linear Combination Lemma and its corollary: if two matrices are related by row operations A −→ · · · −→ B then

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each row of B is a linear combination of the rows of A. That is, Gauss’ method works by taking linear combinations of rows. Therefore, the right setting in which to study row operations in general, and Gauss’ method in particular, is the following vector space. 3.1 Deﬁnition The row space of a matrix is the span of the set of its rows. The row rank is the dimension of the row space, the number of linearly independent rows. 3.2 Example If A= 2 4 3 6

then Rowspace(A) is this subspace of the space of two-component row vectors. {c1 · 2 3 + c2 · 4 6 c1 , c2 ∈ R}

The linear dependence of the second on the ﬁrst is obvious and so we can simplify this description to {c · 2 3 c ∈ R}. 3.3 Lemma If the matrices A and B are related by a row operation A −→ B
ρi ↔ρj

or A −→ B

kρi

or A −→ B

kρi +ρj

(for i = j and k = 0) then their row spaces are equal. Hence, row-equivalent matrices have the same row space, and hence also, the same row rank.
Proof. By the Linear Combination Lemma’s corollary, each row of B is in the

row space of A. Further, Rowspace(B) ⊆ Rowspace(A) because a member of the set Rowspace(B) is a linear combination of the rows of B, which means it is a combination of a combination of the rows of A, and hence, by the Linear Combination Lemma, is also a member of Rowspace(A). For the other containment, recall that row operations are reversible: A −→ B if and only if B −→ A. With that, Rowspace(A) ⊆ Rowspace(B) also follows from the prior paragraph, and so the two sets are equal. QED Thus, row operations leave the row space unchanged. But of course, Gauss’ method performs the row operations systematically, with a speciﬁc goal in mind, echelon form. 3.4 Lemma The nonzero rows of an echelon form matrix make up a linearly independent set.
Proof. A result in the ﬁrst chapter, Lemma III.2.5, states that in an echelon

form matrix, no nonzero row is a linear combination of the other rows. This is a restatement of that result into new terminology. QED Thus, in the language of this chapter, Gaussian reduction works by eliminating linear dependences among rows, leaving the span unchanged, until no nontrivial linear relationships remain (among the nonzero rows). That is, Gauss’ method produces a basis for the row space.

Section III. Basis and Dimension

125

3.5 Example From any matrix, we can produce a basis for the row space by performing Gauss’ method and taking the nonzero rows of the resulting echelon form matrix. For instance,     1 3 1 1 3 1 1 4 1 −ρ1 +ρ2 6ρ2 +ρ3 0 1 0 −→ −→ −2ρ1 +ρ3 2 0 5 0 0 3 produces the basis 1 3 1 , 0 1 0 , 0 0 3 for the row space. This is a basis for the row space of both the starting and ending matrices, since the two row spaces are equal. Using this technique, we can also ﬁnd bases for spans not directly involving row vectors. 3.6 Deﬁnition The column space of a matrix is the span of the set of its columns. The column rank is the dimension of the column space, the number of linearly independent columns. Our interest in column spaces stems from our study of linear systems. An example is that this system c1 + 3c2 + 7c3 = d1 2c1 + 3c2 + 8c3 = d2 c2 + 2c3 = d3 4c1 + 4c3 = d4 has a solution if and only if the vector of d’s is a linear combination of the other column vectors,         d1 7 3 1 8 d2  3 2 c1   + c2   + c3   =   2 d3  1 0 d4 4 0 4 meaning that the vector of d’s is in the column space of the matrix of coeﬃcients. 3.7 Example Given this matrix,  1 2  0 4

3 3 1 0

 7 8  2 4

to get a basis for the column space, temporarily turn the columns into rows and reduce.     1 2 0 4 1 2 0 4 −3ρ1 +ρ2 −2ρ2 +ρ3 3 3 1 0 0 −3 1 −12 −→ −→ −7ρ1 +ρ3 7 8 2 4 0 0 0 0

126 Now turn the rows back to columns.     0 1 2  −3    , 0  1  −12 4

Chapter Two. Vector Spaces

The result is a basis for the column space of the given matrix. 3.8 Deﬁnition The transpose of a matrix is the result of interchanging the rows and columns of that matrix. That is, column j of the matrix A is row j of Atrans , and vice versa. So the instructions for the prior example are “transpose, reduce, and transpose back”. We can even, at the price of tolerating the as-yet-vague idea of vector spaces being “the same”, use Gauss’ method to ﬁnd bases for spans in other types of vector spaces. 3.9 Example To get a basis for the span of {x2 + x4 , 2x2 + 3x4 , −x2 − 3x4 } in the space P4 , think of these three polynomials as “the same” as the row vectors 0 0 1 0 1 , 0 0 2 0 3 , and 0 0 −1 0 −3 , apply Gauss’ method     0 0 1 0 1 0 0 1 0 1 0 0 2 0 3  −2ρ1 +ρ2 2ρ2 +ρ3 0 0 0 0 1 −→ −→ ρ1 +ρ3 0 0 −1 0 −3 0 0 0 0 0 and translate back to get the basis x2 + x4 , x4 . (As mentioned earlier, we will make the phrase “the same” precise at the start of the next chapter.) Thus, our ﬁrst point in this subsection is that the tools of this chapter give us a more conceptual understanding of Gaussian reduction. For the second point of this subsection, consider the eﬀect on the column space of this row reduction. 1 2 2 4
−2ρ1 +ρ2

−→

1 0

2 0

The column space of the left-hand matrix contains vectors with a second component that is nonzero. But the column space of the right-hand matrix is diﬀerent because it contains only vectors whose second component is zero. It is this knowledge that row operations can change the column space that makes next result surprising. 3.10 Lemma Row operations do not change the column rank.
Proof. Restated, if A reduces to B then the column rank of B equals the

column rank of A.

Section III. Basis and Dimension

127

We will be done if we can show that row operations do not aﬀect linear relationships among columns because the column rank is just the size of the largest set of unrelated columns. That is, we will show that a relationship exists among columns (such as that the ﬁfth column is twice the second plus the fourth) if and only if that relationship exists after the row operation. But this is exactly the ﬁrst theorem of this book: in a relationship among columns,       0 a1,n a1,1  a2,n  0  a2,1        c1 ·  .  + · · · + cn ·  .  =  .   .  .  .  . . . 0 am,n am,1 row operations leave unchanged the set of solutions (c1 , . . . , cn ).
QED

Another way, besides the prior result, to state that Gauss’ method has something to say about the column space as well as about the row space is to consider again Gauss-Jordan reduction. Recall that it ends with the reduced echelon form of a matrix, as here.     1 3 0 2 1 3 1 6 2 6 3 16 −→ · · · −→ 0 0 1 4 0 0 0 0 1 3 1 6 Consider the row space and the column space of this result. Our ﬁrst point made above says that a basis for the row space is easy to get: simply collect together all of the rows with leading entries. However, because this is a reduced echelon form matrix, a basis for the column space is just as easy: take the columns containing the leading entries, that is, e1 , e2 . (Linear independence is obvious. The other columns are in the span of this set, since they all have a third component of zero.) Thus, for a reduced echelon form matrix, bases for the row and column spaces can be found in essentially the same way — by taking the parts of the matrix, the rows or columns, containing the leading entries. 3.11 Theorem The row rank and column rank of a matrix are equal.
Proof. First bring the matrix to reduced echelon form. At that point, the

row rank equals the number of leading entries since each equals the number of nonzero rows. Also at that point, the number of leading entries equals the column rank because the set of columns containing leading entries consists of some of the ei ’s from a standard basis, and that set is linearly independent and spans the set of columns. Hence, in the reduced echelon form matrix, the row rank equals the column rank, because each equals the number of leading entries. But Lemma 3.3 and Lemma 3.10 show that the row rank and column rank are not changed by using row operations to get to reduced echelon form. Thus the row rank and the column rank of the original matrix are also equal. QED 3.12 Deﬁnition The rank of a matrix is its row rank or column rank.

128

Chapter Two. Vector Spaces

So our second point in this subsection is that the column space and row space of a matrix have the same dimension. Our third and ﬁnal point is that the concepts that we’ve seen arising naturally in the study of vector spaces are exactly the ones that we have studied with linear systems. 3.13 Theorem For linear systems with n unknowns and with matrix of coeﬃcients A, the statements (1) the rank of A is r (2) the space of solutions of the associated homogeneous system has dimension n − r are equivalent. So if the system has at least one particular solution then for the set of solutions, the number of parameters equals n − r, the number of variables minus the rank of the matrix of coeﬃcients.
Proof. The rank of A is r if and only if Gaussian reduction on A ends with r

nonzero rows. That’s true if and only if echelon form matrices row equivalent to A have r-many leading variables. That in turn holds if and only if there are n − r free variables. QED 3.14 Remark [Munkres] Sometimes that result is mistakenly remembered to say that the general solution of an n unknown system of m equations uses n − m parameters. The number of equations is not the relevant ﬁgure, rather, what matters is the number of independent equations (the number of equations in a maximal independent set). Where there are r independent equations, the general solution involves n − r parameters. 3.15 Corollary Where the matrix A is n×n, the statements (1) the rank of A is n (2) A is nonsingular (3) the rows of A form a linearly independent set (4) the columns of A form a linearly independent set (5) any linear system whose matrix of coeﬃcients is A has one and only one solution are equivalent.
Proof. Clearly (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4). The last, (4) ⇐⇒ (5), holds

because a set of n column vectors is linearly independent if and only if it is a basis for Rn , but the system       a1,1 a1,n d1  a2,1   a2,n   d2        c1  .  + · · · + cn  .  =  .   .   .   .  . . . am,1 am,n dm has a unique solution for all choices of d1 , . . . , dn ∈ R if and only if the vectors of a’s form a basis. QED

Section III. Basis and Dimension Exercises
3.16 Transpose each. (a) 2 3 1 1 (b) 2 1 1 3 (c) 1 6 4 7 3 8 (d) 0 0 0

129

(e) −1 −2 3.17 Decide if the vector is in the row space of the matrix. 0 1 3 2 1 (a) , 1 0 (b) −1 0 1 , 1 1 1 3 1 −1 2 7 3.18 Decide if the vector is in the column space. 1 3 1 1 1 1 1 0 4 , 0 (a) , (b) 2 1 1 3 1 −3 −3 0 3.19 Find a basis for the row space of this matrix.

2 0 3 1

0 1 1 0

3 1 0 −4

4 −1 2  −1

3.20 Find the rank of each matrix. 2 1 3 1 −1 2 1 3 2 3 −3 6 (a) 1 −1 2 (b) (c) 5 1 1 1 0 3 −2 2 −4 6 4 3 0 0 0 (d) 0 0 0 0 0 0 3.21 Find a basis for the span of each set. (a) { 1 3 , −1 3 , 1 4 , 2 1 } ⊆ M1×2 1 3 1 (b) { 2 , 1 , −3 } ⊆ R3 1 −1 −3 (c) {1 + x, 1 − x2 , 3 + 2x − x2 } ⊆ P3 1 0 1 1 0 3 −1 0 −5 (d) { , , } ⊆ M2×3 3 1 −1 2 1 4 −1 −1 −9 3.22 Which matrices have rank zero? Rank one? 3.23 Given a, b, c ∈ R, what choice of d will cause this matrix to have the rank of one? a b c d 3.24 Find the column rank of this matrix. 1 2 3 0 −1 1 5 0 0 4 4 1

3.25 Show that a linear system with at least one solution has at most one solution if and only if the matrix of coeﬃcients has rank equal to the number of its columns. 3.26 If a matrix is 5×9, which set must be dependent, its set of rows or its set of columns?

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Chapter Two. Vector Spaces

3.27 Give an example to show that, despite that they have the same dimension, the row space and column space of a matrix need not be equal. Are they ever equal? 3.28 Show that the set {(1, −1, 2, −3), (1, 1, 2, 0), (3, −1, 6, −6)} does not have the same span as {(1, 0, 1, 0), (0, 2, 0, 3)}. What, by the way, is the vector space? 3.29 Show that this set of column vectors 3x + 2y + 4z = d1 d1 − z = d2 d2 there are x, y, and z such that x 2x + 2y + 5z = d3 d3 is a subspace of R3 . Find a basis. 3.30 Show that the transpose operation is linear : (rA + sB)trans = rAtrans + sB trans for r, s ∈ R and A, B ∈ Mm×n , 3.31 In this subsection we have shown that Gaussian reduction ﬁnds a basis for the row space. (a) Show that this basis is not unique — diﬀerent reductions may yield diﬀerent bases. (b) Produce matrices with equal row spaces but unequal numbers of rows. (c) Prove that two matrices have equal row spaces if and only if after GaussJordan reduction they have the same nonzero rows. 3.32 Why is there not a problem with Remark 3.14 in the case that r is bigger than n? 3.33 Show that the row rank of an m×n matrix is at most m. Is there a better bound? 3.34 Show that the rank of a matrix equals the rank of its transpose. 3.35 True or false: the column space of a matrix equals the row space of its transpose. 3.36 We have seen that a row operation may change the column space. Must it? 3.37 Prove that a linear system has a solution if and only if that system’s matrix of coeﬃcients has the same rank as its augmented matrix. 3.38 An m×n matrix has full row rank if its row rank is m, and it has full column rank if its column rank is n. (a) Show that a matrix can have both full row rank and full column rank only if it is square. (b) Prove that the linear system with matrix of coeﬃcients A has a solution for any d1 , . . . , dn ’s on the right side if and only if A has full row rank. (c) Prove that a homogeneous system has a unique solution if and only if its matrix of coeﬃcients A has full column rank. (d) Prove that the statement “if a system with matrix of coeﬃcients A has any solution then it has a unique solution” holds if and only if A has full column rank. 3.39 How would the conclusion of Lemma 3.3 change if Gauss’ method is changed to allow multiplying a row by zero? 3.40 What is the relationship between rank(A) and rank(−A)? Between rank(A) and rank(kA)? What, if any, is the relationship between rank(A), rank(B), and rank(A + B)?

Section III. Basis and Dimension

131

III.4 Combining Subspaces
This subsection is optional. It is required only for the last sections of Chapter Three and Chapter Five and for occasional exercises, and can be passed over without loss of continuity. This chapter opened with the deﬁnition of a vector space, and the middle consisted of a ﬁrst analysis of the idea. This subsection closes the chapter by ﬁnishing the analysis, in the sense that ‘analysis’ means “method of determining the . . . essential features of something by separating it into parts” [Macmillan Dictionary]. A common way to understand things is to see how they can be built from component parts. For instance, we think of R3 as put together, in some way, from the x-axis, the y-axis, and z-axis. In this subsection we will make this precise; we will describe how to decompose a vector space into a combination of some of its subspaces. In developing this idea of subspace combination, we will keep the R3 example in mind as a benchmark model. Subspaces are subsets and sets combine via union. But taking the combination operation for subspaces to be the simple union operation isn’t what we want. For one thing, the union of the x-axis, the y-axis, and z-axis is not all of R3 , so the benchmark model would be left out. Besides, union is all wrong for this reason: a union of subspaces need not be a subspace (it need not be closed; for instance, this R3 vector         1 0 0 1 0 + 1 + 0 = 1 0 0 1 1 is in none of the three axes and hence is not in the union). In addition to the members of the subspaces, we must at least also include all of the linear combinations. 4.1 Deﬁnition Where W1 , . . . , Wk are subspaces of a vector space, their sum is the span of their union W1 + W2 + · · · + Wk = [W1 ∪ W2 ∪ . . . Wk ]. (The notation, writing the ‘+’ between sets in addition to using it between vectors, ﬁts with the practice of using this symbol for any natural accumulation operation.) 4.2 Example The R3 model ﬁts with this operation. Any vector w ∈ R3 can be written as a linear combination c1 v1 + c2 v2 + c3 v3 where v1 is a member of the x-axis, etc., in this way         w1 w1 0 0 w2  = 1 ·  0  + 1 · w2  + 1 ·  0  w3 0 0 w3 and so R3 = x-axis + y-axis + z-axis.

132

Chapter Two. Vector Spaces

4.3 Example A sum of subspaces can be less than the entire space. Inside of P4 , let L be the subspace of linear polynomials {a + bx a, b ∈ R} and let C be the subspace of purely-cubic polynomials {cx3 c ∈ R}. Then L + C is not all of P4 . Instead, it is the subspace L + C = {a + bx + cx3 a, b, c ∈ R}. 4.4 Example A space can be described as a combination of subspaces in more than one way. Besides the decomposition R3 = x-axis + y-axis + z-axis, we can also write R3 = xy-plane + yz-plane. To check this, note that any w ∈ R3 can be written as a linear combination of a member of the xy-plane and a member of the yz-plane; here are two such combinations.             w1 w1 0 w1 w1 0 w2  = 1 · w2  + 1 ·  0  w2  = 1 · w2 /2 + 1 · w2 /2 w3 0 w3 w3 0 w3 The above deﬁnition gives one way in which a space can be thought of as a combination of some of its parts. However, the prior example shows that there is at least one interesting property of our benchmark model that is not captured by the deﬁnition of the sum of subspaces. In the familiar decomposition of R3 , we often speak of a vector’s ‘x part’ or ‘y part’ or ‘z part’. That is, in this model, each vector has a unique decomposition into parts that come from the parts making up the whole space. But in the decomposition used in Example 4.4, we cannot refer to the “xy part” of a vector — these three sums               1 1 0 1 0 1 0 2 = 2 + 0 = 0 + 2 = 1 + 1 3 0 3 0 3 0 3 all describe the vector as comprised of something from the ﬁrst plane plus something from the second plane, but the “xy part” is diﬀerent in each. That is, when we consider how R3 is put together from the three axes “in some way”, we might mean “in such a way that every vector has at least one decomposition”, and that leads to the deﬁnition above. But if we take it to mean “in such a way that every vector has one and only one decomposition” then we need another condition on combinations. To see what this condition is, recall that vectors are uniquely represented in terms of a basis. We can use this to break a space into a sum of subspaces such that any vector in the space breaks uniquely into a sum of members of those subspaces. 4.5 Example The benchmark is R3 with its standard basis E3 = e1 , e2 , e3 . The subspace with the basis B1 = e1 is the x-axis. The subspace with the basis B2 = e2 is the y-axis. The subspace with the basis B3 = e3 is the z-axis. The fact that any member of R3 is expressible as a sum of vectors from these subspaces         x x 0 0 y  =  0  + y  + 0 z 0 0 z

Section III. Basis and Dimension is a reﬂection of the fact that E3 spans the space — this equation         x 1 0 0 y  = c1 0 + c2 1 + c3 0 z 0 0 1

133

has a solution for any x, y, z ∈ R. And, the fact that each such expression is unique reﬂects that fact that E3 is linearly independent — any equation like the one above has a unique solution. 4.6 Example We don’t have to take the basis vectors one at a time, the same idea works if we conglomerate them into larger sequences. Consider again the space R3 and the vectors from the standard basis E3 . The subspace with the basis B1 = e1 , e3 is the xz-plane. The subspace with the basis B2 = e2 is the y-axis. As in the prior example, the fact that any member of the space is a sum of members of the two subspaces in one and only one way       x x 0 y  =  0  + y  z z 0 is a reﬂection of the fact that these vectors form a basis — this system         x 1 0 0 y  = (c1 0 + c3 0) + c2 1 z 0 1 0 has one and only one solution for any x, y, z ∈ R. These examples illustrate a natural way to decompose a space into a sum of subspaces in such a way that each vector decomposes uniquely into a sum of vectors from the parts. The next result says that this way is the only way. 4.7 Deﬁnition The concatenation of the sequences B1 = β1,1 , . . . , β1,n1 , . . . , Bk = βk,1 , . . . , βk,nk is their adjoinment. B1 B2 · · · Bk = β1,1 , . . . , β1,n1 , β2,1 , . . . , βk,nk

4.8 Lemma Let V be a vector space that is the sum of some of its subspaces V = W1 + · · · + Wk . Let B1 , . . . , Bk be any bases for these subspaces. Then the following are equivalent. (1) For every v ∈ V , the expression v = w1 + · · · + wk (with wi ∈ Wi ) is unique. (2) The concatenation B1 · · · Bk is a basis for V . (3) The nonzero members of {w1 , . . . , wk } (with wi ∈ Wi ) form a linearly independent set — among nonzero vectors from diﬀerent Wi ’s, every linear relationship is trivial.

134

Chapter Two. Vector Spaces

Proof. We will show that (1) =⇒ (2), that (2) =⇒ (3), and ﬁnally that

(3) =⇒ (1). For these arguments, observe that we can pass from a combination of w’s to a combination of β’s d1 w1 + · · · + dk wk = d1 (c1,1 β1,1 + · · · + c1,n1 β1,n1 ) + · · · + dk (ck,1 βk,1 + · · · + ck,nk βk,nk ) = d1 c1,1 · β1,1 + · · · + dk ck,nk · βk,nk (∗)

and vice versa. For (1) =⇒ (2), assume that all decompositions are unique. We will show that B1 · · · Bk spans the space and is linearly independent. It spans the space because the assumption that V = W1 + · · · + Wk means that every v can be expressed as v = w1 + · · · + wk , which translates by equation (∗) to an expression of v as a linear combination of the β’s from the concatenation. For linear independence, consider this linear relationship. 0 = c1,1 β1,1 + · · · + ck,nk βk,nk Regroup as in (∗) (that is, take d1 , . . . , dk to be 1 and move from bottom to top) to get the decomposition 0 = w1 + · · · + wk . Because of the assumption that decompositions are unique, and because the zero vector obviously has the decomposition 0 = 0 + · · · + 0, we now have that each wi is the zero vector. This means that ci,1 βi,1 + · · · + ci,ni βi,ni = 0. Thus, since each Bi is a basis, we have the desired conclusion that all of the c’s are zero. For (2) =⇒ (3), assume that B1 · · · Bk is a basis for the space. Consider a linear relationship among nonzero vectors from diﬀerent Wi ’s, 0 = · · · + di wi + · · · in order to show that it is trivial. (The relationship is written in this way because we are considering a combination of nonzero vectors from only some of the Wi ’s; for instance, there might not be a w1 in this combination.) As in (∗), 0 = · · ·+di (ci,1 βi,1 +· · ·+ci,ni βi,ni )+· · · = · · ·+di ci,1 ·βi,1 +· · ·+di ci,ni ·βi,ni +· · · and the linear independence of B1 · · · Bk gives that each coeﬃcient di ci,j is zero. Now, wi is a nonzero vector, so at least one of the ci,j ’s is not zero, and thus di is zero. This holds for each di , and therefore the linear relationship is trivial. Finally, for (3) =⇒ (1), assume that, among nonzero vectors from diﬀerent Wi ’s, any linear relationship is trivial. Consider two decompositions of a vector v = w1 + · · · + wk and v = u1 + · · · + uk in order to show that the two are the same. We have 0 = (w1 + · · · + wk ) − (u1 + · · · + uk ) = (w1 − u1 ) + · · · + (wk − uk ) which violates the assumption unless each wi − ui is the zero vector. Hence, decompositions are unique. QED

Section III. Basis and Dimension

135

4.9 Deﬁnition A collection of subspaces {W1 , . . . , Wk } is independent if no nonzero vector from any Wi is a linear combination of vectors from the other subspaces W1 , . . . , Wi−1 , Wi+1 , . . . , Wk . 4.10 Deﬁnition A vector space V is the direct sum (or internal direct sum) of its subspaces W1 , . . . , Wk if V = W1 + W2 + · · · + Wk and the collection {W1 , . . . , Wk } is independent. We write V = W1 ⊕ W2 ⊕ . . . ⊕ Wk . 4.11 Example The benchmark model ﬁts: R3 = x-axis ⊕ y-axis ⊕ z-axis. 4.12 Example The space of 2×2 matrices is this direct sum. { a 0 0 d a, d ∈ R} ⊕ { 0 0 b 0 b ∈ R} ⊕ { 0 c 0 0 c ∈ R}

It is the direct sum of subspaces in many other ways as well; direct sum decompositions are not unique. 4.13 Corollary The dimension of a direct sum is the sum of the dimensions of its summands.
Proof. In Lemma 4.8, the number of basis vectors in the concatenation equals

the sum of the number of vectors in the subbases that make up the concatenation. QED The special case of two subspaces is worth mentioning separately. 4.14 Deﬁnition When a vector space is the direct sum of two of its subspaces, then they are said to be complements. 4.15 Lemma A vector space V is the direct sum of two of its subspaces W1 and W2 if and only if it is the sum of the two V = W1 +W2 and their intersection is trivial W1 ∩ W2 = {0 }.
Proof. Suppose ﬁrst that V = W1 ⊕ W2 . By deﬁnition, V is the sum of the

two. To show that the two have a trivial intersection, let v be a vector from W1 ∩ W2 and consider the equation v = v. On the left side of that equation is a member of W1 , and on the right side is a linear combination of members (actually, of only one member) of W2 . But the independence of the spaces then implies that v = 0, as desired. For the other direction, suppose that V is the sum of two spaces with a trivial intersection. To show that V is a direct sum of the two, we need only show that the spaces are independent — no nonzero member of the ﬁrst is expressible as a linear combination of members of the second, and vice versa. This is true because any relationship w1 = c1 w2,1 + · · · + dk w2,k (with w1 ∈ W1 and w2,j ∈ W2 for all j) shows that the vector on the left is also in W2 , since the right side is a combination of members of W2 . The intersection of these two spaces is trivial, so w1 = 0. The same argument works for any w2 . QED

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Chapter Two. Vector Spaces

4.16 Example In the space R2 , the x-axis and the y-axis are complements, that is, R2 = x-axis ⊕ y-axis. A space can have more than one pair of complementary subspaces; another pair here are the subspaces consisting of the lines y = x and y = 2x. 4.17 Example In the space F = {a cos θ + b sin θ a, b ∈ R}, the subspaces W1 = {a cos θ a ∈ R} and W2 = {b sin θ b ∈ R} are complements. In addition to the fact that a space like F can have more than one pair of complementary subspaces, inside of the space a single subspace like W1 can have more than one complement — another complement of W1 is W3 = {b sin θ + b cos θ b ∈ R}. 4.18 Example In R3 , the xy-plane and the yz-planes are not complements, which is the point of the discussion following Example 4.4. One complement of the xy-plane is the z-axis. A complement of the yz-plane is the line through (1, 1, 1). 4.19 Example Following Lemma 4.15, here is a natural question: is the simple sum V = W1 + · · · + Wk also a direct sum if and only if the intersection of the subspaces is trivial? The answer is that if there are more than two subspaces then having a trivial intersection is not enough to guarantee unique decomposition (i.e., is not enough to ensure that the spaces are independent). In R3 , let W1 be the x-axis, let W2 be the y-axis, and let W3 be this.   q W3 = {q  q, r ∈ R} r The check that R3 = W1 + W2 + W3 is easy. The intersection W1 ∩ W2 ∩ W3 is trivial, but decompositions aren’t unique.               x 0 0 x x−y 0 y y  = 0 + y − x + x =  0  + 0 + y  z 0 0 z 0 0 z (This example also shows that this requirement is also not enough: that all pairwise intersections of the subspaces be trivial. See Exercise 30.) In this subsection we have seen two ways to regard a space as built up from component parts. Both are useful; in particular, in this book the direct sum deﬁnition is needed to do the Jordan Form construction in the ﬁfth chapter. Exercises
4.20 Decide if R2 is the direct sum of each W1 and W2 . x x (a) W1 = { x ∈ R}, W2 = { x ∈ R} 0 x (b) W1 = { s s s ∈ R}, W2 = { s 1.1s s ∈ R}

(c) W1 = R2 , W2 = {0}

Section III. Basis and Dimension
t t x 0

137

(d) W1 = W2 = { (e) W1 = { 1 0 +

t ∈ R} x ∈ R}, W2 = { −1 0 + 0 y y ∈ R}

4.21 Show that R3 is the direct sum of the xy-plane with each of these. (a) the z-axis (b) the line z z ∈ R} { z z 4.22 Is P2 the direct sum of {a + bx2 a, b ∈ R} and {cx c ∈ R}? 4.23 In Pn , the even polynomials are the members of this set E = {p ∈ Pn p(−x) = p(x) for all x} and the odd polynomials are the members of this set. O = {p ∈ Pn p(−x) = −p(x) for all x} Show that these are complementary subspaces. 4.24 Which of these subspaces of R3 W1 : the x-axis, W2 : the y-axis, W3 : the z-axis, W4 : the plane x + y + z = 0, W5 : the yz-plane can be combined to (a) sum to R3 ? (b) direct sum to R3 ? 4.25 Show that Pn = {a0 a0 ∈ R} ⊕ . . . ⊕ {an xn an ∈ R}. 4.26 What is W1 + W2 if W1 ⊆ W2 ? 4.27 Does Example 4.5 generalize? That is, is this true or false: if a vector space V has a basis β1 , . . . , βn then it is the direct sum of the spans of the one-dimensional subspaces V = [{β1 }] ⊕ . . . ⊕ [{βn }]? 4.28 Can R4 be decomposed as a direct sum in two diﬀerent ways? Can R1 ? 4.29 This exercise makes the notation of writing ‘+’ between sets more natural. Prove that, where W1 , . . . , Wk are subspaces of a vector space, W1 + · · · + Wk = {w1 + w2 + · · · + wk w1 ∈ W1 , . . . , wk ∈ Wk }, and so the sum of subspaces is the subspace of all sums. 4.30 (Refer to Example 4.19. This exercise shows that the requirement that pariwise intersections be trivial is genuinely stronger than the requirement only that the intersection of all of the subspaces be trivial.) Give a vector space and three subspaces W1 , W2 , and W3 such that the space is the sum of the subspaces, the intersection of all three subspaces W1 ∩ W2 ∩ W3 is trivial, but the pairwise intersections W1 ∩ W2 , W1 ∩ W3 , and W2 ∩ W3 are nontrivial. 4.31 Prove that if V = W1 ⊕ . . . ⊕ Wk then Wi ∩ Wj is trivial whenever i = j. This shows that the ﬁrst half of the proof of Lemma 4.15 extends to the case of more than two subspaces. (Example 4.19 shows that this implication does not reverse; the other half does not extend.) 4.32 Recall that no linearly independent set contains the zero vector. Can an independent set of subspaces contain the trivial subspace? 4.33 Does every subspace have a complement? 4.34 Let W1 , W2 be subspaces of a vector space. (a) Assume that the set S1 spans W1 , and that the set S2 spans W2 . Can S1 ∪ S2 span W1 + W2 ? Must it?

138

Chapter Two. Vector Spaces

(b) Assume that S1 is a linearly independent subset of W1 and that S2 is a linearly independent subset of W2 . Can S1 ∪ S2 be a linearly independent subset of W1 + W2 ? Must it? 4.35 When a vector space is decomposed as a direct sum, the dimensions of the subspaces add to the dimension of the space. The situation with a space that is given as the sum of its subspaces is not as simple. This exercise considers the two-subspace special case. (a) For these subspaces of M2×2 ﬁnd W1 ∩ W2 , dim(W1 ∩ W2 ), W1 + W2 , and dim(W1 + W2 ). W1 = { 0 c 0 d c, d ∈ R} W2 = { 0 c b 0 b, c ∈ R}

(b) Suppose that U and W are subspaces of a vector space. Suppose that the sequence β1 , . . . , βk is a basis for U ∩ W . Finally, suppose that the prior sequence has been expanded to give a sequence µ1 , . . . , µj , β1 , . . . , βk that is a basis for U , and a sequence β1 , . . . , βk , ω1 , . . . , ωp that is a basis for W . Prove that this sequence µ1 , . . . , µj , β1 , . . . , βk , ω1 , . . . , ωp is a basis for for the sum U + W . (c) Conclude that dim(U + W ) = dim(U ) + dim(W ) − dim(U ∩ W ). (d) Let W1 and W2 be eight-dimensional subspaces of a ten-dimensional space. List all values possible for dim(W1 ∩ W2 ). 4.36 Let V = W1 ⊕ . . . ⊕ Wk and for each index i suppose that Si is a linearly independent subset of Wi . Prove that the union of the Si ’s is linearly independent. 4.37 A matrix is symmetric if for each pair of indices i and j, the i, j entry equals the j, i entry. A matrix is antisymmetric if each i, j entry is the negative of the j, i entry. (a) Give a symmetric 2×2 matrix and an antisymmetric 2×2 matrix. (Remark. For the second one, be careful about the entries on the diagional.) (b) What is the relationship between a square symmetric matrix and its transpose? Between a square antisymmetric matrix and its transpose? (c) Show that Mn×n is the direct sum of the space of symmetric matrices and the space of antisymmetric matrices. 4.38 Let W1 , W2 , W3 be subspaces of a vector space. Prove that (W1 ∩ W2 ) + (W1 ∩ W3 ) ⊆ W1 ∩ (W2 + W3 ). Does the inclusion reverse? 4.39 The example of the x-axis and the y-axis in R2 shows that W1 ⊕ W2 = V does not imply that W1 ∪ W2 = V . Can W1 ⊕ W2 = V and W1 ∪ W2 = V happen? 4.40 Consider Corollary 4.13. Does it work both ways — that is, supposing that V = W1 + · · · + Wk , is V = W1 ⊕ . . . ⊕ Wk if and only if dim(V ) = dim(W1 ) + · · · + dim(Wk )? 4.41 We know that if V = W1 ⊕ W2 then there is a basis for V that splits into a basis for W1 and a basis for W2 . Can we make the stronger statement that every basis for V splits into a basis for W1 and a basis for W2 ? 4.42 We can ask about the algebra of the ‘+’ operation. (a) Is it commutative; is W1 + W2 = W2 + W1 ? (b) Is it associative; is (W1 + W2 ) + W3 = W1 + (W2 + W3 )? (c) Let W be a subspace of some vector space. Show that W + W = W . (d) Must there be an identity element, a subspace I such that I + W = W + I = W for all subspaces W ?

Section III. Basis and Dimension

139

(e) Does left-cancelation hold: if W1 + W2 = W1 + W3 then W2 = W3 ? Right cancelation? 4.43 Consider the algebraic properties of the direct sum operation. (a) Does direct sum commute: does V = W1 ⊕ W2 imply that V = W2 ⊕ W1 ? (b) Prove that direct sum is associative: (W1 ⊕ W2 ) ⊕ W3 = W1 ⊕ (W2 ⊕ W3 ). (c) Show that R3 is the direct sum of the three axes (the relevance here is that by the previous item, we needn’t specify which two of the threee axes are combined ﬁrst). (d) Does the direct sum operation left-cancel: does W1 ⊕ W2 = W1 ⊕ W3 imply W2 = W3 ? Does it right-cancel? (e) There is an identity element with respect to this operation. Find it. (f ) Do some, or all, subspaces have inverses with respect to this operation: is there a subspace W of some vector space such that there is a subspace U with the property that U ⊕ W equals the identity element from the prior item?

140

Chapter Two. Vector Spaces

Topic: Fields
Linear combinations involving only fractions or only integers are much easier for computations than combinations involving real numbers, because computing with irrational numbers is awkward. Could other number systems, like the rationals or the integers, work in the place of R in the deﬁnition of a vector space? Yes and no. If we take “work” to mean that the results of this chapter remain true then an analysis of which properties of the reals we have used in this chapter gives the following list of conditions an algebraic system needs in order to “work” in the place of R. Deﬁnition. A ﬁeld is a set F with two operations ‘+’ and ‘·’ such that (1) for any a, b ∈ F the result of a + b is in F and • a+b=b+a • if c ∈ F then a + (b + c) = (a + b) + c (2) for any a, b ∈ F the result of a · b is in F and • a·b=b·a • if c ∈ F then a · (b · c) = (a · b) · c (3) if a, b, c ∈ F then a · (b + c) = a · b + a · c (4) there is an element 0 ∈ F such that • if a ∈ F then a + 0 = a • for each a ∈ F there is an element −a ∈ F such that (−a) + a = 0 (5) there is an element 1 ∈ F such that • if a ∈ F then a · 1 = a • for each element a = 0 of F there is an element a−1 ∈ F such that a−1 · a = 1. The number system consisting of the set of real numbers along with the usual addition and multiplication operation is a ﬁeld, naturally. Another ﬁeld is the set of rational numbers with its usual addition and multiplication operations. An example of an algebraic structure that is not a ﬁeld is the integer number system—it fails the ﬁnal condition. Some examples are surprising. The set {0, 1} under these operations: + 0 1 is a ﬁeld (see Exercise 4). 0 0 1 1 1 0 · 0 1 0 0 0 1 0 1

Topic: Fields

141

We could develop Linear Algebra as the theory of vector spaces with scalars from an arbitrary ﬁeld, instead of sticking to taking the scalars only from R. In that case, almost all of the statements in this book would carry over by replacing ‘R’ with ‘F’, and thus by taking coeﬃcients, vector entries, and matrix entries to be elements of F (“almost” because statements involving distances or angles are exceptions). Here are some examples; each applies to a vector space V over a ﬁeld F. ∗ For any v ∈ V and a ∈ F, (i) 0 · v = 0, and (ii) −1 · v + v = 0, and (iii) a · 0 = 0. ∗ The span (the set of linear combinations) of a subset of V is a subspace of V . ∗ Any subset of a linearly independent set is also linearly independent. ∗ In a ﬁnite-dimensional vector space, any two bases have the same number of elements. (Even statements that don’t explicitly mention F use ﬁeld properties in their proof.) We won’t develop vector spaces in this more general setting because the additional abstraction can be a distraction. The ideas we want to bring out already appear when we stick to the reals. The only exception is in Chapter Five. In that chapter we must factor polynomials, so we will switch to considering vector spaces over the ﬁeld of complex numbers. We will discuss this more, including a brief review of complex arithmetic, when we get there. Exercises
1 Show that the real numbers form a ﬁeld. 2 Prove that these are ﬁelds. (a) The rational numbers Q (b) The complex numbers C 3 Give an example that shows that the integer number system is not a ﬁeld. 4 Consider the set {0, 1} subject to the operations given above. Show that it is a ﬁeld. 5 Give suitable operations to make the set {0, 1, 2} a ﬁeld.

142

Chapter Two. Vector Spaces

Topic: Crystals
Everyone has noticed that table salt comes in little cubes.

Remarkably, the explanation for the cubical external shape is the simplest one: the internal shape, the way the atoms lie, is also cubical. The internal structure is pictured below. Salt is sodium cloride, and the small spheres shown are sodium while the big ones are cloride. To simplify the view, it only shows the sodiums and clorides on the front, top, and right.

The specks of salt that we see when we spread a little out on the table consist of many repetitions of this fundamental unit. That is, these cubes of atoms stack up to make the larger cubical structure that we see. A solid, such as table salt, with a regular internal structure is a crystal. We can restrict our attention to the front face. There, we have the square repeated many times.

˚ The distance between the corners of the square cell is about 3.34 Angstroms (an ˚ngstrom is 10−10 meters). Obviously that unit is unwieldly. Instead, the A thing to do is to take as a unit the length of each side of the square. That is, we naturally adopt this basis. 3.34 0 , 0 3.34 Then we can describe, say, the corner in the upper right of the picture above as 3 β1 + 2 β2 .

Topic: Crystals

143

Another crystal from everyday experience is pencil lead. It is graphite, formed from carbon atoms arranged in this shape.

This is a single plane of graphite. A piece of graphite consists of many of these planes layered in a stack. (The chemical bonds between the planes are much weaker than the bonds inside the planes, which explains why pencils write — the graphite can be sheared so that the planes slide oﬀ and are left on the paper.) We can get a convienent unit of length by decomposing the hexagonal ring into three regions that are rotations of this unit cell.

Then a natural basis consists of the vectors that form the sides of that unit cell. The distance along the bottom and slant is 1.42 ˚ngstroms, so this A 1.42 1.23 , 0 .71 is a good basis. The selection of convienent bases extends to three dimensions. Another familiar crystal formed from carbon is diamond. Like table salt, it is built from cubes, but the structure inside each cube is more complicated than salt’s. In addition to carbons at each corner,

there are carbons in the middle of each face.

144

Chapter Two. Vector Spaces

(To show the added face carbons clearly, the corner carbons have been reduced to dots.) There are also four more carbons inside the cube, two that are a quarter of the way up from the bottom and two that are a quarter of the way down from the top.

(As before, carbons shown earlier have been reduced here to dots.) The distance along any edge of the cube is 2.18 ˚ngstroms. Thus, a natural basis for A describing the locations of the carbons, and the bonds between them, is this.       0 0 2.18  0  , 2.18 ,  0  0 2.18 0 Even the few examples given here show that the structures of crystals is complicated enough that some organized system to give the locations of the atoms, and how they are chemically bound, is needed. One tool for that organization is a convienent basis. This application of bases is simple, but it shows a context where the idea arises naturally. The work in this chapter just takes this simple idea and develops it. Exercises
1 How many fundamental regions are there in one face of a speck of salt? (With a ruler, we can estimate that face is a square that is 0.1 cm on a side.) 2 In the graphite picture, imagine that we are interested in a point 5.67 ˚ngstroms A over and 3.14 ˚ngstroms up from the origin. A (a) Express that point in terms of the basis given for graphite. (b) How many hexagonal shapes away is this point from the origin? (c) Express that point in terms of a second basis, where the ﬁrst basis vector is the same, but the second is perpendicular to the ﬁrst (going up the plane) and of the same length. 3 Give the locations of the atoms in the diamond cube both in terms of the basis, and in ˚ngstroms. A 4 This illustrates how the dimensions of a unit cell could be computed from the shape in which a substance crystalizes ([Ebbing], p. 462). (a) Recall that there are 6.022×1023 atoms in a mole (this is Avagadro’s number). From that, and the fact that platinum has a mass of 195.08 grams per mole, calculate the mass of each atom. (b) Platinum crystalizes in a face-centered cubic lattice with atoms at each lattice point, that is, it looks like the middle picture given above for the diamond crystal. Find the number of platinums per unit cell (hint: sum the fractions of platinums that are inside of a single cell). (c) From that, ﬁnd the mass of a unit cell. (d) Platinum crystal has a density of 21.45 grams per cubic centimeter. From this, and the mass of a unit cell, calculate the volume of a unit cell.

Topic: Crystals
(e) Find the length of each edge. (f ) Describe a natural three-dimensional basis.

145

146

Chapter Two. Vector Spaces

Imagine that a Political Science class studying the American presidential process holds a mock election. Members of the class are asked to rank, from most preferred to least preferred, the nominees from the Democratic Party, the Republican Party, and the Third Party, and this is the result (> means ‘is preferred to’). preference order Democrat > Republican > Third Democrat > Third > Republican Republican > Democrat > Third Republican > Third > Democrat Third > Democrat > Republican Third > Republican > Democrat total number with that preference 5 4 2 8 8 2 29

What is the preference of the group as a whole? Overall, the group prefers the Democrat to the Republican by ﬁve votes; seventeen voters ranked the Democrat above the Republican versus twelve the other way. And, overall, the group prefers the Republican to the Third’s nominee, ﬁfteen to fourteen. But, strangely enough, the group also prefers the Third to the Democrat, eighteen to eleven.
Democrat
7 voters 5 voters

Third

Republican
1 voter

This is an example of a voting paradox , speciﬁcally, a majority cycle. Voting paradoxes are studied in part because of their implications for practical politics. For instance, the instructor can manipulate the class into choosing the Democrat as the overall winner by ﬁrst asking the class to choose between the Republican and the Third, and then asking the class to choose between the winner of that contest, the Republican, and the Democrat. By similar manipulations, any of the other two candidates can be made to come out as the winner. (In this Topic we will stick to three-candidate elections, but similar results apply to larger elections.) Voting paradoxes are also studied simply because they are mathematically interesting. One interesting aspect is that the group’s overall majority cycle occurs despite that each single voters’s preference list is rational —in a straightline order. That is, the majority cycle seems to arise in the aggregate, without being present in the elements of that aggregate, the preference lists. Recently,

147

however, linear algebra has been used [Zwicker] to argue that a tendency toward cyclic preference is actually present in each voter’s list, and that it surfaces when there is more adding of the tendency than cancelling. For this argument, abbreviating the choices as D, R, and T , we can describe how a voter with preference order D > R > T contributes to the above cycle.
D −1 voter T 1 voter 1 voter R

(The negative sign is here because the arrow describes T as preferred to D, but this voter likes them the other way.) The descriptions for the other preference lists are in the table on page 148. Now, to conduct the election we linearly combine these descriptions; for instance, the Political Science mock election
D
−1 1 −1

D
1 1

D
−1

T
1

R

+4·

T
−1

R

+ ··· + 2 ·

T
−1

R

yields the circular group preference shown earlier. Of course, taking linear combinations is linear algebra. The above cycle notation is suggestive but inconvienent, so we temporarily switch to using column vectors by starting at the D and taking the numbers from the cycle in counterclockwise order. Thus, the mock election and a single D > R > T vote are represented in this way.     −1 7 1 and  1  1 5 We will decompose vote vectors into two parts, one cyclic and the other acyclic. For the ﬁrst part, we say that a vector is purely cyclic if it is in this subspace of R3 .     k 1 C = {k  k ∈ R} = {k · 1 k ∈ R} k 1 For the second part, consider the subspace (see Exercise 6) of vectors that are perpendicular to all of the vectors in C.       c1 c1 k = {c2  c2  k  = 0 for all k ∈ R} c3 c3 k       c1 −1 −1 = {c2  c1 + c2 + c3 = 0} = {c2  1  + c3  0  c2 , c3 ∈ R} c3 0 1

C⊥

148

Chapter Two. Vector Spaces

(Read that aloud as “C perp”.) So we are led to this basis for R3 .       1 −1 −1 1 ,  1  ,  0  1 0 1 We can represent votes with respect to this basis, and thereby decompose them into a cyclic part and an acyclic part. (Note for readers who have covered the optional section in this chapter: that is, the space is the direct sum of C and C ⊥ .) For example, consider the D > R > T voter discussed above. The representation in terms of the basis is easily found, c1 − c2 − c3 = −1 c1 + c2 = 1 c1 + c3 = 1
−ρ1 +ρ2 (−1/2)ρ2 +ρ3 −ρ1 +ρ3

−→

−→

c1 − c2 − 2c2 +

c3 = −1 c3 = 2 (3/2)c3 = 1

so that c1 = 1/3, c2 = 2/3, and c3 = 2/3. Then             1 −1 −1 1/3 −4/3 −1 1   2   2       1 = · 1 + · 1 + · 0 = 1/3 + 2/3  3 3 3 1 0 1 1/3 2/3 1 gives the desired decomposition into a cyclic part and and an acyclic part.
D
−1 1 1/3

D
1/3 −4/3

D
2/3

T
1

R

=

T
1/3

R

+

T
2/3

R

Thus, this D > R > T voter’s rational preference list can indeed be seen to have a cyclic part. The T > R > D voter is opposite to the one just considered in that the ‘>’ symbols are reversed. This voter’s decomposition
D
1 −1 −1/3

D
−1/3 4/3

D
−2/3

T
−1

R

=

T
−1/3

R

+

T
−2/3

R

shows that these opposite preferences have decompositions that are opposite. We say that the ﬁrst voter has positive spin since the cycle part is with the direction we have chosen for the arrows, while the second voter’s spin is negative. The fact that that these opposite voters cancel each other is reﬂected in the fact that their vote vectors add to zero. This suggests an alternate way to tally an election. We could ﬁrst cancel as many opposite preference lists as possible, and then determine the outcome by adding the remaining lists. The rows of the table below contain the three pairs of opposite preference lists. The columns group those pairs by spin. For instance, the ﬁrst row contains the two voters just considered.

149

positive spin Democrat > Republican > Third
D
−1 1 1/3

negative spin Third > Republican > Democrat
D D
2/3 1 −1 −1/3

D
1/3 −4/3

D
−1/3 4/3

D
−2/3

T
1

R

=

T
1/3

R

+

T
2/3

R

T
−1

R

=

T
−1/3

R

+

T
−2/3

R

Republican > Third > Democrat
D
1 −1 1/3

Democrat > Third > Republican
D D
−4/3 −1 1 −1/3

D
1/3 2/3

D
−1/3−2/3

D
4/3

T
1

R

=

T
1/3

R

+

T
2/3

R

T
−1

R

=

T
−1/3

R

+

T
−2/3

R

Third > Democrat > Republican
D
1 1 1/3

Republican > Democrat > Third
D D
2/3 −1 −1 −1/3

D
1/3 2/3

D
−1/3−2/3

D
−2/3

T
−1

R

=

T
1/3

R

+

T
−4/3

R

T
1

R

=

T
−1/3

R

+

T
4/3

R

If we conduct the election as just described then after the cancellation of as many opposite pairs of voters as possible, there will be left three sets of preference lists, one set from the ﬁrst row, one set from the second row, and one set from the third row. We will ﬁnish by proving that a voting paradox can happen only if the spins of these three sets are in the same direction. That is, for a voting paradox to occur, the three remaining sets must all come from the left of the table or all come from the right (see Exercise 3). This shows that there is some connection between the majority cycle and the decomposition that we are using—a voting paradox can happen only when the tendencies toward cyclic preference reinforce each other. For the proof, assume that opposite preference orders have been cancelled, and we are left with one set of preference lists from each of the three rows. Consider the sum of these three (here, the numbers a, b, and c could be positive, negative, or zero).
D
−a a b

D
−b c

D
c −a + b + c

D
a−b+c

T
a

R

+

T
b

R

+

T
−c

R

=

T

R

a+b−c

A voting paradox occurs when the three numbers on the right, a − b + c and a + b − c and −a + b + c, are all nonnegative or all nonpositive. On the left, at least two of the three numbers, a and b and c, are both nonnegative or both nonpositive. We can assume that they are a and b. That makes four cases: the cycle is nonnegative and a and b are nonnegative, the cycle is nonpositive and a and b are nonpositive, etc. We will do only the ﬁrst case, since the second is similar and the other two are also easy. So assume that the cycle is nonnegative and that a and b are nonnegative. The conditions 0 ≤ a − b + c and 0 ≤ −a + b + c add to give that 0 ≤ 2c, which implies that c is also nonnegative, as desired. That ends the proof. This result says only that having all three spin in the same direction is a necessary condition for a majority cycle. It is not suﬃcient; see Exercise 4.

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Chapter Two. Vector Spaces

Voting theory and associated topics are the subject of current research. There are many intriguing results, most notably the one produced by K. Arrow [Arrow], who won the Nobel Prize in part for this work, showing that no voting system is entirely fair (for a reasonable deﬁnition of “fair”). For more information, some good introductory articles are [Gardner, 1970], [Gardner, 1974], [Gardner, 1980], and [Neimi & Riker]. A quite readable recent book is [Taylor]. The long list of cases from recent American political history given in [Poundstone] show that manipulation of these paradoxes is routine in practice (and the author proposes a solution). This Topic is largely drawn from [Zwicker]. (Author’s Note: I would like to thank Professor Zwicker for his kind and illuminating discussions.) Exercises
1 Here is a reasonable way in which a voter could have a cyclic preference. Suppose that this voter ranks each candidate on each of three criteria. (a) Draw up a table with the rows labelled ‘Democrat’, ‘Republican’, and ‘Third’, and the columns labelled ‘character’, ‘experience’, and ‘policies’. Inside each column, rank some candidate as most preferred, rank another as in the middle, and rank the remaining one as least preferred. (b) In this ranking, is the Democrat preferred to the Republican in (at least) two out of three criteria, or vice versa? Is the Republican preferred to the Third? (c) Does the table that was just constructed have a cyclic preference order? If not, make one that does. So it is possible for a voter to have a cyclic preference among candidates. The paradox described above, however, is that even if each voter has a straight-line preference list, a cyclic preference can still arise for the entire group. 2 Compute the values in the table of decompositions. 3 Do the cancellations of opposite preference orders for the Political Science class’s mock election. Are all the remaining preferences from the left three rows of the table or from the right? 4 The necessary condition that is proved above—a voting paradox can happen only if all three preference lists remaining after cancellation have the same spin—is not also suﬃcient. (a) Continuing the positive cycle case considered in the proof, use the two inequalities 0 ≤ a − b + c and 0 ≤ −a + b + c to show that |a − b| ≤ c. (b) Also show that c ≤ a + b, and hence that |a − b| ≤ c ≤ a + b. (c) Give an example of a vote where there is a majority cycle, and addition of one more voter with the same spin causes the cycle to go away. (d) Can the opposite happen; can addition of one voter with a “wrong” spin cause a cycle to appear? (e) Give a condition that is both necessary and suﬃcient to get a majority cycle. 5 A one-voter election cannot have a majority cycle because of the requirement that we’ve imposed that the voter’s list must be rational. (a) Show that a two-voter election may have a majority cycle. (We consider the group preference a majority cycle if all three group totals are nonnegative or if all three are nonpositive—that is, we allow some zero’s in the group preference.) (b) Show that for any number of voters greater than one, there is an election involving that many voters that results in a majority cycle.

151

6 Let U be a subspace of R3 . Prove that the set U ⊥ = {v v u = 0 for all u ∈ U } of vectors that are perpendicular to each vector in U is also a subspace of R3 .

152

Chapter Two. Vector Spaces

Topic: Dimensional Analysis
“You can’t add apples and oranges,” the old saying goes. It reﬂects our experience that in applications the quantities have units and keeping track of those units is worthwhile. Everyone has done calculations such as this one that use the units as a check. 60 min hr day sec sec · 60 · 24 · 365 = 31 536 000 min hr day year year

However, the idea of including the units can be taken beyond bookkeeping. It can be used to draw conclusions about what relationships are possible among the physical quantities. To start, consider the physics equation: distance = 16 · (time)2 . If the distance is in feet and the time is in seconds then this is a true statement about falling bodies. However it is not correct in other unit systems; for instance, it is not correct in the meter-second system. We can ﬁx that by making the 16 a dimensional constant. ft · (time)2 dist = 16 sec2 For instance, the above equation holds in the yard-second system. distance in yards = 16 (1/3) yd 16 yd · (time in sec)2 = · (time in sec)2 2 sec 3 sec2

So our ﬁrst point is that by “including the units” we mean that we are restricting our attention to equations that use dimensional constants. By using dimensional constants, we can be vague about units and say only that all quantities are measured in combinations of some units of length L, mass M , and time T . We shall refer to these three as dimensions (these are the only three dimensions that we shall need in this Topic). For instance, velocity could be measured in feet/second or fathoms/hour, but in all events it involves some unit of length divided by some unit of time so the dimensional formula of velocity is L/T . Similarly, the dimensional formula of density is M/L3 . We shall prefer using negative exponents over the fraction bars and we shall include the dimensions with a zero exponent, that is, we shall write the dimensional formula of velocity as L1 M 0 T −1 and that of density as L−3 M 1 T 0 . In this context, “You can’t add apples to oranges” becomes the advice to check that all of an equation’s terms have the same dimensional formula. An example is this version of the falling body equation: d − gt2 = 0. The dimensional formula of the d term is L1 M 0 T 0 . For the other term, the dimensional formula of g is L1 M 0 T −2 (g is the dimensional constant given above as 16 ft/sec2 ) and the dimensional formula of t is L0 M 0 T 1 , so that of the entire gt2 term is L1 M 0 T −2 (L0 M 0 T 1 )2 = L1 M 0 T 0 . Thus the two terms have the same dimensional formula. An equation with this property is dimensionally homogeneous. Quantities with dimensional formula L0 M 0 T 0 are dimensionless. For example, we measure an angle by taking the ratio of the subtended arc to the radius

Topic: Dimensional Analysis

153

arc r

which is the ratio of a length to a length L1 M 0 T 0 /L1 M 0 T 0 and thus angles have the dimensional formula L0 M 0 T 0 . The classic example of using the units for more than bookkeeping, using them to draw conclusions, considers the formula for the period of a pendulum. p = –some expression involving the length of the string, etc.– The period is in units of time L0 M 0 T 1 . So the quantities on the other side of the equation must have dimensional formulas that combine in such a way that their L’s and M ’s cancel and only a single T remains. The table on page 154 has the quantities that an experienced investigator would consider possibly relevant. The only dimensional formulas involving L are for the length of the string and the acceleration due to gravity. For the L’s of these two to cancel, when they appear in the equation they must be in ratio, e.g., as ( /g)2 , or as cos( /g), or as ( /g)−1 . Therefore the period is a function of /g. This is a remarkable result: with a pencil and paper analysis, before we ever took out the pendulum and made measurements, we have determined something about the relationship among the quantities. To do dimensional analysis systematically, we need to know two things (arguments for these are in [Bridgman], Chapter II and IV). The ﬁrst is that each equation relating physical quantities that we shall see involves a sum of terms, where each term has the form
p m11 mp2 · · · mpk 2 k

for numbers m1 , . . . , mk that measure the quantities. For the second, observe that an easy way to construct a dimensionally homogeneous expression is by taking a product of dimensionless quantities or by adding such dimensionless terms. Buckingham’s Theorem states that any complete relationship among quantities with dimensional formulas can be algebraically manipulated into a form where there is some function f such that f (Π1 , . . . , Πn ) = 0 for a complete set {Π1 , . . . , Πn } of dimensionless products. (The ﬁrst example below describes what makes a set of dimensionless products ‘complete’.) We usually want to express one of the quantities, m1 for instance, in terms of the others, and for that we will assume that the above equality can be rewritten ˆ m1 = m−p2 · · · m−pk · f (Π2 , . . . , Πn ) 2 k where Π1 = m1 mp2 · · · mpk is dimensionless and the products Π2 , . . . , Πn don’t 2 k ˆ involve m1 (as with f , here f is just some function, this time of n−1 arguments). Thus, to do dimensional analysis we should ﬁnd which dimensionless products are possible. For example, consider again the formula for a pendulum’s period.

154

Chapter Two. Vector Spaces dimensional formula L0 M 0 T 1 L1 M 0 T 0 L0 M 1 T 0 L1 M 0 T −2 L0 M 0 T 0

quantity period p length of string mass of bob m acceleration due to gravity g arc of swing θ

By the ﬁrst fact cited above, we expect the formula to have (possibly sums of terms of) the form pp1 p2 mp3 g p4 θp5 . To use the second fact, to ﬁnd which combinations of the powers p1 , . . . , p5 yield dimensionless products, consider this equation. (L0 M 0 T 1 )p1 (L1 M 0 T 0 )p2 (L0 M 1 T 0 )p3 (L1 M 0 T −2 )p4 (L0 M 0 T 0 )p5 = L0 M 0 T 0 It gives three conditions on the powers. p2 p3 p1 + p4 = 0 =0 − 2p4 = 0

Note that p3 is 0 and so the mass of the bob does not aﬀect the period. Gaussian reduction and parametrization of that system gives this       p1 0 1 p2  −1/2 0       {p3  =  0  p1 + 0 p5 p1 , p5 ∈ R}       p4   1/2  0 1 0 p5 (we’ve taken p1 as one of the parameters in order to express the period in terms of the other quantities). Here is the linear algebra. The set of dimensionless products contains all terms pp1 p2 mp3 ap4 θp5 subject to the conditions above. This set forms a vector space under the ‘+’ operation of multiplying two such products and the ‘·’ operation of raising such a product to the power of the scalar (see Exercise 5). The term ‘complete set of dimensionless products’ in Buckingham’s Theorem means a basis for this vector space. We can get a basis by ﬁrst taking p1 = 1, p5 = 0 and then p1 = 0, p5 = 1. The associated dimensionless products are Π1 = p −1/2 g 1/2 and Π2 = θ. Because the set {Π1 , Π2 } is complete, Buckingham’s Theorem says that p=
1/2 −1/2

g

ˆ · f (θ) =

ˆ /g · f (θ)

ˆ where f is a function that we cannot determine from this analysis (a ﬁrst year physics text will show by other means that for small angles it is approximately ˆ the constant function f (θ) = 2π).

Topic: Dimensional Analysis

155

Thus, analysis of the relationships that are possible between the quantities with the given dimensional formulas has produced a fair amount of information: a pendulum’s period does not depend on the mass of the bob, and it rises with the square root of the length of the string. For the next example we try to determine the period of revolution of two bodies in space orbiting each other under mutual gravitational attraction. An experienced investigator could expect that these are the relevant quantities. quantity period p mean separation r ﬁrst mass m1 second mass m2 grav. constant G dimensional formula L0 M 0 T 1 L1 M 0 T 0 L0 M 1 T 0 L0 M 1 T 0 L3 M −1 T −2

To get the complete set of dimensionless products we consider the equation (L0 M 0 T 1 )p1 (L1 M 0 T 0 )p2 (L0 M 1 T 0 )p3 (L0 M 1 T 0 )p4 (L3 M −1 T −2 )p5 = L0 M 0 T 0 which results in a system p2 p1 with this solution.    0 1 0 −3/2     { 1/2  p1 + −1 p4 p1 , p4 ∈ R}     1  0  0 1/2  As earlier, the linear algebra here is that the set of dimensionless products of these quantities forms a vector space, and we want to produce a basis for that space, a ‘complete’ set of dimensionless products. One such set, gotten from setting p1 = 1 and p4 = 0, and also setting p1 = 0 and p4 = 1 1/2 is {Π1 = pr−3/2 m1 G1/2 , Π2 = m−1 m2 }. With that, Buckingham’s Theorem 1 says that any complete relationship among these quantities is stateable this form. r3/2 −1/2 ˆ ˆ p = r3/2 m1 G−1/2 · f (m−1 m2 ) = √ · f (m2 /m1 ) 1 Gm1 Remark. An important application of the prior formula is when m1 is the mass of the sun and m2 is the mass of a planet. Because m1 is very much greater ˆ than m2 , the argument to f is approximately 0, and we can wonder whether this part of the formula remains approximately constant as m2 varies. One way to see that it does is this. The sun is so much larger than the planet that the + 3p5 = 0 p3 + p4 − p5 = 0 − 2p5 = 0

156

Chapter Two. Vector Spaces

mutual rotation is approximately about the sun’s center. If we vary the planet’s mass m2 by a factor of x (e.g., Venus’s mass is x = 0.815 times Earth’s mass), then the force of attraction is multiplied by x, and x times the force acting on x times the mass gives, since F = ma, the same acceleration, about the same center (approximately). Hence, the orbit will be the same and so its period will be the same, and thus the right side of the above equation also remains ˆ unchanged (approximately). Therefore, f (m2 /m1 ) is approximately constant as m2 varies. This is Kepler’s Third Law: the square of the period of a planet is proportional to the cube of the mean radius of its orbit about the sun. The ﬁnal example was one of the ﬁrst explicit applications of dimensional analysis. Lord Raleigh considered the speed of a wave in deep water and suggested these as the relevant quantities. quantity velocity of the wave v density of the water d acceleration due to gravity g wavelength λ The equation (L1 M 0 T −1 )p1 (L−3 M 1 T 0 )p2 (L1 M 0 T −2 )p3 (L1 M 0 T 0 )p4 = L0 M 0 T 0 gives this system p1 − 3p2 + p3 + p4 = 0 p2 =0 −p1 − 2p3 =0 with this solution space  1  0   { −1/2 p1 p1 ∈ R} −1/2  (as in the pendulum example, one of the quantities d turns out not to be involved in the relationship). There is one dimensionless product, Π1 = vg −1/2 λ−1/2 , and √ ˆ so v is λg times a constant (f is constant since it is a function of no arguments). As the three examples above show, dimensional analysis can bring us far toward expressing the relationship among the quantities. For further reading, the classic reference is [Bridgman]—this brief book is delightful. Another source is [Giordano, Wells, Wilde]. A description of dimensional analysis’s place in modeling is in [Giordano, Jaye, Weir]. Exercises
1 Consider a projectile, launched with initial velocity v0 , at an angle θ. An investigation of this motion might start with the guess that these are the relevant

dimensional formula L1 M 0 T −1 L−3 M 1 T 0 L1 M 0 T −2 L1 M 0 T 0

Topic: Dimensional Analysis
quantities. [de Mestre] quantity horizontal position x vertical position y initial speed v0 angle of launch θ acceleration due to gravity g time t dimensional formula L1 M 0 T 0 L1 M 0 T 0 L1 M 0 T −1 L0 M 0 T 0 L1 M 0 T −2 L0 M 0 T 1

157

2 2 (a) Show that {gt/v0 , gx/v0 , gy/v0 , θ} is a complete set of dimensionless products. (Hint. This can be done by ﬁnding the appropriate free variables in the linear system that arises, but there is a shortcut that uses the properties of a basis.) (b) These two equations of motion for projectiles are familiar: x = v0 cos(θ)t and y = v0 sin(θ)t − (g/2)t2 . Manipulate each to rewrite it as a relationship among the dimensionless products of the prior item. 2 [Einstein] conjectured that the infrared characteristic frequencies of a solid may be determined by the same forces between atoms as determine the solid’s ordanary elastic behavior. The relevant quantities are

quantity characteristic frequency ν compressibility k number of atoms per cubic cm N mass of an atom m

dimensional formula L0 M 0 T −1 L1 M −1 T 2 L−3 M 0 T 0 L0 M 1 T 0

Show that there is one dimensionless product. Conclude that, in any complete relationship among quantities with these dimensional formulas, k is a constant times ν −2 N −1/3 m−1 . This conclusion played an important role in the early study of quantum phenomena. 3 The torque produced by an engine has dimensional formula L2 M 1 T −2 . We may ﬁrst guess that it depends on the engine’s rotation rate (with dimensional formula L0 M 0 T −1 ), and the volume of air displaced (with dimensional formula L3 M 0 T 0 ). [Giordano, Wells, Wilde] (a) Try to ﬁnd a complete set of dimensionless products. What goes wrong? (b) Adjust the guess by adding the density of the air (with dimensional formula L−3 M 1 T 0 ). Now ﬁnd a complete set of dimensionless products. 4 Dominoes falling make a wave. We may conjecture that the wave speed v depends on the the spacing d between the dominoes, the height h of each domino, and the acceleration due to gravity g. [Tilley] (a) Find the dimensional formula for each of the four quantities. (b) Show that {Π1 = h/d, Π2 = dg/v 2 } is a complete set of dimensionless products. (c) Show that if h/d is ﬁxed then the propagation speed is proportional to the square root of d. 5 Prove that the dimensionless products form a vector space under the + operation of multiplying two such products and the · operation of raising such the product to the power of the scalar. (The vector arrows are a precaution against confusion.) That is, prove that, for any particular homogeneous system, this set of products

158
of powers of m1 , . . . , mk {mp1 . . . mkk 1 is a vector space under:
p p

Chapter Two. Vector Spaces

p1 , . . . , pk satisfy the system}
q p +qk

p p m11 . . . mkk +mq1 . . . mkk = m11 +q1 . . . mkk 1

and r·(mp1 . . . mkk ) = mrp1 . . . mk 1 1
p rpk

(assume that all variables represent real numbers). 6 The advice about apples and oranges is not right. Consider the familiar equations for a circle C = 2πr and A = πr2 . (a) Check that C and A have diﬀerent dimensional formulas. (b) Produce an equation that is not dimensionally homogeneous (i.e., it adds apples and oranges) but is nonetheless true of any circle. (c) The prior item asks for an equation that is complete but not dimensionally homogeneous. Produce an equation that is dimensionally homogeneous but not complete. (Just because the old saying isn’t strictly right, doesn’t keep it from being a useful strategy. Dimensional homogeneity is often used as a check on the plausibility of equations used in models. For an argument that any complete equation can easily be made dimensionally homogeneous, see [Bridgman], Chapter I, especially page 15.)

Chapter Three

Maps Between Spaces
I Isomorphisms

In the examples following the deﬁnition of a vector space we developed the intuition that some spaces are “the same” as others. For instance, the space of two-tall column vectors and the space of two-wide row vectors are not equal because their elements — column vectors and row vectors — are not equal, but we have the idea that these spaces diﬀer only in how their elements appear. We will now make this idea precise. This section illustrates a common aspect of a mathematical investigation. With the help of some examples, we’ve gotten an idea. We will next give a formal deﬁnition, and then we will produce some results backing our contention that the deﬁnition captures the idea. We’ve seen this happen already, for instance, in the ﬁrst section of the Vector Space chapter. There, the study of linear systems led us to consider collections closed under linear combinations. We deﬁned such a collection as a vector space, and we followed it with some supporting results. Of course, that deﬁnition wasn’t an end point, instead it led to new insights such as the idea of a basis. Here too, after producing a deﬁnition, and supporting it, we will get two surprises (pleasant ones). First, we will ﬁnd that the deﬁnition applies to some unforeseen, and interesting, cases. Second, the study of the deﬁnition will lead to new ideas. In this way, our investigation will build a momentum.

I.1 Definition and Examples
We start with two examples that suggest the right deﬁnition. 1.1 Example Consider the example mentioned above, the space of two-wide row vectors and the space of two-tall column vectors. They are “the same” in that if we associate the vectors that have the same components, e.g., 1 2 ←→ 159 1 2

160

Chapter Three. Maps Between Spaces

then this correspondence preserves the operations, for instance this addition 1 2 + 3 4 = 4 6 ←→ 1 3 + 2 4 = 4 6

and this scalar multiplication. 5· 1 2 = 5 10 ←→ 5· 1 2 = 5 10

More generally stated, under the correspondence a0 both operations are preserved: a0 and r · a0 a1 = ra0 ra1 ←→ r· a0 a1 = ra0 ra1 a1 + b0 b1 = a0 + b0 a1 + b1 ←→ a0 a1 + b0 b1 = a0 + b0 a1 + b1 a1 ←→ a0 a1

(all of the variables are real numbers). 1.2 Example Another two spaces we can think of as “the same” are P2 , the space of quadratic polynomials, and R3 . A natural correspondence is this. a0 + a1 x + a2 x2   a0 a1  a2   1 (e.g., 1 + 2x + 3x2 ←→ 2) 3

←→

The structure is preserved: corresponding elements add in a corresponding way a0 + a1 x + a2 x2 + b0 + b1 x + b2 x2 (a0 + b0 ) + (a1 + b1 )x + (a2 + b2 )x2       a0 b0 a0 + b0 a1  + b1  = a1 + b1  a2 b2 a2 + b2

←→

and scalar multiplication corresponds also.     a0 ra0 r · a1  = ra1  a2 ra2

r · (a0 + a1 x + a2 x2 ) = (ra0 ) + (ra1 )x + (ra2 )x2

←→

Section I. Isomorphisms

161

1.3 Deﬁnition An isomorphism between two vector spaces V and W is a map f : V → W that (1) is a correspondence: f is one-to-one and onto;∗ (2) preserves structure: if v1 , v2 ∈ V then f (v1 + v2 ) = f (v1 ) + f (v2 ) and if v ∈ V and r ∈ R then f (rv) = r f (v) (we write V ∼ W , read “V is isomorphic to W ”, when such a map exists). = (“Morphism” means map, so “isomorphism” means a map expressing sameness.) 1.4 Example The vector space G = {c1 cos θ + c2 sin θ c1 , c2 ∈ R} of functions of θ is isomorphic to the vector space R2 under this map. c1 cos θ + c2 sin θ −→
f

c1 c2

We will check this by going through the conditions in the deﬁnition. We will ﬁrst verify condition (1), that the map is a correspondence between the sets underlying the spaces. To establish that f is one-to-one, we must prove that f (a) = f (b) only when a = b. If f (a1 cos θ + a2 sin θ) = f (b1 cos θ + b2 sin θ) then, by the deﬁnition of f , a1 a2 = b1 b2

from which we can conclude that a1 = b1 and a2 = b2 because column vectors are equal only when they have equal components. We’ve proved that f (a) = f (b) implies that a = b, which shows that f is one-to-one. To check that f is onto we must check that any member of the codomain R2 is the image of some member of the domain G. But that’s clear — any x y ∈ R2

is the image under f of x cos θ + y sin θ ∈ G. Next we will verify condition (2), that f preserves structure.
∗ More

information on one-to-one and onto maps is in the appendix.

162

Chapter Three. Maps Between Spaces This computation shows that f preserves addition.

f (a1 cos θ + a2 sin θ) + (b1 cos θ + b2 sin θ) = f (a1 + b1 ) cos θ + (a2 + b2 ) sin θ = = a 1 + b1 a 2 + b2 a1 a2 + b1 b2

= f (a1 cos θ + a2 sin θ) + f (b1 cos θ + b2 sin θ) A similar computation shows that f preserves scalar multiplication. f r · (a1 cos θ + a2 sin θ) = f ( ra1 cos θ + ra2 sin θ ) = ra1 ra2 a1 a2

=r·

= r · f (a1 cos θ + a2 sin θ) With that, conditions (1) and (2) are veriﬁed, so we know that f is an isomorphism and we can say that the spaces are isomorphic G ∼ R2 . = 1.5 Example Let V be the space {c1 x + c2 y + c3 z c1 , c2 , c3 ∈ R} of linear combinations of three variables x, y, and z, under the natural addition and scalar multiplication operations. Then V is isomorphic to P2 , the space of quadratic polynomials. To show this we will produce an isomorphism map. There is more than one possibility; for instance, here are four. −→ c1 x + c2 y + c3 z −→ −→ −→
f4 f3 f2 f1

c1 + c2 x + c3 x2 c2 + c3 x + c1 x2 −c1 − c2 x − c3 x2 c1 + (c1 + c2 )x + (c1 + c3 )x2

The ﬁrst map is the more natural correspondence in that it just carries the coeﬃcients over. However, below we shall verify that the second one is an isomorphism, to underline that there are isomorphisms other than just the obvious one (showing that f1 is an isomorphism is Exercise 12). To show that f2 is one-to-one, we will prove that if f2 (c1 x + c2 y + c3 z) = f2 (d1 x + d2 y + d3 z) then c1 x + c2 y + c3 z = d1 x + d2 y + d3 z. The assumption that f2 (c1 x + c2 y + c3 z) = f2 (d1 x + d2 y + d3 z) gives, by the deﬁnition of f2 , that c2 + c3 x + c1 x2 = d2 + d3 x + d1 x2 . Equal polynomials have equal coeﬃcients, so c2 = d2 , c3 = d3 , and c1 = d1 . Thus f2 (c1 x + c2 y + c3 z) = f2 (d1 x + d2 y + d3 z) implies that c1 x + c2 y + c3 z = d1 x + d2 y + d3 z and therefore f2 is one-to-one.

Section I. Isomorphisms

163

The map f2 is onto because any member a + bx + cx2 of the codomain is the image of some member of the domain, namely it is the image of cx + ay + bz. For instance, 2 + 3x − 4x2 is f2 (−4x + 2y + 3z). The computations for structure preservation are like those in the prior example. This map preserves addition f2 (c1 x + c2 y + c3 z) + (d1 x + d2 y + d3 z) = f2 (c1 + d1 )x + (c2 + d2 )y + (c3 + d3 )z = (c2 + d2 ) + (c3 + d3 )x + (c1 + d1 )x2 = (c2 + c3 x + c1 x2 ) + (d2 + d3 x + d1 x2 ) = f2 (c1 x + c2 y + c3 z) + f2 (d1 x + d2 y + d3 z) and scalar multiplication. f2 r · (c1 x + c2 y + c3 z) = f2 (rc1 x + rc2 y + rc3 z) = rc2 + rc3 x + rc1 x2 = r · (c2 + c3 x + c1 x2 ) = r · f2 (c1 x + c2 y + c3 z) Thus f2 is an isomorphism and we write V ∼ P2 . = We are sometimes interested in an isomorphism of a space with itself, called an automorphism. An identity map is an automorphism. The next two examples show that there are others. 1.6 Example A dilation map ds : R2 → R2 that multiplies all vectors by a nonzero scalar s is an automorphism of R2 .
d1.5 (u) u v

−→

d1.5

d1.5 (v)

A rotation or turning map tθ : R2 → R2 that rotates all vectors through an angle θ is an automorphism.
tπ/6 (u)

−→
u

tπ/6

A third type of automorphism of R2 is a map f : R2 → R2 that ﬂips or reﬂects all vectors over a line through the origin.

164

Chapter Three. Maps Between Spaces
f (u) u

−→

f

See Exercise 29. 1.7 Example Consider the space P5 of polynomials of degree 5 or less and the map f that sends a polynomial p(x) to p(x − 1). For instance, under this map x2 → (x−1)2 = x2 −2x+1 and x3 +2x → (x−1)3 +2(x−1) = x3 −3x2 +5x−3. This map is an automorphism of this space; the check is Exercise 21. This isomorphism of P5 with itself does more than just tell us that the space is “the same” as itself. It gives us some insight into the space’s structure. For instance, below is shown a family of parabolas, graphs of members of P5 . Each has a vertex at y = −1, and the left-most one has zeroes at −2.25 and −1.75, the next one has zeroes at −1.25 and −0.75, etc.

p0

p1

Geometrically, the substitution of x − 1 for x in any function’s argument shifts its graph to the right by one. Thus, f (p0 ) = p1 and f ’s action is to shift all of the parabolas to the right by one. Notice that the picture before f is applied is the same as the picture after f is applied, because while each parabola moves to the right, another one comes in from the left to take its place. This also holds true for cubics, etc. So the automorphism f gives us the insight that P5 has a certain horizontal-homogeneity; this space looks the same near x = 1 as near x = 0. As described in the preamble to this section, we will next produce some results supporting the contention that the deﬁnition of isomorphism above captures our intuition of vector spaces being the same. Of course the deﬁnition itself is persuasive: a vector space consists of two components, a set and some structure, and the deﬁnition simply requires that the sets correspond and that the structures correspond also. Also persuasive are the examples above. In particular, Example 1.1, which gives an isomorphism between the space of two-wide row vectors and the space of two-tall column vectors, dramatizes our intuition that isomorphic spaces are the same in all relevant respects. Sometimes people say, where V ∼ W , that “W is just V = painted green” — any diﬀerences are merely cosmetic. Further support for the deﬁnition, in case it is needed, is provided by the following results that, taken together, suggest that all the things of interest in a

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vector space correspond under an isomorphism. Since we studied vector spaces to study linear combinations, “of interest” means “pertaining to linear combinations”. Not of interest is the way that the vectors are presented typographically (or their color!). As an example, although the deﬁnition of isomorphism doesn’t explicitly say that the zero vectors must correspond, it is a consequence of that deﬁnition. 1.8 Lemma An isomorphism maps a zero vector to a zero vector.
Proof. Where f : V → W is an isomorphism, ﬁx any v ∈ V . Then f (0V ) = f (0 · v) = 0 · f (v) = 0W . QED

The deﬁnition of isomorphism requires that sums of two vectors correspond and that so do scalar multiples. We can extend that to say that all linear combinations correspond. 1.9 Lemma For any map f : V → W between vector spaces these statements are equivalent. (1) f preserves structure f (v1 + v2 ) = f (v1 ) + f (v2 ) and f (cv) = c f (v)

(2) f preserves linear combinations of two vectors f (c1 v1 + c2 v2 ) = c1 f (v1 ) + c2 f (v2 ) (3) f preserves linear combinations of any ﬁnite number of vectors f (c1 v1 + · · · + cn vn ) = c1 f (v1 ) + · · · + cn f (vn )
Proof. Since the implications (3) =⇒ (2) and (2) =⇒ (1) are clear, we need

only show that (1) =⇒ (3). Assume statement (1). We will prove statement (3) by induction on the number of summands n. The one-summand base case, that f (cv1 ) = c f (v1 ), is covered by the assumption of statement (1). For the inductive step assume that statement (3) holds whenever there are k or fewer summands, that is, whenever n = 1, or n = 2, . . . , or n = k. Consider the k + 1-summand case. The ﬁrst half of (1) gives f (c1 v1 + · · · + ck vk + ck+1 vk+1 ) = f (c1 v1 + · · · + ck vk ) + f (ck+1 vk+1 ) by breaking the sum along the ﬁnal ‘+’. Then the inductive hypothesis lets us break up the k-term sum. = f (c1 v1 ) + · · · + f (ck vk ) + f (ck+1 vk+1 ) Finally, the second half of statement (1) gives = c1 f (v1 ) + · · · + ck f (vk ) + ck+1 f (vk+1 ) when applied k + 1 times.
QED

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In addition to adding to the intuition that the deﬁnition of isomorphism does indeed preserve the things of interest in a vector space, that lemma’s second item is an especially handy way of checking that a map preserves structure. We close with a summary. The material in this section augments the chapter on Vector Spaces. There, after giving the deﬁnition of a vector space, we informally looked at what diﬀerent things can happen. Here, we deﬁned the relation ‘∼ between vector spaces and we have argued that it is the right way to split the =’ collection of vector spaces into cases because it preserves the features of interest in a vector space — in particular, it preserves linear combinations. That is, we have now said precisely what we mean by ‘the same’, and by ‘diﬀerent’, and so we have precisely classiﬁed the vector spaces. Exercises
1.10 Verify, using Example 1.4 as a model, that the two correspondences given before the deﬁnition are isomorphisms. (a) Example 1.1 (b) Example 1.2 1.11 For the map f : P1 → R2 given by a + bx −→
f

a−b b

Find the image of each of these elements of the domain. (a) 3 − 2x (b) 2 + 2x (c) x Show that this map is an isomorphism. 1.12 Show that the natural map f1 from Example 1.5 is an isomorphism. 1.13 Decide whether each map is an isomorphism (if it is an isomorphism then prove it and if it isn’t then state a condition that it fails to satisfy). (a) f : M2×2 → R given by a c (b) f : M2×2 → R4 given by a c (c) f : M2×2 → P3 given by a c a c b d b d → c + (d + c)x + (b + a)x2 + ax3 b d a+b+c+d  a+b+c  →  a+b a b d → ad − bc

(d) f : M2×2 → P3 given by → c + (d + c)x + (b + a + 1)x2 + ax3

1.14 Show that the map f : R1 → R1 given by f (x) = x3 is one-to-one and onto. Is it an isomorphism? 1.15 Refer to Example 1.1. Produce two more isomorphisms (of course, that they satisfy the conditions in the deﬁnition of isomorphism must be veriﬁed). 1.16 Refer to Example 1.2. Produce two more isomorphisms (and verify that they satisfy the conditions).

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1.17 Show that, although R2 is not itself a subspace of R3 , it is isomorphic to the xy-plane subspace of R3 . 1.18 Find two isomorphisms between R16 and M4×4 . 1.19 For what k is Mm×n isomorphic to Rk ? 1.20 For what k is Pk isomorphic to Rn ? 1.21 Prove that the map in Example 1.7, from P5 to P5 given by p(x) → p(x − 1), is a vector space isomorphism. 1.22 Why, in Lemma 1.8, must there be a v ∈ V ? That is, why must V be nonempty? 1.23 Are any two trivial spaces isomorphic? 1.24 In the proof of Lemma 1.9, what about the zero-summands case (that is, if n is zero)? 1.25 Show that any isomorphism f : P0 → R1 has the form a → ka for some nonzero real number k. 1.26 These prove that isomorphism is an equivalence relation. (a) Show that the identity map id : V → V is an isomorphism. Thus, any vector space is isomorphic to itself. (b) Show that if f : V → W is an isomorphism then so is its inverse f −1 : W → V . Thus, if V is isomorphic to W then also W is isomorphic to V . (c) Show that a composition of isomorphisms is an isomorphism: if f : V → W is an isomorphism and g : W → U is an isomorphism then so also is g ◦ f : V → U . Thus, if V is isomorphic to W and W is isomorphic to U , then also V is isomorphic to U . 1.27 Suppose that f : V → W preserves structure. Show that f is one-to-one if and only if the unique member of V mapped by f to 0W is 0V . 1.28 Suppose that f : V → W is an isomorphism. Prove that the set {v1 , . . . , vk } ⊆ V is linearly dependent if and only if the set of images {f (v1 ), . . . , f (vk )} ⊆ W is linearly dependent. 1.29 Show that each type of map from Example 1.6 is an automorphism. (a) Dilation ds by a nonzero scalar s. (b) Rotation tθ through an angle θ. (c) Reﬂection f over a line through the origin. Hint. For the second and third items, polar coordinates are useful. 1.30 Produce an automorphism of P2 other than the identity map, and other than a shift map p(x) → p(x − k). 1.31 (a) Show that a function f : R1 → R1 is an automorphism if and only if it has the form x → kx for some k = 0. (b) Let f be an automorphism of R1 such that f (3) = 7. Find f (−2). (c) Show that a function f : R2 → R2 is an automorphism if and only if it has the form x ax + by → y cx + dy for some a, b, c, d ∈ R with ad − bc = 0. Hint. Exercises in prior subsections have shown that b a is not a multiple of d c if and only if ad − bc = 0.

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(d) Let f be an automorphism of R2 with f( Find f( 0 ). −1 1 )= 3 2 −1 and f( 1 )= 4 0 . 1

1.32 Refer to Lemma 1.8 and Lemma 1.9. Find two more things preserved by isomorphism. 1.33 We show that isomorphisms can be tailored to ﬁt in that, sometimes, given vectors in the domain and in the range we can produce an isomorphism associating those vectors. (a) Let B = β1 , β2 , β3 be a basis for P2 so that any p ∈ P2 has a unique representation as p = c1 β1 + c2 β2 + c3 β3 , which we denote in this way. RepB (p) = c1 c2 c3

Show that the RepB (·) operation is a function from P2 to R3 (this entails showing that with every domain vector v ∈ P2 there is an associated image vector in R3 , and further, that with every domain vector v ∈ P2 there is at most one associated image vector). (b) Show that this RepB (·) function is one-to-one and onto. (c) Show that it preserves structure. (d) Produce an isomorphism from P2 to R3 that ﬁts these speciﬁcations. x + x2 → 1 0 0 and 1−x→ 0 1 0

1.34 Prove that a space is n-dimensional if and only if it is isomorphic to Rn . Hint. Fix a basis B for the space and consider the map sending a vector over to its representation with respect to B. 1.35 (Requires the subsection on Combining Subspaces, which is optional.) Let U and W be vector spaces. Deﬁne a new vector space, consisting of the set U × W = {(u, w) u ∈ U and w ∈ W } along with these operations. (u1 , w1 ) + (u2 , w2 ) = (u1 + u2 , w1 + w2 ) and r · (u, w) = (ru, rw)

This is a vector space, the external direct sum of U and W . (a) Check that it is a vector space. (b) Find a basis for, and the dimension of, the external direct sum P2 × R2 . (c) What is the relationship among dim(U ), dim(W ), and dim(U × W )? (d) Suppose that U and W are subspaces of a vector space V such that V = U ⊕ W (in this case we say that V is the internal direct sum of U and W ). Show that the map f : U × W → V given by (u, w) −→ u + w is an isomorphism. Thus if the internal direct sum is deﬁned then the internal and external direct sums are isomorphic.
f

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169

I.2 Dimension Characterizes Isomorphism
In the prior subsection, after stating the deﬁnition of an isomorphism, we gave some results supporting the intuition that such a map describes spaces as “the same”. Here we will formalize this intuition. While two spaces that are isomorphic are not equal, we think of them as almost equal — as equivalent. In this subsection we shall show that the relationship ‘is isomorphic to’ is an equivalence relation.∗ 2.1 Theorem Isomorphism is an equivalence relation between vector spaces.
Proof. We must prove that this relation has the three properties of being sym-

metric, reﬂexive, and transitive. For each of the three we will use item (2) of Lemma 1.9 and show that the map preserves structure by showing that it preserves linear combinations of two members of the domain. To check reﬂexivity, that any space is isomorphic to itself, consider the identity map. It is clearly one-to-one and onto. The calculation showing that it preserves linear combinations is easy. id(c1 · v1 + c2 · v2 ) = c1 v1 + c2 v2 = c1 · id(v1 ) + c2 · id(v2 ) To check symmetry, that if V is isomorphic to W via some map f : V → W then there is an isomorphism going the other way, consider the inverse map f −1 : W → V . As stated in the appendix, such an inverse function exists and it is also a correspondence. Thus we have reduced the symmetry issue to checking that, because f preserves linear combinations, so also does f −1 . Assume that w1 = f (v1 ) and w2 = f (v2 ), i.e., that f −1 (w1 ) = v1 and f −1 (w2 ) = v2 . f −1 (c1 · w1 + c2 · w2 ) = f −1 c1 · f (v1 ) + c2 · f (v2 ) = f −1 ( f c1 v1 + c2 v2 ) = c1 v1 + c2 v2 = c1 · f −1 (w1 ) + c2 · f −1 (w2 ) Finally, we must check transitivity, that if V is isomorphic to W via some map f and if W is isomorphic to U via some map g then also V is isomorphic to U . Consider the composition g ◦ f : V → U . The appendix notes that the composition of two correspondences is a correspondence, so we need only check that the composition preserves linear combinations. g ◦ f c1 · v1 + c2 · v2 = g f (c1 · v1 + c2 · v2 ) = g c1 · f (v1 ) + c2 · f (v2 ) = c1 · g f (v1 )) + c2 · g(f (v2 ) = c1 · (g ◦ f ) (v1 ) + c2 · (g ◦ f ) (v2 ) Thus g ◦ f : V → U is an isomorphism.

QED

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As a consequence of that result, we know that the universe of vector spaces is partitioned into classes: every space is in one and only one isomorphism class.
All ﬁnite dimensional vector spaces:

V W ...

V ∼W =

2.2 Theorem Vector spaces are isomorphic if and only if they have the same dimension. This follows from the next two lemmas. 2.3 Lemma If spaces are isomorphic then they have the same dimension.
Proof. We shall show that an isomorphism of two spaces gives a correspondence

between their bases. That is, where f : V → W is an isomorphism and a basis for the domain V is B = β1 , . . . , βn , then the image set D = f (β1 ), . . . , f (βn ) is a basis for the codomain W . (The other half of the correspondence — that for any basis of W the inverse image is a basis for V — follows on recalling that if f is an isomorphism then f −1 is also an isomorphism, and applying the prior sentence to f −1 .) To see that D spans W , ﬁx any w ∈ W , note that f is onto and so there is a v ∈ V with w = f (v), and expand v as a combination of basis vectors. w = f (v) = f (v1 β1 + · · · + vn βn ) = v1 · f (β1 ) + · · · + vn · f (βn ) For linear independence of D, if 0W = c1 f (β1 ) + · · · + cn f (βn ) = f (c1 β1 + · · · + cn βn ) then, since f is one-to-one and so the only vector sent to 0W is 0V , we have that 0V = c1 β1 + · · · + cn βn , implying that all of the c’s are zero. QED 2.4 Lemma If spaces have the same dimension then they are isomorphic.
Proof. To show that any two spaces of dimension n are isomorphic, we can

simply show that any one is isomorphic to Rn . Then we will have shown that they are isomorphic to each other, by the transitivity of isomorphism (which was established in Theorem 2.1). Let V be n-dimensional. Fix a basis B = β1 , . . . , βn for the domain V . Consider the representation of the members of that domain with respect to the basis as a function from V to Rn   v1 RepB  .  v = v1 β1 + · · · + vn βn −→  .  . vn

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171

(it is well-deﬁned∗ since every v has one and only one such representation — see Remark 2.5 below). This function is one-to-one because if RepB (u1 β1 + · · · + un βn ) = RepB (v1 β1 + · · · + vn βn ) then     u1 v1  .  .  . = .  . . un vn

and so u1 = v1 , . . . , un = vn , and therefore the original arguments u1 β1 + · · · + un βn and v1 β1 + · · · + vn βn are equal. This function is onto; any n-tall vector   w1  .  w= .  . wn is the image of some v ∈ V , namely w = RepB (w1 β1 + · · · + wn βn ). Finally, this function preserves structure. RepB (r · u + s · v) = RepB ( (ru1 + sv1 )β1 + · · · + (run + svn )βn )   ru1 + sv1   . . =  . run + svn     u1 v1  .  . =r· . +s· .  . . un vn = r · RepB (u) + s · RepB (v) Thus the RepB function is an isomorphism and thus any n-dimensional space is isomorphic to the n-dimensional space Rn . Consequently, any two spaces with the same dimension are isomorphic. QED 2.5 Remark The parenthetical comment in that proof about the role played by the ‘one and only one representation’ result requires some explanation. We need to show that (for a ﬁxed B) each vector in the domain is associated by RepB with one and only one vector in the codomain. A contrasting example, where an association doesn’t have this property, is illuminating. Consider this subset of P2 , which is not a basis. A = {1 + 0x + 0x2 , 0 + 1x + 0x2 , 0 + 0x + 1x2 , 1 + 1x + 2x2 }

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Call those four polynomials α1 , . . . , α4 . If, mimicing above proof, we try to write the members of P2 as p = c1 α1 + c2 α2 + c3 α3 + c4 α4 , and associate p with the four-tall vector with components c1 , . . . , c4 then there is a problem. For, consider p(x) = 1 + x + x2 . The set A spans the space P2 , so there is at least one four-tall vector associated with p. But A is not linearly independent and so vectors do not have unique decompositions. In this case, both p(x) = 1α1 + 1α2 + 1α3 + 0α4 and p(x) = 0α1 + 0α2 − 1α3 + 1α4

and so there is more than one four-tall vector associated with p.     0 1 0 1   and   −1 1 1 0 That is, with input p this association does not have a well-deﬁned (i.e., single) output value. Any map whose deﬁnition appears possibly ambiguous must be checked to see that it is well-deﬁned. For RepB in the above proof that check is Exercise 18. That ends the proof of Theorem 2.2. We say that the isomorphism classes are characterized by dimension because we can describe each class simply by giving the number that is the dimension of all of the spaces in that class. This subsection’s results give us a collection of representatives of the isomorphism classes.∗ 2.6 Corollary A ﬁnite-dimensional vector space is isomorphic to one and only one of the Rn . The proofs above pack many ideas into a small space. Through the rest of this chapter we’ll consider these ideas again, and ﬁll them out. For a taste of this, we will expand here on the proof of Lemma 2.4. 2.7 Example The space M2×2 of 2×2 matrices is isomorphic to R4 . With this basis for the domain B= 1 0 0 0 , 0 0 1 0 , 0 1 0 0 , 0 0 0 1

the isomorphism given in the lemma, the representation map f1 = RepB , simply carries the entries over.   a a b f1  b  −→   c c d d

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173

One way to think of the map f1 is: ﬁx the basis B for the domain and the basis E4 for the codomain, and associate β1 with e1 , and β2 with e2 , etc. Then extend this association to all of the members of two spaces.   a b f1 a b = aβ1 + bβ2 + cβ3 + dβ4 −→ ae1 + be2 + ce3 + de4 =   c c d d We say that the map has been extended linearly from the bases to the spaces. We can do the same thing with diﬀerent bases, for instance, taking this basis for the domain. A= 2 0 0 0 , 0 0 2 0 , 0 2 0 0 , 0 0 0 2

Associating corresponding members of A and E4 and extending linearly a c b d = (a/2)α1 + (b/2)α2 + (c/2)α3 + (d/2)α4   a/2  b/2  f2  −→ (a/2)e1 + (b/2)e2 + (c/2)e3 + (d/2)e4 =   c/2  d/2 gives rise to an isomorphism that is diﬀerent than f1 . The prior map arose by changing the basis for the domain. We can also change the basis for the codomain. Starting with         0 0 0 1 0 1 0 0 B and D =   ,   ,   ,   0 0 0 1 0 1 0 0 associating β1 with δ1 , etc., and then linearly extending that correspondence to all of the two spaces   a b f3 a b = aβ1 + bβ2 + cβ3 + dβ4 −→ aδ1 + bδ2 + cδ3 + dδ4 =   d c d c gives still another isomorphism. So there is a connection between the maps between spaces and bases for those spaces. Later sections will explore that connection. We will close this section with a summary. Recall that in the ﬁrst chapter we deﬁned two matrices as row equivalent if they can be derived from each other by elementary row operations (this was

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the meaning of same-ness that was of interest there). We showed that is an equivalence relation and so the collection of matrices is partitioned into classes, where all the matrices that are row equivalent fall together into a single class. Then, for insight into which matrices are in each class, we gave representatives for the classes, the reduced echelon form matrices. In this section, except that the appropriate notion of same-ness here is vector space isomorphism, we have followed much the same outline. First we deﬁned isomorphism, saw some examples, and established some properties. Then we showed that it is an equivalence relation, and now we have a set of class representatives, the real vector spaces R1 , R2 , etc. All ﬁnite dimensional vector spaces:
R0 R2 R1 R3 ...

One representative per class

As before, the list of representatives helps us to understand the partition. It is simply a classiﬁcation of spaces by dimension. In the second chapter, with the deﬁnition of vector spaces, we seemed to have opened up our studies to many examples of new structures besides the familiar Rn ’s. We now know that isn’t the case. Any ﬁnite-dimensional vector space is actually “the same” as a real space. We are thus considering exactly the structures that we need to consider. The rest of the chapter ﬁlls out the work in this section. In particular, in the next section we will consider maps that preserve structure, but are not necessarily correspondences. Exercises
2.8 Decide if the spaces are isomorphic. (a) R2 , R4 (b) P5 , R5 (c) M2×3 , R6 (d) P5 , M2×3 (e) M2×k , Ck 2 2.9 Consider the isomorphism RepB (·) : P1 → R where B = 1, 1 + x . Find the image of each of these elements of the domain. (a) 3 − 2x; (b) 2 + 2x; (c) x 2.10 Show that if m = n then Rm ∼ Rn . = 2.11 Is Mm×n ∼ Mn×m ? = 2.12 Are any two planes through the origin in R3 isomorphic? 2.13 Find a set of equivalence class representatives other than the set of Rn ’s. 2.14 True or false: between any n-dimensional space and Rn there is exactly one isomorphism. 2.15 Can a vector space be isomorphic to one of its (proper) subspaces? 2.16 This subsection shows that for any isomorphism, the inverse map is also an isomorphism. This subsection also shows that for a ﬁxed basis B of an n-dimensional vector space V , the map RepB : V → Rn is an isomorphism. Find the inverse of this map. 2.17 Prove these facts about matrices. (a) The row space of a matrix is isomorphic to the column space of its transpose.

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175

(b) The row space of a matrix is isomorphic to its column space. 2.18 Show that the function from Theorem 2.2 is well-deﬁned. 2.19 Is the proof of Theorem 2.2 valid when n = 0? 2.20 For each, decide if it is a set of isomorphism class representatives. (a) {Ck k ∈ N} (b) {Pk k ∈ {−1, 0, 1, . . .}} (c) {Mm×n m, n ∈ N} 2.21 Let f be a correspondence between vector spaces V and W (that is, a map that is one-to-one and onto). Show that the spaces V and W are isomorphic via f if and only if there are bases B ⊂ V and D ⊂ W such that corresponding vectors have the same coordinates: RepB (v) = RepD (f (v)). 2.22 Consider the isomorphism RepB : P3 → R4 . (a) Vectors in a real space are orthogonal if and only if their dot product is zero. Give a deﬁnition of orthogonality for polynomials. (b) The derivative of a member of P3 is in P3 . Give a deﬁnition of the derivative of a vector in R4 . 2.23 Does every correspondence between bases, when extended to the spaces, give an isomorphism? 2.24 (Requires the subsection on Combining Subspaces, which is optional.) Suppose that V = V1 ⊕ V2 and that V is isomorphic to the space U under the map f . Show that U = f (V1 ) ⊕ f (U2 ). 2.25 Show that this is not a well-deﬁned function from the rational numbers to the integers: with each fraction, associate the value of its numerator.

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Chapter Three. Maps Between Spaces

II

Homomorphisms

The deﬁnition of isomorphism has two conditions. In this section we will consider the second one, that the map must preserve the algebraic structure of the space. We will focus on this condition by studying maps that are required only to preserve structure; that is, maps that are not required to be correspondences. Experience shows that this kind of map is tremendously useful in the study of vector spaces. For one thing, as we shall see in the second subsection below, while isomorphisms describe how spaces are the same, these maps describe how spaces can be thought of as alike.

II.1 Deﬁnition

1.1 Deﬁnition A function between vector spaces h : V → W that preserves the operations of addition if v1 , v2 ∈ V then h(v1 + v2 ) = h(v1 ) + h(v2 ) and scalar multiplication if v ∈ V and r ∈ R then h(r · v) = r · h(v) is a homomorphism or linear map. 1.2 Example The projection map π : R3 → R2   x π y  −→ x y z is a homomorphism. It preserves addition       x1 x2 x1 + x2 x1 + x2 π( y1  +  y2 ) = π( y1 + y2 ) = y1 + y2 z1 z2 z1 + z2 and scalar multiplication.     x1 rx1 π(r ·  y1 ) = π( ry1 ) = z1 rz1

    x1 x2 = π( y1 ) + π( y2 ) z1 z2

rx1 ry1

  x1 = r · π( y1 ) z1

This map is not an isomorphism since it is not one-to-one. For instance, both 0 and e3 in R3 are mapped to the zero vector in R2 .

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177

1.3 Example Of course, the domain and codomain might be other than spaces of column vectors. Both of these are homomorphisms; the veriﬁcations are straightforward. (1) f1 : P2 → P3 given by a0 + a1 x + a2 x2 → a0 x + (a1 /2)x2 + (a2 /3)x3 (2) f2 : M2×2 → R given by a c b d →a+d

1.4 Example Between any two spaces there is a zero homomorphism, mapping every vector in the domain to the zero vector in the codomain. 1.5 Example These two suggest why we use the term ‘linear map’. (1) The map g : R3 → R given by   x g y  −→ 3x + 2y − 4.5z z is linear (i.e., is a homomorphism). In contrast, the map g : R3 → R given ˆ by   x g ˆ y  −→ 3x + 2y − 4.5z + 1 z is not; for instance,     0 1 g (0 + 0) = 4 ˆ 0 0

while

    0 1 g (0) + g (0) = 5 ˆ ˆ 0 0

(to show that a map is not linear we need only produce one example of a linear combination that is not preserved). (2) The ﬁrst of these two maps t1 , t2 : R3 → R2 is linear while the second is not.     x x 5x − 2y t1 t2 y  −→ 5x − 2y and y  −→ x+y xy z z Finding an example that the second fails to preserve structure is easy. What distinguishes the homomorphisms is that the coordinate functions are linear combinations of the arguments. See also Exercise 23.

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Obviously, any isomorphism is a homomorphism — an isomorphism is a homomorphism that is also a correspondence. So, one way to think of the ‘homomorphism’ idea is that it is a generalization of ‘isomorphism’, motivated by the observation that many of the properties of isomorphisms have only to do with the map’s structure preservation property and not to do with it being a correspondence. As examples, these two results from the prior section do not use one-to-one-ness or onto-ness in their proof, and therefore apply to any homomorphism. 1.6 Lemma A homomorphism sends a zero vector to a zero vector. 1.7 Lemma Each of these is a necessary and suﬃcient condition for f : V → W to be a homomorphism. (1) f (c1 · v1 + c2 · v2 ) = c1 · f (v1 ) + c2 · f (v2 ) for any c1 , c2 ∈ R and v1 , v2 ∈ V (2) f (c1 · v1 + · · · + cn · vn ) = c1 · f (v1 ) + · · · + cn · f (vn ) for any c1 , . . . , cn ∈ R and v1 , . . . , vn ∈ V Part (1) is often used to check that a function is linear. 1.8 Example The map f : R2 → R4 given by   x/2 f  0  x  −→  x + y  y 3y satisﬁes (1) of the prior result       x2 /2 x1 /2 r1 (x1 /2) + r2 (x2 /2)  0   0    0       r1 (x1 + y1 ) + r2 (x2 + y2 ) = r1 x1 + y1  + r2 x2 + y2  r1 (3y1 ) + r2 (3y2 ) 3y2 3y1 and so it is a homomorphism. However, some of the results that we have seen for isomorphisms fail to hold for homomorphisms in general. Consider the theorem that an isomorphism between spaces gives a correspondence between their bases. Homomorphisms do not give any such correspondence; Example 1.2 shows that there is no such correspondence, and another example is the zero map between any two nontrivial spaces. Instead, for homomorphisms a weaker but still very useful result holds. 1.9 Theorem A homomorphism is determined by its action on a basis. That is, if β1 , . . . , βn is a basis of a vector space V and w1 , . . . , wn are (perhaps not distinct) elements of a vector space W then there exists a homomorphism from V to W sending β1 to w1 , . . . , and βn to wn , and that homomorphism is unique.

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179

Proof. We will deﬁne the map by associating β1 with w1 , etc., and then extending linearly to all of the domain. That is, where v = c1 β1 + · · · + cn βn , the map h : V → W is given by h(v) = c1 w1 + · · · + cn wn . This is well-deﬁned because, with respect to the basis, the representation of each domain vector v is unique. This map is a homomorphism since it preserves linear combinations; where v1 = c1 β1 + · · · + cn βn and v2 = d1 β1 + · · · + dn βn , we have this.

h(r1 v1 + r2 v2 ) = h((r1 c1 + r2 d1 )β1 + · · · + (r1 cn + r2 dn )βn ) = (r1 c1 + r2 d1 )w1 + · · · + (r1 cn + r2 dn )wn = r1 h(v1 ) + r2 h(v2 ) ˆ And, this map is unique since if h : V → W is another homomorphism such ˆ ˆ that h(βi ) = wi for each i then h and h agree on all of the vectors in the domain. ˆ ˆ h(v) = h(c1 β1 + · · · + cn βn ) ˆ ˆ = c1 h(β1 ) + · · · + cn h(βn ) = c1 w1 + · · · + cn wn = h(v) ˆ Thus, h and h are the same map.
QED

1.10 Example This result says that we can construct a homomorphism by ﬁxing a basis for the domain and specifying where the map sends those basis vectors. For instance, if we specify a map h : R2 → R2 that acts on the standard basis E2 in this way h( 1 )= 0 −1 1 and h( 0 )= 1 −4 4

then the action of h on any other member of the domain is also speciﬁed. For instance, the value of h on this argument h( 3 1 0 1 0 ) = h(3 · −2· ) = 3 · h( ) − 2 · h( )= −2 0 1 0 1 5 −5

is a direct consequence of the value of h on the basis vectors. Later in this chapter we shall develop a scheme, using matrices, that is convienent for computations like this one. Just as the isomorphisms of a space with itself are useful and interesting, so too are the homomorphisms of a space with itself. 1.11 Deﬁnition A linear map from a space into itself t : V → V is a linear transformation.

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1.12 Remark In this book we use ‘linear transformation’ only in the case where the codomain equals the domain, but it is widely used in other texts as a general synonym for ‘homomorphism’. 1.13 Example The map on R2 that projects all vectors down to the x-axis x y is a linear transformation. 1.14 Example The derivative map d/dx : Pn → Pn a0 + a1 x + · · · + an xn −→ a1 + 2a2 x + 3a3 x2 + · · · + nan xn−1 is a linear transformation, as this result from calculus notes: d(c1 f + c2 g)/dx = c1 (df /dx) + c2 (dg/dx). 1.15 Example The matrix transpose map a c b d → a b c d
d/dx

x 0

is a linear transformation of M2×2 . Note that this transformation is one-to-one and onto, and so in fact it is an automorphism. We ﬁnish this subsection about maps by recalling that we can linearly combine maps. For instance, for these maps from R2 to itself x y −→
f

2x 3x − 2y

and

x y

−→

g

0 5x

the linear combination 5f − 2g is also a map from R2 to itself. x y
5f −2g

−→

10x 5x − 10y

1.16 Lemma For vector spaces V and W , the set of linear functions from V to W is itself a vector space, a subspace of the space of all functions from V to W . It is denoted L(V, W ).
Proof. This set is non-empty because it contains the zero homomorphism. So

to show that it is a subspace we need only check that it is closed under linear combinations. Let f, g : V → W be linear. Then their sum is linear (f + g)(c1 v1 + c2 v2 ) = c1 f (v1 ) + c2 f (v2 ) + c1 g(v1 ) + c2 g(v2 ) = c1 f + g (v1 ) + c2 f + g (v2 ) and any scalar multiple is also linear. (r · f )(c1 v1 + c2 v2 ) = r(c1 f (v1 ) + c2 f (v2 )) = c1 (r · f )(v1 ) + c2 (r · f )(v2 ) Hence L(V, W ) is a subspace.
QED

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181

We started this section by isolating the structure preservation property of isomorphisms. That is, we deﬁned homomorphisms as a generalization of isomorphisms. Some of the properties that we studied for isomorphisms carried over unchanged, while others were adapted to this more general setting. It would be a mistake, though, to view this new notion of homomorphism as derived from, or somehow secondary to, that of isomorphism. In the rest of this chapter we shall work mostly with homomorphisms, partly because any statement made about homomorphisms is automatically true about isomorphisms, but more because, while the isomorphism concept is perhaps more natural, experience shows that the homomorphism concept is actually more fruitful and more central to further progress. Exercises
1.17 Decide if each h : R3 → R2 is linear. x x x (a) h( y ) = (b) h( y ) = x+y+z z z x 2x + y (d) h( y ) = 3y − 4z z 1.18 Decide if each map h : M2×2 → R is linear. a b (a) h( )=a+d c d (b) h( (c) h( (d) h( a c a c b ) = ad − bc d b ) = 2a + 3b + c − d d 0 0 x (c) h( y ) = z 1 1

a b ) = a2 + b2 c d 1.19 Show that these two maps are homomorphisms. (a) d/dx : P3 → P2 given by a0 + a1 x + a2 x2 + a3 x3 maps to a1 + 2a2 x + 3a3 x2 (b) : P2 → P3 given by b0 + b1 x + b2 x2 maps to b0 x + (b1 /2)x2 + (b2 /3)x3 Are these maps inverse to each other? 1.20 Is (perpendicular) projection from R3 to the xz-plane a homomorphism? Projection to the yz-plane? To the x-axis? The y-axis? The z-axis? Projection to the origin? 1.21 Show that, while the maps from Example 1.3 preserve linear operations, they are not isomorphisms. 1.22 Is an identity map a linear transformation? 1.23 Stating that a function is ‘linear’ is diﬀerent than stating that its graph is a line. (a) The function f1 : R → R given by f1 (x) = 2x − 1 has a graph that is a line. Show that it is not a linear function. (b) The function f2 : R2 → R given by x y → x + 2y

does not have a graph that is a line. Show that it is a linear function.

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1.24 Part of the deﬁnition of a linear function is that it respects addition. Does a linear function respect subtraction? 1.25 Assume that h is a linear transformation of V and that β1 , . . . , βn is a basis of V . Prove each statement. (a) If h(βi ) = 0 for each basis vector then h is the zero map. (b) If h(βi ) = βi for each basis vector then h is the identity map. (c) If there is a scalar r such that h(βi ) = r · βi for each basis vector then h(v) = r · v for all vectors in V . 1.26 Consider the vector space R+ where vector addition and scalar multiplication are not the ones inherited from R but rather are these: a + b is the product of a and b, and r · a is the r-th power of a. (This was shown to be a vector space in an earlier exercise.) Verify that the natural logarithm map ln : R+ → R is a homomorphism between these two spaces. Is it an isomorphism? 1.27 Consider this transformation of R2 . x x/2 → y y/3 Find the image under this map of this ellipse. { x y (x2 /4) + (y 2 /9) = 1}

1.28 Imagine a rope wound around the earth’s equator so that it ﬁts snugly (suppose that the earth is a sphere). How much extra rope must be added to raise the circle to a constant six feet oﬀ the ground? 1.29 Verify that this map h : R3 → R x x 3 y −1 = 3x − y − z → y z z −1 is linear. Generalize. 1.30 Show that every homomorphism from R1 to R1 acts via multiplication by a scalar. Conclude that every nontrivial linear transformation of R1 is an isomorphism. Is that true for transformations of R2 ? Rn ? 1.31 (a) Show that for any scalars a1,1 , . . . , am,n this map h : Rn → Rm is a homomorphism.     x1 a1,1 x1 + · · · + a1,n xn .   . .  . →  . . xn am,1 x1 + · · · + am,n xn (b) Show that for each i, the i-th derivative operator di /dxi is a linear transformation of Pn . Conclude that for any scalars ck , . . . , c0 this map is a linear transformation of that space. dk dk−1 d f + ck−1 k−1 f + · · · + c1 f + c0 f k dx dx dx 1.32 Lemma 1.16 shows that a sum of linear functions is linear and that a scalar multiple of a linear function is linear. Show also that a composition of linear functions is linear. 1.33 Where f : V → W is linear, suppose that f (v1 ) = w1 , . . . , f (vn ) = wn for some vectors w1 , . . . , wn from W . (a) If the set of w ’s is independent, must the set of v ’s also be independent? f→

Section II. Homomorphisms

183

(b) If the set of v ’s is independent, must the set of w ’s also be independent? (c) If the set of w ’s spans W , must the set of v ’s span V ? (d) If the set of v ’s spans V , must the set of w ’s span W ? 1.34 Generalize Example 1.15 by proving that the matrix transpose map is linear. What is the domain and codomain? 1.35 (a) Where u, v ∈ Rn , the line segment connecting them is deﬁned to be the set = {t · u + (1 − t) · v t ∈ [0..1]}. Show that the image, under a homomorphism h, of the segment between u and v is the segment between h(u) and h(v). (b) A subset of Rn is convex if, for any two points in that set, the line segment joining them lies entirely in that set. (The inside of a sphere is convex while the skin of a sphere is not.) Prove that linear maps from Rn to Rm preserve the property of set convexity. 1.36 Let h : Rn → Rm be a homomorphism. (a) Show that the image under h of a line in Rn is a (possibly degenerate) line in Rn . (b) What happens to a k-dimensional linear surface? 1.37 Prove that the restriction of a homomorphism to a subspace of its domain is another homomorphism. 1.38 Assume that h : V → W is linear. (a) Show that the rangespace of this map {h(v) v ∈ V } is a subspace of the codomain W . (b) Show that the nullspace of this map {v ∈ V h(v) = 0W } is a subspace of the domain V . (c) Show that if U is a subspace of the domain V then its image {h(u) u ∈ U } is a subspace of the codomain W . This generalizes the ﬁrst item. (d) Generalize the second item. 1.39 Consider the set of isomorphisms from a vector space to itself. Is this a subspace of the space L(V, V ) of homomorphisms from the space to itself? 1.40 Does Theorem 1.9 need that β1 , . . . , βn is a basis? That is, can we still get a well-deﬁned and unique homomorphism if we drop either the condition that the set of β’s be linearly independent, or the condition that it span the domain? 1.41 Let V be a vector space and assume that the maps f1 , f2 : V → R1 are linear. (a) Deﬁne a map F : V → R2 whose component functions are the given linear ones. f1 (v) v→ f2 (v) Show that F is linear. (b) Does the converse hold — is any linear map from V to R2 made up of two linear component maps to R1 ? (c) Generalize.

II.2 Rangespace and Nullspace
Isomorphisms and homomorphisms both preserve structure. The diﬀerence is

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that homomorphisms needn’t be onto and needn’t be one-to-one. This means that homomorphisms are a more general kind of map, subject to fewer restrictions than isomorphisms. We will examine what can happen with homomorphisms that is prevented by the extra restrictions satisﬁed by isomorphisms. We ﬁrst consider the eﬀect of dropping the onto requirement, of not requiring as part of the deﬁnition that a homomorphism be onto its codomain. For instance, the injection map ι : R2 → R3   x x → y  y 0 is not an isomorphism because it is not onto. Of course, being a function, a homomorphism is onto some set, namely its range; the map ι is onto the xyplane subset of R3 . 2.1 Lemma Under a homomorphism, the image of any subspace of the domain is a subspace of the codomain. In particular, the image of the entire space, the range of the homomorphism, is a subspace of the codomain.
Proof. Let h : V → W be linear and let S be a subspace of the domain V .

The image h(S) is a subset of the codomain W . It is nonempty because S is nonempty and thus to show that h(S) is a subspace of W we need only show that it is closed under linear combinations of two vectors. If h(s1 ) and h(s2 ) are members of h(S) then c1 ·h(s1 )+c2 ·h(s2 ) = h(c1 ·s1 )+h(c2 ·s2 ) = h(c1 ·s1 +c2 ·s2 ) is also a member of h(S) because it is the image of c1 · s1 + c2 · s2 from S. QED 2.2 Deﬁnition The rangespace of a homomorphism h : V → W is R(h) = {h(v) v ∈ V } sometimes denoted h(V ). The dimension of the rangespace is the map’s rank. (We shall soon see the connection between the rank of a map and the rank of a matrix.) 2.3 Example Recall that the derivative map d/dx : P3 → P3 given by a0 + a1 x + a2 x2 + a3 x3 → a1 + 2a2 x + 3a3 x2 is linear. The rangespace R(d/dx) is the set of quadratic polynomials {r + sx + tx2 r, s, t ∈ R}. Thus, the rank of this map is three. 2.4 Example With this homomorphism h : M2×2 → P3 a c b d → (a + b + 2d) + 0x + cx2 + cx3

an image vector in the range can have any constant term, must have an x coeﬃcient of zero, and must have the same coeﬃcient of x2 as of x3 . That is, the rangespace is R(h) = {r + 0x + sx2 + sx3 r, s ∈ R} and so the rank is two.

Section II. Homomorphisms

185

The prior result shows that, in passing from the deﬁnition of isomorphism to the more general deﬁnition of homomorphism, omitting the ‘onto’ requirement doesn’t make an essential diﬀerence. Any homomorphism is onto its rangespace. However, omitting the ‘one-to-one’ condition does make a diﬀerence. A homomorphism may have many elements of the domain that map to one element of the codomain. Below is a “bean” sketch of a many-to-one map between sets.∗ It shows three elements of the codomain that are each the image of many members of the domain.

Recall that for any function h : V → W , the set of elements of V that are mapped to w ∈ W is the inverse image h−1 (w) = {v ∈ V h(v) = w}. Above, the three sets of many elements on the left are inverse images. 2.5 Example Consider the projection π : R3 → R2   x π y  −→ x y z which is a homomorphism that is many-to-one. In this instance, an inverse image set is a vertical line of vectors in the domain.

R2 R
3

w 2 1

2.6 Example This homomorphism h : R → R x y −→ x + y
h

is also many-to-one; for a ﬁxed w ∈ R1 , the inverse image h−1 (w)

R2 w

R1

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is the set of plane vectors whose components add to w. The above examples have only to do with the fact that we are considering functions, speciﬁcally, many-to-one functions. They show the inverse images as sets of vectors that are related to the image vector w. But these are more than just arbitrary functions, they are homomorphisms; what do the two preservation conditions say about the relationships? In generalizing from isomorphisms to homomorphisms by dropping the oneto-one condition, we lose the property that we’ve stated intuitively as: the domain is “the same as” the range. That is, we lose that the domain corresponds perfectly to the range in a one-vector-by-one-vector way. What we shall keep, as the examples below illustrate, is that a homomorphism describes a way in which the domain is “like”, or “analgous to”, the range. 2.7 Example We think of R3 as being like R2 , except that vectors have an extra component. That is, we think of the vector with components x, y, and z as like the vector with components x and y. In deﬁning the projection map π, we make precise which members of the domain we are thinking of as related to which members of the codomain. Understanding in what way the preservation conditions in the deﬁnition of homomorphism show that the domain elements are like the codomain elements is easiest if we draw R2 as the xy-plane inside of R3 . (Of course, R2 is a set of two-tall vectors while the xy-plane is a set of three-tall vectors with a third component of zero, but there is an obvious correspondence.) Then, π(v) is the “shadow” of v in the plane and the preservation of addition property says that

x1 y1 z1

above

x1 y1

plus

x2 y2 z2

above

x2 y2

equals

x1 + y1 y1 + y2 z1 + z2

above

x1 + x2 y1 + y2

Brieﬂy, the shadow of a sum π(v1 + v2 ) equals the sum of the shadows π(v1 ) + π(v2 ). (Preservation of scalar multiplication has a similar interpretation.) Redrawing by separating the two spaces, moving the codomain R2 to the right, gives an uglier picture but one that is more faithful to the “bean” sketch.

w2 w1 + w 2 w1

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187

Again in this drawing, the vectors that map to w1 lie in the domain in a vertical line (only one such vector is shown, in gray). Call any such member of this inverse image a “w1 vector”. Similarly, there is a vertical line of “w2 vectors” and a vertical line of “w1 + w2 vectors”. Now, π has the property that if π(v1 ) = w1 and π(v2 ) = w2 then π(v1 + v2 ) = π(v1 ) + π(v2 ) = w1 + w2 . This says that the vector classes add, in the sense that any w1 vector plus any w2 vector equals a w1 + w2 vector, (A similar statement holds about the classes under scalar multiplication.) Thus, although the two spaces R3 and R2 are not isomorphic, π describes a way in which they are alike: vectors in R3 add as do the associated vectors in R2 — vectors add as their shadows add. 2.8 Example A homomorphism can be used to express an analogy between spaces that is more subtle than the prior one. For the map x y −→ x + y
h

from Example 2.6 ﬁx two numbers w1 , w2 in the range R. A v1 that maps to w1 has components that add to w1 , that is, the inverse image h−1 (w1 ) is the set of vectors with endpoint on the diagonal line x + y = w1 . Call these the “w1 vectors”. Similarly, we have the “w2 vectors” and the “w1 + w2 vectors”. Then the addition preservation property says that
v1 + v2 v1 v2

a “w1 vector”

plus

a “w2 vector”

equals

a “w1 + w2 vector”.

Restated, if a w1 vector is added to a w2 vector then the result is mapped by h to a w1 + w2 vector. Brieﬂy, the image of a sum is the sum of the images. Even more brieﬂy, h(v1 + v2 ) = h(v1 ) + h(v2 ). (The preservation of scalar multiplication condition has a similar restatement.) 2.9 Example The inverse images can be structures other than lines. For the linear map h : R3 → R2   x y  → x x z the inverse image sets are planes x = 0, x = 1, etc., perpendicular to the x-axis.

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We won’t describe how every homomorphism that we will use is an analogy because the formal sense that we make of “alike in that . . . ” is ‘a homomorphism exists such that . . . ’. Nonetheless, the idea that a homomorphism between two spaces expresses how the domain’s vectors fall into classes that act like the the range’s vectors is a good way to view homomorphisms. Another reason that we won’t treat all of the homomorphisms that we see as above is that many vector spaces are hard to draw (e.g., a space of polynomials). However, there is nothing bad about gaining insights from those spaces that we are able to draw, especially when those insights extend to all vector spaces. We derive two such insights from the three examples 2.7, 2.8, and 2.9. First, in all three examples, the inverse images are lines or planes, that is, linear surfaces. In particular, the inverse image of the range’s zero vector is a line or plane through the origin — a subspace of the domain. 2.10 Lemma For any homomorphism, the inverse image of a subspace of the range is a subspace of the domain. In particular, the inverse image of the trivial subspace of the range is a subspace of the domain.
Proof. Let h : V → W be a homomorphism and let S be a subspace of the

rangespace h. Consider h−1 (S) = {v ∈ V h(v) ∈ S}, the inverse image of the set S. It is nonempty because it contains 0V , since h(0V ) = 0W , which is an element S, as S is a subspace. To show that h−1 (S) is closed under linear combinations, let v1 and v2 be elements, so that h(v1 ) and h(v2 ) are elements of S, and then c1 v1 + c2 v2 is also in the inverse image because h(c1 v1 + c2 v2 ) = c1 h(v1 ) + c2 h(v2 ) is a member of the subspace S. QED 2.11 Deﬁnition The nullspace or kernel of a linear map h : V → W is the inverse image of 0W N (h) = h−1 (0W ) = {v ∈ V h(v) = 0W }.

The dimension of the nullspace is the map’s nullity.

0V

0W

2.12 Example The map from Example 2.3 has this nullspace N (d/dx) = {a0 + 0x + 0x2 + 0x3 a0 ∈ R}. 2.13 Example The map from Example 2.4 has this nullspace. N (h) = { a 0 b −(a + b)/2 a, b ∈ R}

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189

Now for the second insight from the above pictures. In Example 2.7, each of the vertical lines is squashed down to a single point — π, in passing from the domain to the range, takes all of these one-dimensional vertical lines and “zeroes them out”, leaving the range one dimension smaller than the domain. Similarly, in Example 2.8, the two-dimensional domain is mapped to a one-dimensional range by breaking the domain into lines (here, they are diagonal lines), and compressing each of those lines to a single member of the range. Finally, in Example 2.9, the domain breaks into planes which get “zeroed out”, and so the map starts with a three-dimensional domain but ends with a one-dimensional range — this map “subtracts” two from the dimension. (Notice that, in this third example, the codomain is two-dimensional but the range of the map is only one-dimensional, and it is the dimension of the range that is of interest.) 2.14 Theorem A linear map’s rank plus its nullity equals the dimension of its domain.
Proof. Let h : V → W be linear and let BN = β1 , . . . , βk be a basis for the

nullspace. Extend that to a basis BV = β1 , . . . , βk , βk+1 , . . . , βn for the entire domain. We shall show that BR = h(βk+1 ), . . . , h(βn ) is a basis for the rangespace. Then counting the size of these bases gives the result. To see that BR is linearly independent, consider the equation ck+1 h(βk+1 ) + · · · + cn h(βn ) = 0W . This gives that h(ck+1 βk+1 + · · · + cn βn ) = 0W and so ck+1 βk+1 +· · ·+cn βn is in the nullspace of h. As BN is a basis for this nullspace, there are scalars c1 , . . . , ck ∈ R satisfying this relationship. c1 β1 + · · · + ck βk = ck+1 βk+1 + · · · + cn βn But BV is a basis for V so each scalar equals zero. Therefore BR is linearly independent. To show that BR spans the rangespace, consider h(v) ∈ R(h) and write v as a linear combination v = c1 β1 + · · · + cn βn of members of BV . This gives h(v) = h(c1 β1 +· · ·+cn βn ) = c1 h(β1 )+· · ·+ck h(βk )+ck+1 h(βk+1 )+· · ·+cn h(βn ) and since β1 , . . . , βk are in the nullspace, we have that h(v) = 0 + · · · + 0 + ck+1 h(βk+1 ) + · · · + cn h(βn ). Thus, h(v) is a linear combination of members of BR , and so BR spans the space. QED 2.15 Example Where h : R3 → R4 is     x x h 0 y  −→   y  z 0 the rangespace and nullspace are   a 0 R(h) = {  a, b ∈ R} b 0

  0 and N (h) = {0 z ∈ R} z

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and so the rank of h is two while the nullity is one. 2.16 Example If t : R → R is the linear transformation x → −4x, then the range is R(t) = R1 , and so the rank of t is one and the nullity is zero. 2.17 Corollary The rank of a linear map is less than or equal to the dimension of the domain. Equality holds if and only if the nullity of the map is zero. We know that an isomorphism exists between two spaces if and only if their dimensions are equal. Here we see that for a homomorphism to exist, the dimension of the range must be less than or equal to the dimension of the domain. For instance, there is no homomorphism from R2 onto R3 . There are many homomorphisms from R2 into R3 , but none is onto all of three-space. The rangespace of a linear map can be of dimension strictly less than the dimension of the domain (Example 2.3’s derivative transformation on P3 has a domain of dimension four but a range of dimension three). Thus, under a homomorphism, linearly independent sets in the domain may map to linearly dependent sets in the range (for instance, the derivative sends {1, x, x2 , x3 } to {0, 1, 2x, 3x2 }). That is, under a homomorphism, independence may be lost. In contrast, dependence stays. 2.18 Lemma Under a linear map, the image of a linearly dependent set is linearly dependent.
Proof. Suppose that c1 v1 + · · · + cn vn = 0V , with some ci nonzero. Then,

because h(c1 v1 + · · · + cn vn ) = c1 h(v1 ) + · · · + cn h(vn ) and because h(0V ) = 0W , we have that c1 h(v1 ) + · · · + cn h(vn ) = 0W with some nonzero ci . QED When is independence not lost? One obvious suﬃcient condition is when the homomorphism is an isomorphism. This condition is also necessary; see Exercise 35. We will ﬁnish this subsection comparing homomorphisms with isomorphisms by observing that a one-to-one homomorphism is an isomorphism from its domain onto its range. 2.19 Deﬁnition A linear map that is one-to-one is nonsingular. (In the next section we will see the connection between this use of ‘nonsingular’ for maps and its familiar use for matrices.) 2.20 Example This nonsingular homomorphism ι : R2 → R3   x x ι −→ y  y 0 gives the obvious correspondence between R2 and the xy-plane inside of R3 . The prior observation allows us to adapt some results about isomorphisms to this setting.

Section II. Homomorphisms

191

2.21 Theorem In an n-dimensional vector space V , these: (1) h is nonsingular, that is, one-to-one (2) h has a linear inverse (3) N (h) = {0 }, that is, nullity(h) = 0 (4) rank(h) = n (5) if β1 , . . . , βn is a basis for V then h(β1 ), . . . , h(βn ) is a basis for R(h) are equivalent statements about a linear map h : V → W .
Proof. We will ﬁrst show that (1) ⇐⇒ (2). We will then show that (1) =⇒

(3) =⇒ (4) =⇒ (5) =⇒ (2). For (1) =⇒ (2), suppose that the linear map h is one-to-one, and so has an inverse. The domain of that inverse is the range of h and so a linear combination of two members of that domain has the form c1 h(v1 ) + c2 h(v2 ). On that combination, the inverse h−1 gives this. h−1 (c1 h(v1 ) + c2 h(v2 )) = h−1 (h(c1 v1 + c2 v2 )) = h−1 ◦ h (c1 v1 + c2 v2 ) = c1 v1 + c2 v2 = c1 h−1 ◦ h (v1 ) + c2 h−1 ◦ h (v2 ) = c1 · h−1 (h(v1 )) + c2 · h−1 (h(v2 )) Thus the inverse of a one-to-one linear map is automatically linear. But this also gives the (2) =⇒ (1) implication, because the inverse itself must be one-to-one. Of the remaining implications, (1) =⇒ (3) holds because any homomorphism maps 0V to 0W , but a one-to-one map sends at most one member of V to 0W . Next, (3) =⇒ (4) is true since rank plus nullity equals the dimension of the domain. For (4) =⇒ (5), to show that h(β1 ), . . . , h(βn ) is a basis for the rangespace we need only show that it is a spanning set, because by assumption the range has dimension n. Consider h(v) ∈ R(h). Expressing v as a linear combination of basis elements produces h(v) = h(c1 β1 + c2 β2 + · · · + cn βn ), which gives that h(v) = c1 h(β1 ) + · · · + cn h(βn ), as desired. Finally, for the (5) =⇒ (2) implication, assume that β1 , . . . , βn is a basis for V so that h(β1 ), . . . , h(βn ) is a basis for R(h). Then every w ∈ R(h) a the unique representation w = c1 h(β1 ) + · · · + cn h(βn ). Deﬁne a map from R(h) to V by w → c1 β1 + c2 β2 + · · · + cn βn (uniqueness of the representation makes this well-deﬁned). Checking that it is linear and that it is the inverse of h are easy. QED We’ve now seen that a linear map shows how the structure of the domain is like that of the range. Such a map can be thought to organize the domain space into inverse images of points in the range. In the special case that the map is

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one-to-one, each inverse image is a single point and the map is an isomorphism between the domain and the range. Exercises
2.22 Let h : P3 → P4 be given by p(x) → x · p(x). Which of these are in the nullspace? Which are in the rangespace? (a) x3 (b) 0 (c) 7 (d) 12x − 0.5x3 (e) 1 + 3x2 − x3 2.23 Find the nullspace, nullity, rangespace, and rank of each map. (a) h : R2 → P3 given by a → a + ax + ax2 b (b) h : M2×2 → R given by a c (c) h : M2×2 → P2 given by a c b d → a + b + c + dx2 b d →a+d

(d) the zero map Z : R3 → R4 2.24 Find the nullity of each map. (a) h : R5 → R8 of rank ﬁve (b) h : P3 → P3 of rank one (c) h : R6 → R3 , an onto map (d) h : M3×3 → M3×3 , onto 2.25 What is the nullspace of the diﬀerentiation transformation d/dx : Pn → Pn ? What is the nullspace of the second derivative, as a transformation of Pn ? The k-th derivative? 2.26 Example 2.7 restates the ﬁrst condition in the deﬁnition of homomorphism as ‘the shadow of a sum is the sum of the shadows’. Restate the second condition in the same style. 2.27 For the homomorphism h : P3 → P3 given by h(a0 + a1 x + a2 x2 + a3 x3 ) = a0 + (a0 + a1 )x + (a2 + a3 )x3 ﬁnd these. (a) N (h) (b) h−1 (2 − x3 ) (c) h−1 (1 + x2 ) 2 2.28 For the map f : R → R given by f( x ) = 2x + y y

sketch these inverse image sets: f −1 (−3), f −1 (0), and f −1 (1). 2.29 Each of these transformations of P3 is nonsingular. Find the inverse function of each. (a) a0 + a1 x + a2 x2 + a3 x3 → a0 + a1 x + 2a2 x2 + 3a3 x3 (b) a0 + a1 x + a2 x2 + a3 x3 → a0 + a2 x + a1 x2 + a3 x3 (c) a0 + a1 x + a2 x2 + a3 x3 → a1 + a2 x + a3 x2 + a0 x3 (d) a0 +a1 x+a2 x2 +a3 x3 → a0 +(a0 +a1 )x+(a0 +a1 +a2 )x2 +(a0 +a1 +a2 +a3 )x3 2.30 Describe the nullspace and rangespace of a transformation given by v → 2v. 2.31 List all pairs (rank(h), nullity(h)) that are possible for linear maps from R5 to R3 . 2.32 Does the diﬀerentiation map d/dx : Pn → Pn have an inverse?

Section II. Homomorphisms
2.33 Find the nullity of the map h : Pn → R given by
x=1

193

a0 + a1 x + · · · + an xn →
x=0

a0 + a1 x + · · · + an xn dx.

2.34 (a) Prove that a homomorphism is onto if and only if its rank equals the dimension of its codomain. (b) Conclude that a homomorphism between vector spaces with the same dimension is one-to-one if and only if it is onto. 2.35 Show that a linear map is nonsingular if and only if it preserves linear independence. 2.36 Corollary 2.17 says that for there to be an onto homomorphism from a vector space V to a vector space W , it is necessary that the dimension of W be less than or equal to the dimension of V . Prove that this condition is also suﬃcient; use Theorem 1.9 to show that if the dimension of W is less than or equal to the dimension of V , then there is a homomorphism from V to W that is onto. 2.37 Let h : V → R be a homomorphism, but not the zero homomorphism. Prove that if β1 , . . . , βn is a basis for the nullspace and if v ∈ V is not in the nullspace then v, β1 , . . . , βn is a basis for the entire domain V . 2.38 Recall that the nullspace is a subset of the domain and the rangespace is a subset of the codomain. Are they necessarily distinct? Is there a homomorphism that has a nontrivial intersection of its nullspace and its rangespace? 2.39 Prove that the image of a span equals the span of the images. That is, where h : V → W is linear, prove that if S is a subset of V then h([S]) equals [h(S)]. This generalizes Lemma 2.1 since it shows that if U is any subspace of V then its image {h(u) u ∈ U } is a subspace of W , because the span of the set U is U . 2.40 (a) Prove that for any linear map h : V → W and any w ∈ W , the set h−1 (w) has the form {v + n n ∈ N (h)} for v ∈ V with h(v) = w (if h is not onto then this set may be empty). Such a set is a coset of N (h) and is denoted v + N (h). (b) Consider the map t : R2 → R2 given by x y −→
t

ax + by cx + dy

for some scalars a, b, c, and d. Prove that t is linear. (c) Conclude from the prior two items that for any linear system of the form ax + by = e cx + dy = f the solution set can be written (the vectors are members of R2 ) {p + h h satisﬁes the associated homogeneous system} where p is a particular solution of that linear system (if there is no particular solution then the above set is empty). (d) Show that this map h : Rn → Rm is linear

x1 a1,1 x1 + · · · + a1,n xn .   . . → . .  . xn am,1 x1 + · · · + am,n xn for any scalars a1,1 , . . . , am,n . Extend the conclusion made in the prior item.

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Chapter Three. Maps Between Spaces
(e) Show that the k-th derivative map is a linear transformation of Pn for each k. Prove that this map is a linear transformation of that space

dk dk−1 d f + ck−1 k−1 f + · · · + c1 f + c0 f k dx dx dx for any scalars ck , . . . , c0 . Draw a conclusion as above. 2.41 Prove that for any transformation t : V → V that is rank one, the map given by composing the operator with itself t ◦ t : V → V satisﬁes t ◦ t = r · t for some real number r. 2.42 Show that for any space V of dimension n, the dual space f→ L(V, R) = {h : V → R h is linear} is isomorphic to Rn . It is often denoted V ∗ . Conclude that V ∗ ∼ V . = 2.43 Show that any linear map is the sum of maps of rank one. 2.44 Is ‘is homomorphic to’ an equivalence relation? (Hint: the diﬃculty is to decide on an appropriate meaning for the quoted phrase.) 2.45 Show that the rangespaces and nullspaces of powers of linear maps t : V → V form descending V ⊇ R(t) ⊇ R(t2 ) ⊇ . . . and ascending {0} ⊆ N (t) ⊆ N (t2 ) ⊆ . . . chains. Also show that if k is such that R(tk ) = R(tk+1 ) then all following rangespaces are equal: R(tk ) = R(tk+1 ) = R(tk+2 ) . . . . Similarly, if N (tk ) = N (tk+1 ) then N (tk ) = N (tk+1 ) = N (tk+2 ) = . . . .

Section III. Computing Linear Maps

195

III

Computing Linear Maps

The prior section shows that a linear map is determined by its action on a basis. In fact, the equation h(v) = h(c1 · β1 + · · · + cn · βn ) = c1 · h(β1 ) + · · · + cn · h(βn ) shows that, if we know the value of the map on the vectors in a basis, then we can compute the value of the map on any vector v at all. We just need to ﬁnd the c’s to express v with respect to the basis. This section gives the scheme that computes, from the representation of a vector in the domain RepB (v), the representation of that vector’s image in the codomain RepD (h(v)), using the representations of h(β1 ), . . . , h(βn ).

III.1 Representing Linear Maps with Matrices

1.1 Example Consider a map h with domain R2 and codomain R3 (ﬁxing 2 1 , 0 4       1 0 1 and D = 0 , −2 , 0 0 0 1

B=

as the bases for these spaces) that is determined by this action on the vectors in the domain’s basis.     1 1 2 1 h h −→ 1 −→ 2 0 4 1 0 To compute the action of this map on any vector at all from the domain, we ﬁrst express h(β1 ) and h(β2 ) with respect to the codomain’s basis:         0 1 1 1 1  1 = 0 0 − −2 + 1 0 2 0 1 1 0 and         1 1 0 1 2 = 1 0 − 1 −2 + 0 0 0 0 0 1  1 RepD (h(β2 )) = −1 0 D   0 RepD (h(β1 )) = −1/2 1 D 

so

so

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Chapter Three. Maps Between Spaces

(these are easy to check). Then, as described in the preamble, for any member v of the domain, we can express the image h(v) in terms of the h(β)’s. h(v) = h(c1 · = c1 · h( 2 1 + c2 · ) 0 4

2 1 ) + c2 · h( ) 0 4             0 1 1 0 1 1 1  −2 + 1 0) + c2 · (1 0 − 1 −2 + 0 0) = c1 · (0 0 − 2 0 1 0 0 1 0       0 1 1 1 = (0c1 + 1c2 ) · 0 + (− c1 − 1c2 ) · −2 + (1c1 + 0c2 ) · 0 2 0 1 0 Thus,  0c1 + 1c2 then RepD ( h(v) ) = −(1/2)c1 − 1c2 . 1c1 + 0c2 

with RepB (v) =

c1 c2

For instance,  2 4 then RepD ( h( ) ) = −5/2. 8 1 

with RepB (

4 )= 8

1 2

B

We will express computations like the one above with a matrix notation.  0 −1/2 1  1 c1 −1 c 0 B,D 2  0c1 + 1c2 = (−1/2)c1 − 1c2  1c1 + 0c2 D 

B

In the middle is the argument v to the map, represented with respect to the domain’s basis B by a column vector with components c1 and c2 . On the right is the value h(v) of the map on that argument, represented with respect to the codomain’s basis D by a column vector with components 0c1 + 1c2 , etc. The matrix on the left is the new thing. It consists of the coeﬃcients from the vector on the right, 0 and 1 from the ﬁrst row, −1/2 and −1 from the second row, and 1 and 0 from the third row. This notation simply breaks the parts from the right, the coeﬃcients and the c’s, out separately on the left, into a vector that represents the map’s argument and a matrix that we will take to represent the map itself.

Section III. Computing Linear Maps

197

1.2 Deﬁnition Suppose that V and W are vector spaces of dimensions n and m with bases B and D, and that h : V → W is a linear map. If     h1,1 h1,n  h2,1   h2,n      RepD (h(β1 )) =  .  . . . RepD (h(βn )) =  .   .   .  . . hm,1 D hm,n D then h1,1  h2,1  RepB,D (h) =   hm,1  h1,2 h2,2 . . . hm,2 ... ... ...  h1,n h2,n     hm,n B,D

is the matrix representation of h with respect to B, D. Brieﬂy, the vectors representing the h(β)’s are adjoined to make the matrix representing the map.   . . . . . .   RepB,D (h) =  RepD ( h(β1 ) ) ··· RepD ( h(βn ) )    . . . . . . Observe that the number of columns n of the matrix is the dimension of the domain of the map, and the number of rows m is the dimension of the codomain. 1.3 Example If h : R3 → P1 is given by   a1 h a2  −→ (2a1 + a2 ) + (−a3 )x a3 then where       0 0 2 B = 0 , 2 , 0 1 0 0 the action of h on B is given by   0 h 0 −→ −x 1 and a simple calculation gives RepD (−x) = −1/2 −1/2 RepD (2) =
D

and D = 1 + x, −1 + x

  0 h 2 −→ 2 0

  2 h 0 −→ 4 0

1 −1

RepD (4) =
D

2 −2

D

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Chapter Three. Maps Between Spaces

showing that this is the matrix representing h with respect to the bases. RepB,D (h) = −1/2 −1/2 1 −1 2 −2

B,D

We will use lower case letters for a map, upper case for the matrix, and lower case again for the entries of the matrix. Thus for the map h, the matrix representing it is H, with entries hi,j . 1.4 Theorem Assume that V and W are vector spaces of dimensions n and m with bases B and D, and that h : V → W is a linear map. If h is represented by   h1,1 h1,2 . . . h1,n  h2,1 h2,2 . . . h2,n    RepB,D (h) =   . .   . hm,1 and v ∈ V is represented by   c1  c2    RepB (v) =  .  . . cn B then the representation of the image of v is this.  h1,1 c1 + h1,2 c2 + · · · + h1,n cn  h2,1 c1 + h2,2 c2 + · · · + h2,n cn  RepD ( h(v) ) =  . .  . hm,1 c1 + hm,2 c2 + · · · + hm,n cn
Proof. Exercise 28.

hm,2

...

hm,n

B,D

    
D

QED

We will think of the matrix RepB,D (h) and the vector RepB (v) as combining to make the vector RepD (h(v)). 1.5 Deﬁnition The matrix-vector product of a m×n matrix and a n×1 vector is this.       a1,1 a1,2 . . . a1,n a1,1 c1 + a1,2 c2 + · · · + a1,n cn c1  a2,1 a2,2 . . . a2,n       .   a2,1 c1 + a2,2 c2 + · · · + a2,n cn    .  =   . . . .   .   . . cn am,1 am,2 . . . am,n am,1 c1 + am,2 c2 + · · · + am,n cn The point of Deﬁnition 1.2 is to generalize Example 1.1, that is, the point of the deﬁnition is Theorem 1.4, that the matrix describes how to get from

Section III. Computing Linear Maps

199

the representation of a domain vector with respect to the domain’s basis to the representation of its image in the codomain with respect to the codomain’s basis. With Deﬁnition 1.5, we can restate this as: application of a linear map is represented by the matrix-vector product of the map’s representative and the vector’s representative. 1.6 Example With the matrix from Example 1.3 we can calculate where that map sends this vector.   4 v = 1 0 This vector is represented, with respect to the domain basis B, by   0 RepB (v) = 1/2 2 B and so this is the representation of the value h(v) with respect to the codomain basis D.   0 −1/2 1 2 1/2 RepD (h(v)) = −1/2 −1 −2 B,D 2 B = (−1/2) · 0 + 1 · (1/2) + 2 · 2 (−1/2) · 0 − 1 · (1/2) − 2 · 2 =
D

9/2 −9/2

D

To ﬁnd h(v) itself, not its representation, take (9/2)(1 + x) − (9/2)(−1 + x) = 9. 1.7 Example Let π : R3 → R2 be projection onto the xy-plane. To give a matrix representing this map, we ﬁrst ﬁx bases.       1 1 −1 2 1 B = 0 , 1 ,  0  D= , 1 1 0 0 1 For each vector in the domain’s basis, we ﬁnd its image under the map.       1 1 −1 1 1 π π π 0 −→ 1 −→  0  −→ −1 0 1 0 0 0 1 Then we ﬁnd the representation of each image with respect to the codomain’s basis RepD ( 1 )= 0 1 −1 RepD ( 1 )= 1 0 1 RepD ( −1 )= 0 −1 1

(these are easily checked). Finally, adjoining these representations gives the matrix representing π with respect to B, D. RepB,D (π) = 1 −1 0 −1 1 1

B,D

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Chapter Three. Maps Between Spaces

We can illustrate Theorem 1.4 by computing the matrix-vector product representing the following statement about the projection map.   2 2 π(2) = 2 1 Representing this vector from the domain with respect to the domain’s basis     2 1 RepB (2) = 2 1 1 B gives this matrix-vector product.   2 1 RepD ( π(1) ) = −1 1

  1 0 −1 2 = 1 1 B,D 1 B

0 2

D

Expanding this representation into a linear combination of vectors from D 0· 2 1 +2· 1 1 = 2 2

checks that the map’s action is indeed reﬂected in the operation of the matrix. (We will sometimes compress these three displayed equations into one     2 1 2 h 2 = 2 −→ 0 = 2 2 D H 1 1 B in the course of a calculation.) We now have two ways to compute the eﬀect of projection, the straightforward formula that drops each three-tall vector’s third component to make a two-tall vector, and the above formula that uses representations and matrixvector multiplication. Compared to the ﬁrst way, the second way might seem complicated. However, it has advantages. The next example shows that giving a formula for some maps is simpliﬁed by this new scheme. 1.8 Example To represent a rotation map tθ : R2 → R2 that turns all vectors in the plane counterclockwise through an angle θ
tπ/6 (u)

−→
u

tπ/6

Section III. Computing Linear Maps

201

we start by ﬁxing bases. Using E2 both as a domain basis and as a codomain basis is natural, Now, we ﬁnd the image under the map of each vector in the domain’s basis. 1 0
θ −→

t

cos θ sin θ

0 1

θ −→

t

− sin θ cos θ

Then we represent these images with respect to the codomain’s basis. Because this basis is E2 , vectors are represented by themselves. Finally, adjoining the representations gives the matrix representing the map. RepE2 ,E2 (tθ ) = cos θ sin θ − sin θ cos θ

The advantage of this scheme is that just by knowing how to represent the image of the two basis vectors, we get a formula that tells us the image of any vector at all; here a vector rotated by θ = π/6. √ tπ/6 3 3/2 √ −1/2 3 3.598 −→ ≈ −2 −2 −0.232 1/2 3/2 (Again, we are using the fact that, with respect to E2 , vectors represent themselves.) We have already seen the addition and scalar multiplication operations of matrices and the dot product operation of vectors. Matrix-vector multiplication is a new operation in the arithmetic of vectors and matrices. Nothing in Deﬁnition 1.5 requires us to view it in terms of representations. We can get some insight into this operation by turning away from what is being represented, and instead focusing on how the entries combine. 1.9 Example In the deﬁnition the width of the matrix equals the height of the vector. Hence, the ﬁrst product below is deﬁned while the second is not.   1 1 1 0 0 1 1 0 0   0 = 6 4 3 1 0 4 3 1 2 One reason that this product is not deﬁned is purely formal: the deﬁnition requires that the sizes match, and these sizes don’t match. Behind the formality, though, is a reason why we will leave it undeﬁned — the matrix represents a map with a three-dimensional domain while the vector represents a member of a twodimensional space. A good way to view a matrix-vector product is as the dot products of the rows of the matrix with the column vector.   c    . . 1 . . . .    c2    ai,1 ai,2 . . . ai,n   .  = ai,1 c1 + ai,2 c2 + . . . + ai,n cn    .    . . . . . . . cn

202 Looked at in this Matrix-vector  h1,1 h1,2  h2,1 h2,2   . .  . hm,1 hm,2

Chapter Three. Maps Between Spaces row-by-row way, this new operation generalizes dot product. product can also be viewed column-by-column.     h1,1 c1 + h1,2 c2 + · · · + h1,n cn c1 . . . h1,n . . . h2,n   c2   h2,1 c1 + h2,2 c2 + · · · + h2,n cn        .  =  . .   .   . . ... hm,n cn hm,1 c1 + hm,2 c2 + · · · + hm,n cn     h1,n h1,1  h2,n   h2,1      = c1  .  + · · · + cn  .   .   .  . . hm,1 hm,n

1.10 Example 1 2 0 0  2 1 0 −1 −1   −1 = 2 −1 +1 2 0 3 3 1  = 1 7

The result has the columns of the matrix weighted by the entries of the vector. This way of looking at it brings us back to the objective stated at the start of this section, to compute h(c1 β1 + · · · + cn βn ) as c1 h(β1 ) + · · · + cn h(βn ). We began this section by noting that the equality of these two enables us to compute the action of h on any argument knowing only h(β1 ), . . . , h(βn ). We have developed this into a scheme to compute the action of the map by taking the matrix-vector product of the matrix representing the map and the vector representing the argument. In this way, any linear map is represented with respect to some bases by a matrix. In the next subsection, we will show the converse, that any matrix represents a linear map. Exercises
1.11 Multiply the matrix 1 0 1 3 −1 1 1 2 0

by each vector (or state “not deﬁned”). 2 0 −2 (a) 1 (b) (c) 0 −2 0 0 1.12 Perform, if possible, each matrix-vector multiplication. 1 2 1 4 1 1 0 3 (a) (b) (c) 3 −1/2 2 −2 1 0 1 1.13 Solve this matrix equation. 8 2 1 1 x 0 1 3 y = 4 1 −1 2 z 4

1 −2

1 1

1 3 1

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203

1.14 For a homomorphism from P2 to P3 that sends 1 → 1 + x, x → 1 + 2x, and x2 → x − x3 where does 1 − 3x + 2x2 go? 1.15 Assume that h : R2 → R3 is determined by this action. 0 2 0 1 1 → → 2 1 0 −1 0 Using the standard bases, ﬁnd (a) the matrix representing this map; (b) a general formula for h(v). 1.16 Let d/dx : P3 → P3 be the derivative transformation. (a) Represent d/dx with respect to B, B where B = 1, x, x2 , x3 . (b) Represent d/dx with respect to B, D where D = 1, 2x, 3x2 , 4x3 . 1.17 Represent each linear map with respect to each pair of bases. (a) d/dx : Pn → Pn with respect to B, B where B = 1, x, . . . , xn , given by a0 + a1 x + a2 x2 + · · · + an xn → a1 + 2a2 x + · · · + nan xn−1 (b) : Pn → Pn+1 with respect to Bn , Bn+1 where Bi = 1, x, . . . , xi , given by a1 2 an n+1 a0 + a1 x + a2 x2 + · · · + an xn → a0 x + x + ··· + x 2 n+1 1 n (c) 0 : Pn → R with respect to B, E1 where B = 1, x, . . . , x and E1 = 1 , given by an a1 + ··· + a0 + a1 x + a2 x2 + · · · + an xn → a0 + 2 n+1 (d) eval3 : Pn → R with respect to B, E1 where B = 1, x, . . . , xn and E1 = 1 , given by a0 + a1 x + a2 x2 + · · · + an xn → a0 + a1 · 3 + a2 · 32 + · · · + an · 3n (e) slide−1 : Pn → Pn with respect to B, B where B = 1, x, . . . , xn , given by a0 + a1 x + a2 x2 + · · · + an xn → a0 + a1 · (x + 1) + · · · + an · (x + 1)n 1.18 Represent the identity map on any nontrivial space with respect to B, B, where B is any basis. 1.19 Represent, with respect to the natural basis, the transpose transformation on the space M2×2 of 2×2 matrices. 1.20 Assume that B = β1 , β2 , β3 , β4 is a basis for a vector space. Represent with respect to B, B the transformation that is determined by each. (a) β1 → β2 , β2 → β3 , β3 → β4 , β4 → 0 (b) β1 → β2 , β2 → 0, β3 → β4 , β4 → 0 (c) β1 → β2 , β2 → β3 , β3 → 0, β4 → 0 1.21 Example 1.8 shows how to represent the rotation transformation of the plane with respect to the standard basis. Express these other transformations also with respect to the standard basis. (a) the dilation map ds , which multiplies all vectors by the same scalar s (b) the reﬂection map f , which reﬂects all all vectors across a line through the origin 1.22 Consider a linear transformation of R2 determined by these two. 1 2 1 −1 → → 1 0 0 0 (a) Represent this transformation with respect to the standard bases.

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(b) Where does the transformation send this vector? 0 5 (c) Represent this transformation with respect to these bases. B= 1 1 , −1 1 D= 2 −1 , 2 1

(d) Using B from the prior item, represent the transformation with respect to B, B. 1.23 Suppose that h : V → W is nonsingular so that by Theorem 2.21, for any basis B = β1 , . . . , βn ⊂ V the image h(B) = h(β1 ), . . . , h(βn ) is a basis for W. (a) Represent the map h with respect to B, h(B). (b) For a member v of the domain, where the representation of v has components c1 , . . . , cn , represent the image vector h(v) with respect to the image basis h(B). 1.24 Give a formula for the product of a matrix and ei , the column vector that is all zeroes except for a single one in the i-th position. 1.25 For each vector space of functions of one real variable, represent the derivative transformation with respect to B, B. (a) {a cos x + b sin x a, b ∈ R}, B = cos x, sin x (b) {aex + be2x a, b ∈ R}, B = ex , e2x (c) {a + bx + cex + dxex a, b, c, d ∈ R}, B = 1, x, ex , xex 1.26 Find the range of the linear transformation of R2 represented with respect to the standard bases by each matrix. 1 0 0 0 a b (a) (b) (c) a matrix of the form 0 0 3 2 2a 2b 1.27 Can one matrix represent two diﬀerent linear maps? That is, can RepB,D (h) = ˆ RepB,D (h)? ˆ ˆ 1.28 Prove Theorem 1.4. 1.29 Example 1.8 shows how to represent rotation of all vectors in the plane through an angle θ about the origin, with respect to the standard bases. (a) Rotation of all vectors in three-space through an angle θ about the x-axis is a transformation of R3 . Represent it with respect to the standard bases. Arrange the rotation so that to someone whose feet are at the origin and whose head is at (1, 0, 0), the movement appears clockwise. (b) Repeat the prior item, only rotate about the y-axis instead. (Put the person’s head at e2 .) (c) Repeat, about the z-axis. (d) Extend the prior item to R4 . (Hint: ‘rotate about the z-axis’ can be restated as ‘rotate parallel to the xy-plane’.) 1.30 (Schur’s Triangularization Lemma) (a) Let U be a subspace of V and ﬁx bases BU ⊆ BV . What is the relationship between the representation of a vector from U with respect to BU and the representation of that vector (viewed as a member of V ) with respect to BV ? (b) What about maps? (c) Fix a basis B = β1 , . . . , βn for V and observe that the spans [{0}] = {0} ⊂ [{β1 }] ⊂ [{β1 , β2 }] ⊂ ··· ⊂ [B] = V

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205

form a strictly increasing chain of subspaces. Show that for any linear map h : V → W there is a chain W0 = {0} ⊆ W1 ⊆ · · · ⊆ Wm = W of subspaces of W such that h([{β1 , . . . , βi }]) ⊂ Wi for each i. (d) Conclude that for every linear map h : V → W there are bases B, D so the matrix representing h with respect to B, D is upper-triangular (that is, each entry hi,j with i > j is zero). (e) Is an upper-triangular representation unique?

III.2 Any Matrix Represents a Linear Map
The prior subsection shows that the action of a linear map h is described by a matrix H, with respect to appropriate bases, in this way.     h1,1 v1 + · · · + h1,n vn v1 h   . . . v =  .  −→   = h(v) . .
H

vn

B

hm,1 v1 + · · · + hm,n vn

D

In this subsection, we will show the converse, that each matrix represents a linear map. Recall that, in the deﬁnition of the matrix representation of a linear map, the number of columns of the matrix is the dimension of the map’s domain and the number of rows of the matrix is the dimension of the map’s codomain. Thus, for instance, a 2×3 matrix cannot represent a map from R5 to R4 . The next result says that, beyond this restriction on the dimensions, there are no other limitations: the 2×3 matrix represents a map from any three-dimensional space to any two-dimensional space. 2.1 Theorem Any matrix represents a homomorphism between vector spaces of appropriate dimensions, with respect to any pair of bases.
Proof. For the matrix

h1,1  h2,1  H=  hm,1

h1,2 h2,2 . . . hm,2

... ... ...

 h1,n h2,n     hm,n

ﬁx any n-dimensional domain space V and any m-dimensional codomain space W . Also ﬁx bases B = β1 , . . . , βn and D = δ1 , . . . , δm for those spaces. Deﬁne a function h : V → W by: where v in the domain is represented as   v1 . RepB (v) =  .  . vn
B

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then its image h(v) is the member the codomain represented by   h1,1 v1 + · · · + h1,n vn   . . RepD ( h(v) ) =   . hm,1 v1 + · · · + hm,n vn D that is, h(v) = h(v1 β1 + · · · + vn βn ) is deﬁned to be (h1,1 v1 + · · · + h1,n vn ) · δ1 + · · · + (hm,1 v1 + · · · + hm,n vn ) · δm . (This is well-deﬁned by the uniqueness of the representation RepB (v).) Observe that h has simply been deﬁned to make it the map that is represented with respect to B, D by the matrix H. So to ﬁnish, we need only check that h is linear. If v, u ∈ V are such that     v1 u1 .  .  RepB (v) =  .  and RepB (u) =  .  . . vn and c, d ∈ R then the calculation h(cv + du) = h1,1 (cv1 + du1 ) + · · · + h1,n (cvn + dun ) · δ1 + · · · + hm,1 (cv1 + du1 ) + · · · + hm,n (cvn + dun ) · δm = c · h(v) + d · h(u) provides this veriﬁcation.
QED

un

2.2 Example Which map the matrix represents depends on which bases are used. If H= 1 0 0 , 0 B1 = D1 = 1 0 , 0 1 , and B2 = D2 = 0 1 , 1 0 ,

then h1 : R2 → R2 represented by H with respect to B1 , D1 maps c1 c2 = c1 c2 →
B1

c1 0

=
D1

c1 0

while h2 : R2 → R2 represented by H with respect to B2 , D2 is this map. c1 c2 = c2 c1 →
B2

c2 0

=
D2

0 c2

These two are diﬀerent. The ﬁrst is projection onto the x axis, while the second is projection onto the y axis. So not only is any linear map described by a matrix but any matrix describes a linear map. This means that we can, when convenient, handle linear maps entirely as matrices, simply doing the computations, without have to worry that

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207

a matrix of interest does not represent a linear map on some pair of spaces of interest. (In practice, when we are working with a matrix but no spaces or bases have been speciﬁed, we will often take the domain and codomain to be Rn and Rm and use the standard bases. In this case, because the representation is transparent — the representation with respect to the standard basis of v is v — the column space of the matrix equals the range of the map. Consequently, the column space of H is often denoted by R(H).) With the theorem, we have characterized linear maps as those maps that act in this matrix way. Each linear map is described by a matrix and each matrix describes a linear map. We ﬁnish this section by illustrating how a matrix can be used to tell things about its maps. 2.3 Theorem The rank of a matrix equals the rank of any map that it represents.
Proof. Suppose that the matrix H is m×n. Fix domain and codomain spaces

V and W of dimension n and m, with bases B = β1 , . . . , βn and D. Then H represents some linear map h between those spaces with respect to these bases whose rangespace {h(v) v ∈ V } = {h(c1 β1 + · · · + cn βn ) c1 , . . . , cn ∈ R} = {c1 h(β1 ) + · · · + cn h(βn ) c1 , . . . , cn ∈ R} is the span [{h(β1 ), . . . , h(βn )}]. The rank of h is the dimension of this rangespace. The rank of the matrix is its column rank (or its row rank; the two are equal). This is the dimension of the column space of the matrix, which is the span of the set of column vectors [{RepD (h(β1 )), . . . , RepD (h(βn ))}]. To see that the two spans have the same dimension, recall that a representation with respect to a basis gives an isomorphism RepD : W → Rm . Under this isomorphism, there is a linear relationship among members of the rangespace if and only if the same relationship holds in the column space, e.g, 0 = c1 h(β1 ) + · · · + cn h(βn ) if and only if 0 = c1 RepD (h(β1 )) + · · · + cn RepD (h(βn )). Hence, a subset of the rangespace is linearly independent if and only if the corresponding subset of the column space is linearly independent. This means that the size of the largest linearly independent subset of the rangespace equals the size of the largest linearly independent subset of the column space, and so the two spaces have the same dimension. QED 2.4 Example Any map represented  1 1  0 0 by 2 2 0 0  2 1  3 2

must, by deﬁnition, be from a three-dimensional domain to a four-dimensional codomain. In addition, because the rank of this matrix is two (we can spot this

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by eye or get it with Gauss’ method), any map represented by this matrix has a two-dimensional rangespace. 2.5 Corollary Let h be a linear map represented by a matrix H. Then h is onto if and only if the rank of H equals the number of its rows, and h is one-to-one if and only if the rank of H equals the number of its columns.
Proof. For the ﬁrst half, the dimension of the rangespace of h is the rank of h,

which equals the rank of H by the theorem. Since the dimension of the codomain of h is the number of rows in H, if the rank of H equals the number of rows, then the dimension of the rangespace equals the dimension of the codomain. But a subspace with the same dimension as its superspace must equal that superspace (a basis for the rangespace is a linearly independent subset of the codomain, whose size is equal to the dimension of the codomain, and so this set is a basis for the codomain). For the second half, a linear map is one-to-one if and only if it is an isomorphism between its domain and its range, that is, if and only if its domain has the same dimension as its range. But the number of columns in h is the dimension of h’s domain, and by the theorem the rank of H equals the dimension of h’s range. QED The above results end any confusion caused by our use of the word ‘rank’ to mean apparently diﬀerent things when applied to matrices and when applied to maps. We can also justify the dual use of ‘nonsingular’. We’ve deﬁned a matrix to be nonsingular if it is square and is the matrix of coeﬃcients of a linear system with a unique solution, and we’ve deﬁned a linear map to be nonsingular if it is one-to-one. 2.6 Corollary A square matrix represents nonsingular maps if and only if it is a nonsingular matrix. Thus, a matrix represents an isomorphism if and only if it is square and nonsingular.
Proof. Immediate from the prior result. QED

2.7 Example Any map from R2 to P1 represented with respect to any pair of bases by 1 2 0 3 is nonsingular because this matrix has rank two. 2.8 Example Any map g : V → W represented by 1 3 2 6

is not nonsingular because this matrix is not nonsingular.

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209

We’ve now seen that the relationship between maps and matrices goes both ways: ﬁxing bases, any linear map is represented by a matrix and any matrix describes a linear map. That is, by ﬁxing spaces and bases we get a correspondence between maps and matrices. In the rest of this chapter we will explore this correspondence. For instance, we’ve deﬁned for linear maps the operations of addition and scalar multiplication and we shall see what the corresponding matrix operations are. We shall also see the matrix operation that represent the map operation of composition. And, we shall see how to ﬁnd the matrix that represents a map’s inverse. Exercises
2.9 Decide if the vector is in the column space of the matrix. 1 2 1 1 4 −8 0 1 , (b) , (c) (a) 2 5 −3 2 −4 1 −1 −1 1 −1 1 −1 , 1 2 0 0

2.10 Decide if each vector lies in the range of the map from R3 to R2 represented with respect to the standard bases by the matrix. 1 1 3 1 2 0 3 1 (a) , (b) , 0 1 4 3 4 0 6 1 2.11 Consider this matrix, representing a transformation of R2 , and these bases for that space. 1 · 2 1 −1 1 1 B= 0 1 , 1 0 D= 1 1 , 1 −1

(a) To what vector in the codomain is the ﬁrst member of B mapped? (b) The second member? (c) Where is a general vector from the domain (a vector with components x and y) mapped? That is, what transformation of R2 is represented with respect to B, D by this matrix? 2.12 What transformation of F = {a cos θ + b sin θ a, b ∈ R} is represented with respect to B = cos θ − sin θ, sin θ and D = cos θ + sin θ, cos θ by this matrix? 0 1 0 0

2.13 Decide if 1 + 2x is in the range of the map from R3 to P2 represented with respect to E3 and 1, 1 + x2 , x by this matrix. 1 0 1 3 1 0 0 0 1

2.14 Example 2.8 gives a matrix that is nonsingular, and is therefore associated with maps that are nonsingular. (a) Find the set of column vectors representing the members of the nullspace of any map represented by this matrix. (b) Find the nullity of any such map. (c) Find the set of column vectors representing the members of the rangespace of any map represented by this matrix. (d) Find the rank of any such map. (e) Check that rank plus nullity equals the dimension of the domain.

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2.15 Because the rank of a matrix equals the rank of any map it represents, if ˆ one matrix represents two diﬀerent maps H = RepB,D (h) = RepB,D (h) (where ˆ ˆ ˆ : V → W ) then the dimension of the rangespace of h equals the dimension of h, h ˆ the rangespace of h. Must these equal-dimensioned rangespaces actually be the same? 2.16 Let V be an n-dimensional space with bases B and D. Consider a map that sends, for v ∈ V , the column vector representing v with respect to B to the column vector representing v with respect to D. Show that is a linear transformation of Rn . 2.17 Example 2.2 shows that changing the pair of bases can change the map that a matrix represents, even though the domain and codomain remain the same. Could the map ever not change? Is there a matrix H, vector spaces V and W , and associated pairs of bases B1 , D1 and B2 , D2 (with B1 = B2 or D1 = D2 or both) such that the map represented by H with respect to B1 , D1 equals the map represented by H with respect to B2 , D2 ? 2.18 A square matrix is a diagonal matrix if it is all zeroes except possibly for the entries on its upper-left to lower-right diagonal — its 1, 1 entry, its 2, 2 entry, etc. Show that a linear map is an isomorphism if there are bases such that, with respect to those bases, the map is represented by a diagonal matrix with no zeroes on the diagonal. 2.19 Describe geometrically the action on R2 of the map represented with respect to the standard bases E2 , E2 by this matrix. 3 0 Do the same for these. 1 0 0 0 0 1 1 0 1 0 3 1 0 2

2.20 The fact that for any linear map the rank plus the nullity equals the dimension of the domain shows that a necessary condition for the existence of a homomorphism between two spaces, onto the second space, is that there be no gain in dimension. That is, where h : V → W is onto, the dimension of W must be less than or equal to the dimension of V . (a) Show that this (strong) converse holds: no gain in dimension implies that there is a homomorphism and, further, any matrix with the correct size and correct rank represents such a map. (b) Are there bases for R3 such that this matrix H= 1 2 0 0 0 1 0 0 0

represents a map from R3 to R3 whose range is the xy plane subspace of R3 ? 2.21 Let V be an n-dimensional space and suppose that x ∈ Rn . Fix a basis B for V and consider the map hx : V → R given v → x RepB (v) by the dot product. (a) Show that this map is linear. (b) Show that for any linear map g : V → R there is an x ∈ Rn such that g = hx . (c) In the prior item we ﬁxed the basis and varied the x to get all possible linear maps. Can we get all possible linear maps by ﬁxing an x and varying the basis?

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211

2.22 Let V, W, X be vector spaces with bases B, C, D. (a) Suppose that h : V → W is represented with respect to B, C by the matrix H. Give the matrix representing the scalar multiple rh (where r ∈ R) with respect to B, C by expressing it in terms of H. (b) Suppose that h, g : V → W are represented with respect to B, C by H and G. Give the matrix representing h + g with respect to B, C by expressing it in terms of H and G. (c) Suppose that h : V → W is represented with respect to B, C by H and g : W → X is represented with respect to C, D by G. Give the matrix representing g ◦ h with respect to B, D by expressing it in terms of H and G.

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Chapter Three. Maps Between Spaces

IV

Matrix Operations

The prior section shows how matrices represent linear maps. A good strategy, on seeing a new idea, is to explore how it interacts with some already-established ideas. In the ﬁrst subsection we will ask how the representation of the sum of two maps f + g is related to the representations of the two maps, and how the representation of a scalar product r · h of a map is related to the representation of that map. In later subsections we will see how to represent map composition and map inverse.

IV.1 Sums and Scalar Products
Recall that for two maps f and g with the same domain and codomain, the map sum f + g has this deﬁnition. v −→ f (v) + g(v) The easiest way to see how the representations of the maps combine to represent the map sum is with an example. 1.1 Example Suppose that f, g : R2 → R3 are represented with respect to the bases B and D by these matrices.     0 0 1 3 G = RepB,D (g) = −1 −2 F = RepB,D (f ) = 2 0 2 4 B,D 1 0 B,D Then, for any v ∈ V represented with respect to B, computation of the representation of f (v) + g(v)         1 3 0 0 1v1 + 3v2 0v1 + 0v2 2 0 v1 + −1 −2 v1 = 2v1 + 0v2  + −1v1 − 2v2  v2 v2 1 0 2 4 1v1 + 0v2 2v1 + 4v2 gives this representation of f + g (v).     (1 + 0)v1 + (3 + 0)v2 1v1 + 3v2 (2 − 1)v1 + (0 − 2)v2  = 1v1 − 2v2  (1 + 2)v1 + (0 + 4)v2 3v1 + 4v2 Thus, the action of f + g is described by this matrix-vector product.     1 3 1v1 + 3v2 v1 1 −2 = 1v1 − 2v2  v 3 4 B,D 2 B 3v1 + 4v2 D This matrix is the entry-by-entry sum of original matrices, e.g., the 1, 1 entry of RepB,D (f + g) is the sum of the 1, 1 entry of F and the 1, 1 entry of G.
f +g

Section IV. Matrix Operations Representing a scalar multiple of a map works the same way. 1.2 Example If t is a transformation represented by RepB,D (t) = 1 1 0 1 so that
B,D

213

v=

v1 v2

B

v1 v1 + v2

= t(v)
D

then the scalar multiple map 5t acts in this way. v= v1 v2 −→
B

5v1 5v1 + 5v2

= 5 · t(v)
D

Therefore, this is the matrix representing 5t. RepB,D (5t) = 5 5 0 5

B,D

1.3 Deﬁnition The sum of two same-sized matrices is their entry-by-entry sum. The scalar multiple of a matrix is the result of entry-by-entry scalar multiplication. 1.4 Remark These extend the vector addition and scalar multiplication operations that we deﬁned in the ﬁrst chapter. 1.5 Theorem Let h, g : V → W be linear maps represented with respect to bases B, D by the matrices H and G, and let r be a scalar. Then the map h + g : V → W is represented with respect to B, D by H + G, and the map r · h : V → W is represented with respect to B, D by rH.
Proof. Exercise 8; generalize the examples above. QED

A notable special case of scalar multiplication is multiplication by zero. For any map 0 · h is the zero homomorphism and for any matrix 0 · H is the zero matrix. 1.6 Example The zero map from any three-dimensional space to any twodimensional space is represented by the 2×3 zero matrix Z= 0 0 0 0 0 0

no matter which domain and codomain bases are used. Exercises
1.7 Perform the indicated operations, if deﬁned. 5 −1 2 2 1 4 (a) + 6 1 1 3 0 5 (b) 6 · 2 1 −1 2 −1 3

214
2 0 1 3 2 3 1 3 2 0 +5 +2 1 3 1 3 −1 −2 1 0 4 1 4 5

Chapter Three. Maps Between Spaces

(c) (d) 4 (e) 3

+ 2 −1 1 0

1.8 Prove Theorem 1.5. (a) Prove that matrix addition represents addition of linear maps. (b) Prove that matrix scalar multiplication represents scalar multiplication of linear maps. 1.9 Prove each, where the operations are deﬁned, where G, H, and J are matrices, where Z is the zero matrix, and where r and s are scalars. (a) Matrix addition is commutative G + H = H + G. (b) Matrix addition is associative G + (H + J) = (G + H) + J. (c) The zero matrix is an additive identity G + Z = G. (d) 0 · G = Z (e) (r + s)G = rG + sG (f ) Matrices have an additive inverse G + (−1) · G = Z. (g) r(G + H) = rG + rH (h) (rs)G = r(sG) 1.10 Fix domain and codomain spaces. In general, one matrix can represent many diﬀerent maps with respect to diﬀerent bases. However, prove that a zero matrix represents only a zero map. Are there other such matrices? 1.11 Let V and W be vector spaces of dimensions n and m. Show that the space L(V, W ) of linear maps from V to W is isomorphic to Mm×n . 1.12 Show that it follows from the prior questions that for any six transformations t1 , . . . , t6 : R2 → R2 there are scalars c1 , . . . , c6 ∈ R such that c1 t1 + · · · + c6 t6 is the zero map. (Hint: this is a bit of a misleading question.) 1.13 The trace of a square matrix is the sum of the entries on the main diagonal (the 1, 1 entry plus the 2, 2 entry, etc.; we will see the signiﬁcance of the trace in Chapter Five). Show that trace(H + G) = trace(H) + trace(G). Is there a similar result for scalar multiplication? 1.14 Recall that the transpose of a matrix M is another matrix, whose i, j entry is the j, i entry of M . Veriﬁy these identities. (a) (G + H)trans = Gtrans + H trans (b) (r · H)trans = r · H trans 1.15 A square matrix is symmetric if each i, j entry equals the j, i entry, that is, if the matrix equals its transpose. (a) Prove that for any H, the matrix H + H trans is symmetric. Does every symmetric matrix have this form? (b) Prove that the set of n×n symmetric matrices is a subspace of Mn×n . 1.16 (a) How does matrix rank interact with scalar multiplication — can a scalar product of a rank n matrix have rank less than n? Greater? (b) How does matrix rank interact with matrix addition — can a sum of rank n matrices have rank less than n? Greater?

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215

IV.2 Matrix Multiplication
After representing addition and scalar multiplication of linear maps in the prior subsection, the natural next map operation to consider is composition. 2.1 Lemma A composition of linear maps is linear.
Proof. (This argument has appeared earlier, as part of the proof that isomor-

phism is an equivalence relation between spaces.) Let h : V → W and g : W → U be linear. The calculation g ◦ h c1 · v1 + c2 · v2 = g h(c1 · v1 + c2 · v2 ) = g c1 · h(v1 ) + c2 · h(v2 ) = c1 · g h(v1 )) + c2 · g(h(v2 ) = c1 · (g ◦ h)(v1 ) + c2 · (g ◦ h)(v2 ) shows that g ◦ h : V → U preserves linear combinations.
QED

To see how the representation of the composite arises out of the representations of the two compositors, consider an example. 2.2 Example Let h : R4 → R2 and g : R2 → R3 , ﬁx bases B ⊂ R4 , C ⊂ R2 , D ⊂ R3 , and let these be the representations.   1 1 4 6 8 2 H = RepB,C (h) = G = RepC,D (g) = 0 1 5 7 9 3 B,C 1 0 C,D To represent the composition g ◦ h : R4 → R3 we ﬁx a v, represent h of v, and then represent g of that. The representation of h(v) is the product of h’s matrix and v’s vector.   v1 v2  4 6 8 2   = 4v1 + 6v2 + 8v3 + 2v4 RepC ( h(v) ) = 5 7 9 3 B,C v3  5v1 + 7v2 + 9v3 + 3v4 C v4 B The representation of g( h(v) ) is the product of g’s matrix and h(v)’s vector.   1 1 4v1 + 6v2 + 8v3 + 2v4 RepD ( g(h(v)) ) = 0 1 5v1 + 7v2 + 9v3 + 3v4 C 1 0 C,D   1 · (4v1 + 6v2 + 8v3 + 2v4 ) + 1 · (5v1 + 7v2 + 9v3 + 3v4 ) = 0 · (4v1 + 6v2 + 8v3 + 2v4 ) + 1 · (5v1 + 7v2 + 9v3 + 3v4 ) 1 · (4v1 + 6v2 + 8v3 + 2v4 ) + 0 · (5v1 + 7v2 + 9v3 + 3v4 ) D Distributing and regrouping on the v’s gives   (1 · 4 + 1 · 5)v1 + (1 · 6 + 1 · 7)v2 + (1 · 8 + 1 · 9)v3 + (1 · 2 + 1 · 3)v4 = (0 · 4 + 1 · 5)v1 + (0 · 6 + 1 · 7)v2 + (0 · 8 + 1 · 9)v3 + (0 · 2 + 1 · 3)v4  (1 · 4 + 0 · 5)v1 + (1 · 6 + 0 · 7)v2 + (1 · 8 + 0 · 9)v3 + (1 · 2 + 0 · 3)v4 D

216

Chapter Three. Maps Between Spaces

which we recognizing as the result of this matrix-vector product.    v1 1·4+1·5 1·6+1·7 1·8+1·9 1·2+1·3 v2    = 0 · 4 + 1 · 5 0 · 6 + 1 · 7 0 · 8 + 1 · 9 0 · 2 + 1 · 3 v3  1 · 4 + 0 · 5 1 · 6 + 0 · 7 1 · 8 + 0 · 9 1 · 2 + 0 · 3 B,D v4 D  Thus, the matrix representing g◦h has the rows of G combined with the columns of H. 2.3 Deﬁnition The matrix-multiplicative product of the m×r matrix G and the r×n matrix H is the m×n matrix P , where pi,j = gi,1 h1,j + gi,2 h2,j + · · · + gi,r hr,j that is, the i, j-th entry of the product is the j-th column.   . . .   . . . gi,1 gi,2 . . . gi,r   GH =    . . . 2.4 Example The matrices  1·4+1·5 1·6+1·7 0 · 4 + 1 · 5 0 · 6 + 1 · 7 1·4+0·5 1·6+0·7 2.5 Example   2 0 4 6 1 5 8 2 the dot product of the i-th row and h1,j h2,j . . . hr,j 

. . .     = . . . 

. . . pi,j . . .

  . . . 

from Example 2.2 combine in this way.   1·8+1·9 1·2+1·3 9 13 17 0 · 8 + 1 · 9 0 · 2 + 1 · 3 = 5 7 9 1·8+0·9 1·2+0·3 4 6 8

 5 3 2

3 7

   2 2·1+0·5 2·3+0·7 = 4 · 1 + 6 · 5 4 · 3 + 6 · 7 = 34 18 8·1+2·5 8·3+2·7

 6 54 38

2.6 Theorem A composition of linear maps is represented by the matrix product of the representatives.
Proof. (This argument parallels Example 2.2.) Let h : V → W and g : W → X

be represented by H and G with respect to bases B ⊂ V , C ⊂ W , and D ⊂ X, of sizes n, r, and m. For any v ∈ V , the k-th component of RepC ( h(v) ) is hk,1 v1 + · · · + hk,n vn and so the i-th component of RepD ( g ◦ h (v) ) is this. gi,1 · (h1,1 v1 + · · · + h1,n vn ) + gi,2 · (h2,1 v1 + · · · + h2,n vn ) + · · · + gi,r · (hr,1 v1 + · · · + hr,n vn )

Section IV. Matrix Operations Distribute and regroup on the v’s. = (gi,1 h1,1 + gi,2 h2,1 + · · · + gi,r hr,1 ) · v1

217

+ · · · + (gi,1 h1,n + gi,2 h2,n + · · · + gi,r hr,n ) · vn Finish by recognizing that the coeﬃcient of each vj gi,1 h1,j + gi,2 h2,j + · · · + gi,r hr,j matches the deﬁnition of the i, j entry of the product GH.
QED

The theorem is an example of a result that supports a deﬁnition. We can picture what the deﬁnition and theorem together say with this arrow diagram (‘wrt’ abbreviates ‘with respect to’). Wwrt
h H G g◦h
B C

g

Vwrt

GH

Xwrt

D

Above the arrows, the maps show that the two ways of going from V to X, straight over via the composition or else by way of W , have the same eﬀect v −→ g(h(v))
g◦h

v −→ h(v) −→ g(h(v))

h

g

(this is just the deﬁnition of composition). Below the arrows, the matrices indicate that the product does the same thing — multiplying GH into the column vector RepB (v) has the same eﬀect as multiplying the column ﬁrst by H and then multiplying the result by G. RepB,D (g ◦ h) = GH = RepC,D (g) RepB,C (h) The deﬁnition of the matrix-matrix product operation does not restrict us to view it as a representation of a linear map composition. We can get insight into this operation by studying it as a mechanical procedure. The striking thing is the way that rows and columns combine. One aspect of that combination is that the sizes of the matrices involved is signiﬁcant. Brieﬂy, m×r times r×n equals m×n. 2.7 Example This product is not deﬁned −1 0 2 10 0 1.1 0 0 0 2

because the number of columns on the left does not equal the number of rows on the right.

218

Chapter Three. Maps Between Spaces

In terms of the underlying maps, the fact that the sizes must match up reﬂects the fact that matrix multiplication is deﬁned only when a corresponding function composition dimension n space −→ dimension r space −→ dimension m space is possible. 2.8 Remark The order in which these things are written can be confusing. In the ‘m×r times r×n equals m×n’ equation, the number written ﬁrst m is the dimension of g’s codomain and is thus the number that appears last in the map dimension description above. The explanation is that while f is done ﬁrst and then g is applied, that composition is written g ◦ f , from the notation ‘g(f (v))’. (Some people try to lessen confusion by reading ‘g ◦ f ’ aloud as “g following f ”.) That order then carries over to matrices: g ◦ f is represented by GF . Another aspect of the way that rows and columns combine in the matrix product operation is that in the deﬁnition of the i, j entry pi,j = gi,
1 h g

h

1 ,j

+ gi,

2

h

2 ,j

+ · · · + gi,

r

h

r ,j

the boxed subscripts on the g’s are column indicators while those on the h’s indicate rows. That is, summation takes place over the columns of G but over the rows of H; left is treated diﬀerently than right, so GH may be unequal to HG. Matrix multiplication is not commutative. 2.9 Example Matrix multiplication hardly ever commutes. Test that by multiplying randomly chosen matrices both ways. 1 3 2 4 5 7 6 8 = 19 43 22 50 5 7 6 8 1 3 2 4 = 23 31 34 46

2.10 Example Commutativity can fail more dramatically: 5 7 while 1 3 isn’t even deﬁned. 2.11 Remark The fact that matrix multiplication is not commutative may be puzzling at ﬁrst sight, perhaps just because most algebraic operations in elementary mathematics are commutative. But on further reﬂection, it isn’t so surprising. After all, matrix multiplication represents function composition, which is not commutative — if f (x) = 2x and g(x) = x+1 then g ◦f (x) = 2x+1 while f ◦ g(x) = 2(x + 1) = 2x + 2. True, this g is not linear and we might have hoped that linear functions commute, but this perspective shows that the failure of commutativity for matrix multiplication ﬁts into a larger context. 2 4 0 0 5 7 6 8 6 8 1 3 2 4 0 0 = 23 31 34 46 0 0

Section IV. Matrix Operations

219

Except for the lack of commutativity, matrix multiplication is algebraically well-behaved. Below are some nice properties and more are in Exercise 23 and Exercise 24. 2.12 Theorem If F , G, and H are matrices, and the matrix products are deﬁned, then the product is associative (F G)H = F (GH) and distributes over matrix addition F (G + H) = F G + F H and (G + H)F = GF + HF .
Proof. Associativity holds because matrix multiplication represents function

composition, which is associative: the maps (f ◦ g) ◦ h and f ◦ (g ◦ h) are equal as both send v to f (g(h(v))). Distributivity is similar. For instance, the ﬁrst one goes f ◦ (g + h) (v) = f (g + h)(v) = f g(v) + h(v) = f (g(v)) + f (h(v)) = f ◦ g(v) + f ◦ h(v) (the third equality uses the linearity of f ). QED 2.13 Remark We could alternatively prove that result by slogging through the indices. For example, associativity goes: the i, j-th entry of (F G)H is (fi,1 g1,1 + fi,2 g2,1 + · · · + fi,r gr,1 )h1,j + (fi,1 g1,2 + fi,2 g2,2 + · · · + fi,r gr,2 )h2,j . . . + (fi,1 g1,s + fi,2 g2,s + · · · + fi,r gr,s )hs,j (where F , G, and H are m×r, r×s, and s×n matrices), distribute fi,1 g1,1 h1,j + fi,2 g2,1 h1,j + · · · + fi,r gr,1 h1,j + fi,1 g1,2 h2,j + fi,2 g2,2 h2,j + · · · + fi,r gr,2 h2,j . . . + fi,1 g1,s hs,j + fi,2 g2,s hs,j + · · · + fi,r gr,s hs,j and regroup around the f ’s fi,1 (g1,1 h1,j + g1,2 h2,j + · · · + g1,s hs,j ) + fi,2 (g2,1 h1,j + g2,2 h2,j + · · · + g2,s hs,j ) . . . + fi,r (gr,1 h1,j + gr,2 h2,j + · · · + gr,s hs,j ) to get the i, j entry of F (GH). Contrast these two ways of verifying associativity, the one in the proof and the one just above. The argument just above is hard to understand in the sense that, while the calculations are easy to check, the arithmetic seems unconnected to any idea (it also essentially repeats the proof of Theorem 2.6 and so is ineﬃcient). The argument in the proof is shorter, clearer, and says why this property “really” holds. This illustrates the comments made in the preamble to the chapter on vector spaces — at least some of the time an argument from higher-level constructs is clearer.

220

Chapter Three. Maps Between Spaces

We have now seen how the representation of the composition of two linear maps is derived from the representations of the two maps. We have called the combination the product of the two matrices. This operation is extremely important. Before we go on to study how to represent the inverse of a linear map, we will explore it some more in the next subsection. Exercises
2.14 Compute, or state “not deﬁned”. (a) 3 −4 2 7 1 2 −7 4 0 0 1 −1 3 A= 5 0.5 0 1 8 5 1 4 (b) 1 4 (d) 1 0 5 3 −1 3 2 1 2 3 3 −1 3 −1 1 1 2 −5 −1 1 1

(c)

2.15 Where 1 −1 5 2 −2 3 B= C= 2 0 4 4 −4 1 compute or state ‘not deﬁned’. (a) AB (b) (AB)C (c) BC (d) A(BC) 2.16 Which products are deﬁned? (a) 3 × 2 times 2 × 3 (b) 2 × 3 times 3 × 2 (c) 2 × 2 times 3 × 3 (d) 3×3 times 2×2 2.17 Give the size of the product or state “not deﬁned”. (a) a 2×3 matrix times a 3×1 matrix (b) a 1×12 matrix times a 12×1 matrix (c) a 2×3 matrix times a 2×1 matrix (d) a 2×2 matrix times a 2×2 matrix 2.18 Find the system of equations resulting from starting with h1,1 x1 + h1,2 x2 + h1,3 x3 = d1 h2,1 x1 + h2,2 x2 + h2,3 x3 = d2 and making this change of variable (i.e., substitution). x1 = g1,1 y1 + g1,2 y2 x2 = g2,1 y1 + g2,2 y2 x3 = g3,1 y1 + g3,2 y2 2.19 As Deﬁnition 2.3 points out, the matrix product operation generalizes the dot product. Is the dot product of a 1×n row vector and a n×1 column vector the same as their matrix-multiplicative product? 2.20 Represent the derivative map on Pn with respect to B, B where B is the natural basis 1, x, . . . , xn . Show that the product of this matrix with itself is deﬁned; what the map does it represent? 2.21 Show that composition of linear transformations on R1 is commutative. Is this true for any one-dimensional space? 2.22 Why is matrix multiplication not deﬁned as entry-wise multiplication? That would be easier, and commutative too. 2.23 (a) Prove that H p H q = H p+q and (H p )q = H pq for positive integers p, q. (b) Prove that (rH)p = rp · H p for any positive integer p and scalar r ∈ R. 2.24 (a) How does matrix multiplication interact with scalar multiplication: is r(GH) = (rG)H? Is G(rH) = r(GH)?

Section IV. Matrix Operations

221

(b) How does matrix multiplication interact with linear combinations: is F (rG + sH) = r(F G) + s(F H)? Is (rF + sG)H = rF H + sGH? 2.25 We can ask how the matrix product operation interacts with the transpose operation. (a) Show that (GH)trans = H trans Gtrans . (b) A square matrix is symmetric if each i, j entry equals the j, i entry, that is, if the matrix equals its own transpose. Show that the matrices HH trans and H trans H are symmetric. 2.26 Rotation of vectors in R3 about an axis is a linear map. Show that linear maps do not commute by showing geometrically that rotations do not commute. 2.27 In the proof of Theorem 2.12 some maps are used. What are the domains and codomains? 2.28 How does matrix rank interact with matrix multiplication? (a) Can the product of rank n matrices have rank less than n? Greater? (b) Show that the rank of the product of two matrices is less than or equal to the minimum of the rank of each factor. 2.29 Is ‘commutes with’ an equivalence relation among n×n matrices? 2.30 (This will be used in the Matrix Inverses exercises.) Here is another property of matrix multiplication that might be puzzling at ﬁrst sight. (a) Prove that the composition of the projections πx , πy : R3 → R3 onto the x and y axes is the zero map despite that neither one is itself the zero map. (b) Prove that the composition of the derivatives d2 /dx2 , d3 /dx3 : P4 → P4 is the zero map despite that neither is the zero map. (c) Give a matrix equation representing the ﬁrst fact. (d) Give a matrix equation representing the second. When two things multiply to give zero despite that neither is zero, each is said to be a zero divisor. 2.31 Show that, for square matrices, (S + T )(S − T ) need not equal S 2 − T 2 . 2.32 Represent the identity transformation id : V → V with respect to B, B for any basis B. This is the identity matrix I. Show that this matrix plays the role in matrix multiplication that the number 1 plays in real number multiplication: HI = IH = H (for all matrices H for which the product is deﬁned). 2.33 In real number algebra, quadratic equations have at most two solutions. That is not so with matrix algebra. Show that the 2 × 2 matrix equation T 2 = I has more than two solutions, where I is the identity matrix (this matrix has ones in its 1, 1 and 2, 2 entries and zeroes elsewhere; see Exercise 32). 2.34 (a) Prove that for any 2 × 2 matrix T there are scalars c0 , . . . , c4 that are not all 0 such that the combination c4 T 4 + c3 T 3 + c2 T 2 + c1 T + c0 I is the zero matrix (where I is the 2×2 identity matrix, with 1’s in its 1, 1 and 2, 2 entries and zeroes elsewhere; see Exercise 32). (b) Let p(x) be a polynomial p(x) = cn xn + · · · + c1 x + c0 . If T is a square matrix we deﬁne p(T ) to be the matrix cn T n + · · · + c1 T + I (where I is the appropriately-sized identity matrix). Prove that for any square matrix there is a polynomial such that p(T ) is the zero matrix. (c) The minimal polynomial m(x) of a square matrix is the polynomial of least degree, and with leading coeﬃcient 1, such that m(T ) is the zero matrix. Find the minimal polynomial of this matrix. √ 3/2 √ −1/2 1/2 3/2

222

Chapter Three. Maps Between Spaces

(This is the representation with respect to E2 , E2 , the standard basis, of a rotation through π/6 radians counterclockwise.) 2.35 The inﬁnite-dimensional space P of all ﬁnite-degree polynomials gives a memorable example of the non-commutativity of linear maps. Let d/dx : P → P be the usual derivative and let s : P → P be the shift map. s a0 + a1 x + · · · + an xn −→ 0 + a0 x + a1 x2 + · · · + an xn+1 Show that the two maps don’t commute d/dx ◦ s = s ◦ d/dx; in fact, not only is (d/dx ◦ s) − (s ◦ d/dx) not the zero map, it is the identity map. 2.36 Recall the notation for the sum of the sequence of numbers a1 , a2 , . . . , an .
n

ai = a1 + a2 + · · · + an
i=1

In this notation, the i, j entry of the product of G and H is this.
r

pi,j =
k=1

gi,k hk,j

Using this notation, (a) reprove that matrix multiplication is associative; (b) reprove Theorem 2.6.

IV.3 Mechanics of Matrix Multiplication
In this subsection we consider matrix multiplication as a mechanical process, putting aside for the moment any implications about the underlying maps. As described earlier, the striking thing about matrix multiplication is the way rows and columns combine. The i, j entry of the matrix product is the dot product of row i of the left matrix with column j of the right one. For instance, here a second row and a third column combine to make a 2, 3 entry.     1 1 9 13 17 5 8 2   4 6 9 3 = 5 7 0 1 5 7 9 3 4 6 8 2 1 0 We can view this as the left matrix acting by multiplying its rows, one at a time, into the columns of the right matrix. Of course, another perspective is that the right matrix uses its columns to act on the left matrix’s rows. Below, we will examine actions from the left and from the right for some simple matrices. The ﬁrst case, the action of a zero matrix, is very easy. 3.1 Example Multiplying by an appropriately-sized zero matrix from the left or from the right 0 0 0 0 1 −1 3 2 1 −1 = 0 0 0 0 0 0 2 1 3 4 0 0 0 0 = 0 0 0 0

results in a zero matrix.

Section IV. Matrix Operations

223

After zero matrices, the matrices whose actions are easiest to understand are the ones with a single nonzero entry. 3.2 Deﬁnition A matrix with all zeroes except for a one in the i, j entry is an i, j unit matrix. 3.3 Example This is the 1, 2 unit matrix with multiplying from the left.    0 1 7 0 0 5 6 = 0 7 8 0 0 0 three rows and two columns,  8 0 0

Acting from the left, an i, j unit matrix copies row j of the multiplicand into row i of the result. From the right an i, j unit matrix copies column i of the multiplicand into column j of the result.      1 2 3 0 1 0 1 4 5 6 0 0 = 0 4 7 8 9 0 0 0 7 3.4 Example Rescaling these matrices simply rescales the result. This is the action from the left of the matrix that is twice the one in the prior example.     0 2 14 16 5 6 0 0 =0 0 7 8 0 0 0 0 And this is the action of prior example.  1 4 7 the matrix that is minus three times the one from the 2 5 8  3 0 6 0 9 0   −3 0 0  = 0 0 0  −3 −12 −21

Next in complication are matrices with two nonzero entries. There are two cases. If a left-multiplier has entries in diﬀerent rows then their actions don’t interact. 3.5 Example   1 0 0 1 0 0 2 4 0 0 0 7

2 5 8

  3 1 6 = (0 9 0  1 = 0 0  1 = 14 0

  0 0 0 0 + 0 0 0 0 0   2 3 0 0 0 0 + 14 16 0 0 0 0  2 3 16 18 0 0 0 0 0

  0 1 2) 4 0 7  0 18 0

2 5 8

 3 6 9

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Chapter Three. Maps Between Spaces

But if the left-multiplier’s nonzero entries are in the same row then that row of the result is a combination. 3.6 Example   1 0 2 1 0 0 0 4 0 0 0 7

2 5 8

  3 1 6 = (0 9 0  1 = 0 0  15 =0 0

  0 0 0 0 + 0 0 0 0 0   2 3 14 16 0 0 +  0 0 0 0 0 0  18 21 0 0 0 0 0 0 0

  2 1 0) 4 0 7  18 0 0

2 5 8

 3 6 9

Right-multiplication acts in the same way, with columns. These observations about matrices that are mostly zeroes extend to arbitrary matrices. 3.7 Lemma In a product of two matrices G and H, the columns of GH are formed by taking G times the columns of H     . . . . . . . .   . .  . .  G · h1 ··· ··· G · hn  hn  = G · h1     . . . . . . . . . . . . and the rows of GH are formed by taking the rows of G times H     · · · g1 · · · · · · g1 · H · · ·     . .  ·H =  . . . .     · · · gr · · · · · · gr · H · · · (ignoring the extra parentheses).
Proof. We will show the 2×2 case and leave the general case as an exercise.

GH =

g1,1 g2,1

g1,2 g2,2

h1,1 h2,1

h1,2 h2,2

=

g1,1 h1,1 + g1,2 h2,1 g2,1 h1,1 + g2,2 h2,1

g1,1 h1,2 + g1,2 h2,2 g2,1 h1,2 + g2,2 h2,2

The right side of the ﬁrst equation in the result G h1,1 h2,1 G h1,2 h2,2 = g1,1 h1,1 + g1,2 h2,1 g2,1 h1,1 + g2,2 h2,1 g1,1 h1,2 + g1,2 h2,2 g2,1 h1,2 + g2,2 h2,2

is indeed the same as the right side of GH, except for the extra parentheses (the ones marking the columns as column vectors). The other equation is similarly easy to recognize. QED

Section IV. Matrix Operations

225

An application of those observations is that there is a matrix that just copies out the rows and columns. 3.8 Deﬁnition The main diagonal (or principle diagonal or diagonal ) of a square matrix goes from the upper left to the lower right. 3.9 Deﬁnition An identity matrix is square and has with all entries zero except for ones in the main diagonal.   1 0 ... 0 0 1 . . . 0   In×n =   . .   . 0 0 ... 1 3.10 Example The 3×3 the left  1 0 0 1 0 0 and from the right.  identity leaves its multiplicand unchanged both from  0 2 0  1 1 −7   3 6 2 3 8 =  1 1 0 −7  3 6 3 8 1 0

2 1 −7

 3 6 1 3 8 0 1 0 0

0 1 0

  0 2 0 =  1 1 −7

 3 6 3 8 1 0

3.11 Example So does the 2×2 identity matrix.     1 −2 1 −2 0 −2 0 −2 1 0     1 −1 0 1 = 1 −1 4 3 4 3 In short, an identity matrix is the identity element of the set of n×n matrices with respect to the operation of matrix multiplication. We next see two ways to generalize the identity matrix. The ﬁrst is that if the ones are relaxed to arbitrary reals, the resulting matrix will rescale whole rows or columns. 3.12 Deﬁnition A diagonal matrix is square diagonal.  a1,1 0 ... 0  0 a2,2 . . . 0   . .  . 0 0 ... an,n and has zeros oﬀ the main     

226

Chapter Three. Maps Between Spaces

3.13 Example From the left, the action of multiplication by a diagonal matrix is to rescales the rows. 2 0 0 −1 2 −1 1 4 −1 3 4 4 = 4 1 2 −3 8 −4 −2 −4

From the right such a matrix rescales the columns.   3 0 0 1 2 1  3 4 0 2 0 = 2 2 2 6 4 0 0 −2

−2 −4

The second generalization of identity matrices is that we can put a single one in each row and column in ways other than putting them down the diagonal. 3.14 Deﬁnition A permutation matrix is square and is all zeros except for a single one in each row and column. 3.15 Example From the  0 0 1 0 0 1 left these  1 1 0 4 0 7 matrices permute   2 3 7 8 5 6 = 1 2 8 9 4 5 rows.  9 3 6

From the right they permute columns.   0 0 1 2 3 4 5 6 1 0 0 1 7 8 9

  2 1 0 = 5 8 0

3 6 9

 1 4 7

We ﬁnish this subsection by applying these observations to get matrices that perform Gauss’ method and Gauss-Jordan reduction. 3.16 Example We have seen how to produce a matrix Multiplying by this diagonal matrix rescales the second factor of three.     1 0 0 0 2 1 1 0 2 0 3 0 0 1/3 1 −1 = 0 1 0 0 1 1 0 2 0 1 0 that will rescale rows. row of the other by a 1 3 2  1 −3 0

We have seen how to produce a matrix that will swap rows. Multiplying by this permutation matrix swaps the ﬁrst and third rows.      0 0 1 0 2 1 1 1 0 2 0 0 1 0 0 1 3 −3 = 0 1 3 −3 1 0 0 1 0 2 0 0 2 1 1

Section IV. Matrix Operations

227

To see how to perform a pivot, we observe something about those two examples. The matrix that rescales the second row by a factor of three arises in this way from the identity.     1 0 0 1 0 0 3ρ2 0 1 0 −→ 0 3 0 0 0 1 0 0 1 Similarly, the matrix that swaps ﬁrst and third    1 0 0 0 ρ1 ↔ρ3 0 1 0 −→ 0 0 0 1 1 3.17 Example The 3×3  1 0 0 will, when it acts  1 0 0 rows arises in this way.  0 1 1 0 0 0

matrix that arises as   0 0 1 0 −2ρ2 +ρ3 1 0 −→ 0 1 0 1 0 −2

 0 0 1

from the left,  1 0 0 1 0 0 0 −2 1

perform the pivot operation −2ρ2 + ρ3 .    1 0 2 0 0 2 0 1 3 −3 = 0 1 3 −3 0 0 −5 7 2 1 1

3.18 Deﬁnition The elementary reduction matrices are obtained from identity matrices with one Gaussian operation. We denote them: (1) I −→ Mi (k) for k = 0; (2) I −→ Pi,j for i = j; (3) I −→ Ci,j (k) for i = j. 3.19 Lemma Gaussian reduction can be done through matrix multiplication. (1) If H −→ G then Mi (k)H = G. (2) If H −→ G then Pi,j H = G. (3) If H −→ G then Ci,j (k)H = G.
Proof. Clear. QED
kρi +ρj ρi ↔ρj kρi kρi +ρj ρi ↔ρj kρi

228

Chapter Three. Maps Between Spaces

3.20 Example This is the ﬁrst system, from the ﬁrst chapter, on which we performed Gauss’ method. 3x3 = 9 x1 + 5x2 − 2x3 = 2 (1/3)x1 + 2x2 =3 It can be reduced with matrix   0 0 1 0 0 1 0  1 1/3 1 0 0 triple the ﬁrst row,  3 0 0 1 0 0 and then add −1  1 −1 0 multiplication. Swap the ﬁrst and third rows,    1/3 2 0 3 0 3 9 5 −2 2 =  1 5 −2 2 2 0 3 0 0 3 9

 0 1/3 0  1 1 0

2 0 5 −2 0 3

  1 6 3 2 = 1 5 9 0 0 the second.   9 1 6 2 = 0 −1 0 0 9

0 −2 3

 9 2 9

times the ﬁrst row to  1 6 0 0 0 1 0 1 5 −2 0 1 0 0 3

0 −2 3

 9 −7 9

Now back substitution will give the solution. 3.21 Example ending the prior  1 0 0 Gauss-Jordan reduction works the same way. example, ﬁrst adjust the leading entries    0 0 1 6 0 9 1 6 0 −1 0  0 −1 −2 −7 = 0 1 2 0 1/3 0 0 3 9 0 0 1 For the matrix  9 7 3

and to ﬁnish, clear the third column and    1 0 0 1 1 −6 0 0 1 0 0 1 −2 0 0 0 1 0 0 1 0

then the second column.    6 0 9 1 0 0 3 1 2 7 = 0 1 0 1 0 1 3 0 0 1 3

We have observed the following result, which we shall use in the next subsection. 3.22 Corollary For any matrix H there are elementary reduction matrices R1 , . . . , Rr such that Rr · Rr−1 · · · R1 · H is in reduced echelon form. Until now we have taken the point of view that our primary objects of study are vector spaces and the maps between them, and have adopted matrices only for computational convenience. This subsection show that this point of view isn’t the whole story. Matrix theory is a fascinating and fruitful area. In the rest of this book we shall continue to focus on maps as the primary objects, but we will be pragmatic — if the matrix point of view gives some clearer idea then we shall use it.

Section IV. Matrix Operations Exercises

229

3.23 Predict the result of each multiplication by an elementary reduction matrix, and then check by multiplying it out. 3 0 1 2 4 0 1 2 1 0 1 2 (a) (b) (c) 0 0 3 4 0 2 3 4 −2 1 3 4 1 2 1 −1 1 2 0 1 (e) 3 4 0 1 3 4 1 0 3.24 The need to take linear combinations of rows and columns in tables of numbers arises often in practice. For instance, this is a map of part of Vermont and New York. (d)
Swanton

In part because of Lake Champlain, there are no roads directly connecting some pairs of towns. For instance, there is no way to go from Winooski to Grand Isle without going through Colchester. (Of course, many other roads and towns have been left oﬀ to simplify the graph. From top to bottom of this map is about forty miles.)

Grand Isle

Colchester Winooski Burlington

(a) The incidence matrix of a map is the square matrix whose i, j entry is the number of roads from city i to city j. Produce the incidence matrix of this map (take the cities in alphabetical order). (b) A matrix is symmetric if it equals its transpose. Show that an incidence matrix is symmetric. (These are all two-way streets. Vermont doesn’t have many one-way streets.) (c) What is the signiﬁcance of the square of the incidence matrix? The cube? 3.25 This table gives the number of hours of each type done by each worker, and the associated pay rates. Use matrices to compute the wages due. wage regular overtime Alan 40 12 regular \$25.00 Betty 35 6 overtime \$45.00 Catherine 40 18 Donald 28 0 (Remark. This illustrates, as did the prior problem, that in practice we often want to compute linear combinations of rows and columns in a context where we really aren’t interested in any associated linear maps.) 3.26 Find the product of this matrix with its transpose. cos θ sin θ − sin θ cos θ

230

Chapter Three. Maps Between Spaces

3.27 Prove that the diagonal matrices form a subspace of Mn×n . What is its dimension? 3.28 Does the identity matrix represent the identity map if the bases are unequal? 3.29 Show that every multiple of the identity commutes with every square matrix. Are there other matrices that commute with all square matrices? 3.30 Prove or disprove: nonsingular matrices commute. 3.31 Show that the product of a permutation matrix and its transpose is an identity matrix. 3.32 Show that if the ﬁrst and second rows of G are equal then so are the ﬁrst and second rows of GH. Generalize. 3.33 Describe the product of two diagonal matrices. 3.34 Write 1 0 −3 3 as the product of two elementary reduction matrices. 3.35 Show that if G has a row of zeros then GH (if deﬁned) has a row of zeros. Does that work for columns? 3.36 Show that the set of unit matrices forms a basis for Mn×m . 3.37 Find the formula for the n-th power of this matrix. 1 1 1 0

3.38 The trace of a square matrix is the sum of the entries on its diagonal (its signiﬁcance appears in Chapter Five). Show that trace(GH) = trace(HG). 3.39 A square matrix is upper triangular if its only nonzero entries lie above, or on, the diagonal. Show that the product of two upper triangular matrices is upper triangular. Does this hold for lower triangular also? 3.40 A square matrix is a Markov matrix if each entry is between zero and one and the sum along each row is one. Prove that a product of Markov matrices is Markov. 3.41 Give an example of two matrices of the same rank with squares of diﬀering rank. 3.42 Combine the two generalizations of the identity matrix, the one allowing entires to be other than ones, and the one allowing the single one in each row and column to be oﬀ the diagonal. What is the action of this type of matrix? 3.43 On a computer multiplications are more costly than additions, so people are interested in reducing the number of multiplications used to compute a matrix product. (a) How many real number multiplications are needed in formula we gave for the product of a m×r matrix and a r×n matrix? (b) Matrix multiplication is associative, so all associations yield the same result. The cost in number of multiplications, however, varies. Find the association requiring the fewest real number multiplications to compute the matrix product of a 5×10 matrix, a 10×20 matrix, a 20×5 matrix, and a 5×1 matrix. (c) (Very hard.) Find a way to multiply two 2 × 2 matrices using only seven multiplications instead of the eight suggested by the naive approach. ? 3.44 If A and B are square matrices of the same size such that ABAB = 0, does it follow that BABA = 0? [Putnam, 1990, A-5]

Section IV. Matrix Operations

231

3.45 Demonstrate these four assertions to get an alternate proof that column rank equals row rank. [Am. Math. Mon., Dec. 1966] (a) y · y = 0 iﬀ y = 0. (b) Ax = 0 iﬀ Atrans Ax = 0. (c) dim(R(A)) = dim(R(Atrans A)). (d) col rank(A) = col rank(Atrans ) = row rank(A). 3.46 Prove (where A is an n × n matrix and so deﬁnes a transformation of any n-dimensional space V with respect to B, B where B is a basis) that dim(R(A) ∩ N (A)) = dim(R(A)) − dim(R(A2 )). Conclude (a) N (A) ⊂ R(A) iﬀ dim(N (A)) = dim(R(A)) − dim(R(A2 )); (b) R(A) ⊆ N (A) iﬀ A2 = 0; (c) R(A) = N (A) iﬀ A2 = 0 and dim(N (A)) = dim(R(A)) ; (d) dim(R(A) ∩ N (A)) = 0 iﬀ dim(R(A)) = dim(R(A2 )) ; (e) (Requires the Direct Sum subsection, which is optional.) V = R(A) ⊕ N (A) iﬀ dim(R(A)) = dim(R(A2 )). [Ackerson]

IV.4 Inverses
We now consider how to represent the inverse of a linear map. We start by recalling some facts about function inverses.∗ Some functions have no inverse, or have an inverse on the left side or right side only. 4.1 Example Where π : R3 → R2 is the projection map   x y  → x y z and η : R2 → R3 is the embedding x y   x → y  0

the composition π ◦ η is the identity map on R2 .   x η x x π −→ y  −→ y y 0 We say π is a left inverse map of η or, what is the same thing, that η is a right inverse map of π. However, composition in the other order η ◦ π doesn’t give the identity map — here is a vector that is not sent to itself under η ◦ π.     0 0 η π 0 −→ 0 −→ 0 0 1 0

232

Chapter Three. Maps Between Spaces

In fact, the projection π has no left inverse at all. For, if f were to be a left inverse of π then we would have     x x f x π y  −→ −→ y  y z z for all of the inﬁnitely many z’s. But no function f can send a single argument to more than one value. (An example of a function with no inverse on either side is the zero transformation on R2 .) Some functions have a two-sided inverse map, another function that is the inverse of the ﬁrst, both from the left and from the right. For instance, the map given by v → 2 · v has the two-sided inverse v → (1/2) · v. In this subsection we will focus on two-sided inverses. The appendix shows that a function has a two-sided inverse if and only if it is both one-to-one and onto. The appendix also shows that if a function f has a two-sided inverse then it is unique, and so it is called ‘the’ inverse, and is denoted f −1 . So our purpose in this subsection is, where a linear map h has an inverse, to ﬁnd the relationship between RepB,D (h) and RepD,B (h−1 ) (recall that we have shown, in Theorem 2.21 of Section II of this chapter, that if a linear map has an inverse then the inverse is a linear map also). 4.2 Deﬁnition A matrix G is a left inverse matrix of the matrix H if GH is the identity matrix. It is a right inverse matrix if HG is the identity. A matrix H with a two-sided inverse is an invertible matrix. That two-sided inverse is called the inverse matrix and is denoted H −1 . Because of the correspondence between linear maps and matrices, statements about map inverses translate into statements about matrix inverses. 4.3 Lemma If a matrix has both a left inverse and a right inverse then the two are equal. 4.4 Theorem A matrix is invertible if and only if it is nonsingular.
Proof. (For both results.) Given a matrix H, ﬁx spaces of appropriate dimension for the domain and codomain. Fix bases for these spaces. With respect to these bases, H represents a map h. The statements are true about the map and therefore they are true about the matrix. QED

4.5 Lemma A product of invertible matrices is invertible — if G and H are invertible and if GH is deﬁned then GH is invertible and (GH)−1 = H −1 G−1 .
Proof. (This is just like the prior proof except that it requires two maps.) Fix

appropriate spaces and bases and consider the represented maps h and g. Note that h−1 g −1 is a two-sided map inverse of gh since (h−1 g −1 )(gh) = h−1 (id)h = h−1 h = id and (gh)(h−1 g −1 ) = g(id)g −1 = gg −1 = id. This equality is reﬂected in the matrices representing the maps, as required. QED

Section IV. Matrix Operations

233

Here is the arrow diagram giving the relationship between map inverses and matrix inverses. It is a special case of the diagram for function composition and matrix multiplication. Wwrt
h H H id I
−1

C

h−1

Vwrt

B

Vwrt

B

Beyond its place in our general program of seeing how to represent map operations, another reason for our interest in inverses comes from solving linear systems. A linear system is equivalent to a matrix equation, as here. x1 + x2 = 3 2x1 − x2 = 2 ⇐⇒ 1 2 1 −1 x1 x2 = 3 2 (∗)

By ﬁxing spaces and bases (e.g., R2 , R2 and E2 , E2 ), we take the matrix H to represent some map h. Then solving the system is the same as asking: what domain vector x is mapped by h to the result d ? If we could invert h then we could solve the system by multiplying RepD,B (h−1 ) · RepD (d) to get RepB (x). 4.6 Example We can ﬁnd a left inverse for the matrix just given m p n q 1 2 1 −1 = 1 0 0 1

by using Gauss’ method to solve the resulting linear system. m + 2n m− n =1 =0 p + 2q = 0 p− q=1

Answer: m = 1/3, n = 1/3, p = 2/3, and q = −1/3. This matrix is actually the two-sided inverse of H, as can easily be checked. With it we can solve the system (∗) above by applying the inverse. x y = 1/3 2/3 1/3 −1/3 3 2 = 5/3 4/3

4.7 Remark Why solve systems this way, when Gauss’ method takes less arithmetic (this assertion can be made precise by counting the number of arithmetic operations, as computer algorithm designers do)? Beyond its conceptual appeal of ﬁtting into our program of discovering how to represent the various map operations, solving linear systems by using the matrix inverse has at least two advantages.

234

Chapter Three. Maps Between Spaces

First, once the work of ﬁnding an inverse has been done, solving a system with the same coeﬃcients but diﬀerent constants is easy and fast: if we change the entries on the right of the system (∗) then we get a related problem 1 2 wtih a related solution method. x y = 1/3 2/3 1/3 −1/3 5 1 = 2 3 1 −1 x y = 5 1

In applications, solving many systems having the same matrix of coeﬃcients is common. Another advantage of inverses is that we can explore a system’s sensitivity to changes in the constants. For example, tweaking the 3 on the right of the system (∗) to 1 1 x1 3.01 = 2 −1 x2 2 can be solved with the inverse. 1/3 2/3 1/3 −1/3 3.01 2 = (1/3)(3.01) + (1/3)(2) (2/3)(3.01) − (1/3)(2)

to show that x1 changes by 1/3 of the tweak while x2 moves by 2/3 of that tweak. This sort of analysis is used, for example, to decide how accurately data must be speciﬁed in a linear model to ensure that the solution has a desired accuracy. We ﬁnish by describing the computational procedure usually used to ﬁnd the inverse matrix. 4.8 Lemma A matrix is invertible if and only if it can be written as the product of elementary reduction matrices. The inverse can be computed by applying to the identity matrix the same row steps, in the same order, as are used to Gauss-Jordan reduce the invertible matrix.
Proof. A matrix H is invertible if and only if it is nonsingular and thus Gauss-

Jordan reduces to the identity. By Corollary 3.22 this reduction can be done with elementary matrices Rr · Rr−1 . . . R1 · H = I. This equation gives the two halves of the result. First, elementary matrices are invertible and their inverses are also elemen−1 −1 tary. Applying Rr to the left of both sides of that equation, then Rr−1 , etc., −1 −1 gives H as the product of elementary matrices H = R1 · · · Rr · I (the I is here to cover the trivial r = 0 case). Second, matrix inverses are unique and so comparison of the above equation with H −1 H = I shows that H −1 = Rr · Rr−1 . . . R1 · I. Therefore, applying R1 to the identity, followed by R2 , etc., yields the inverse of H. QED

Section IV. Matrix Operations 4.9 Example To ﬁnd the inverse of 1 2 1 −1

235

we do Gauss-Jordan reduction, meanwhile performing the same operations on the identity. For clerical convenience we write the matrix and the identity sideby-side, and do the reduction steps together. 1 2 1 −1 1 0 0 1
−2ρ1 +ρ2

−→

1 0 1 0

1 −3 1 1

1 −2 1 2/3 1/3 2/3

0 1 0 −1/3 1/3 −1/3

−1/3ρ2

−→ −→

−ρ2 +ρ1

1 0 0 1

This calculation has found the inverse. 1 2 4.10 Example This  0 3 −1 1 1 0 1 0 1 −1 0 0 1 −1
−1

=

1/3 2/3

1/3 −1/3

one happens to start with a row swap.    0 0 1 0 1 0 1 0 ρ1 ↔ρ2 0 3 −1 1 0 0 1 0 −→ 1 −1 0 0 0 1 0 1   1 0 1 0 1 0 −ρ1 +ρ3 0 3 −1 1 0 0 −→ 0 −1 −1 0 −1 1 . . .   1 0 0 1/4 1/4 3/4 0 1 0 1/4 1/4 −1/4 −→ 0 0 1 −1/4 3/4 −3/4

4.11 Example A non-invertible matrix is detected by the fact that the left half won’t reduce to the identity. 1 1 2 2 1 0 0 1
−2ρ1 +ρ2

−→

1 0

1 0

1 −2

0 1

This procedure will ﬁnd the inverse of a general n×n matrix. The 2×2 case is handy. 4.12 Corollary The inverse for a 2×2 matrix exists and equals a c if and only if ad − bc = 0. b d
−1

=

d −b −c a

236

Chapter Three. Maps Between Spaces

Proof. This computation is Exercise 22.

QED

We have seen here, as in the Mechanics of Matrix Multiplication subsection, that we can exploit the correspondence between linear maps and matrices. So we can fruitfully study both maps and matrices, translating back and forth to whichever helps us the most. Over the entire four subsections of this section we have developed an algebra system for matrices. We can compare it with the familiar algebra system for the real numbers. Here we are working not with numbers but with matrices. We have matrix addition and subtraction operations, and they work in much the same way as the real number operations, except that they only combine same-sized matrices. We also have a matrix multiplication operation and an operation inverse to multiplication. These are somewhat like the familiar real number operations (associativity, and distributivity over addition, for example), but there are diﬀerences (failure of commutativity, for example). And, we have scalar multiplication, which is in some ways another extension of real number multiplication. This matrix system provides an example that algebra systems other than the elementary one can be interesting and useful. Exercises
4.13 Supply the intermediate steps in Example 4.10. 4.14 Use Corollary 4.12 to decide if each matrix has an inverse. 2 1 0 4 2 −3 (a) (b) (c) −1 1 1 −3 −4 6 4.15 For each invertible matrix in the prior problem, use Corollary 4.12 to ﬁnd its inverse. 4.16 Find the inverse, if it exists, by using the Gauss-Jordan method. Check the answers for the 2×2 matrices with Corollary 4.12. 1 1 3 3 1 2 1/2 2 −4 0 2 4 (a) (b) (c) (d) 0 2 3 1 −1 2 −1 1 0 0 1 5 2 2 3 4 (e) 0 −2 (f ) 1 −2 −3 2 3 −2 4 −2 −3 4.17 What matrix has this one for its inverse? 1 3 2 5 4.18 How does the inverse operation interact with scalar multiplication and addition of matrices? (a) What is the inverse of rH? (b) Is (H + G)−1 = H −1 + G−1 ? 4.19 Is (T k )−1 = (T −1 )k ? 4.20 Is H −1 invertible? 4.21 For each real number θ let tθ : R2 → R2 be represented with respect to the standard bases by this matrix. cos θ − sin θ sin θ cos θ Show that tθ1 +θ2 = tθ1 · tθ2 . Show also that tθ −1 = t−θ .

Section IV. Matrix Operations
4.22 Do the calculations for the proof of Corollary 4.12. 4.23 Show that this matrix 1 0 1 H= 0 1 0

237

has inﬁnitely many right inverses. Show also that it has no left inverse. 4.24 In Example 4.1, how many left inverses has η? 4.25 If a matrix has inﬁnitely many right-inverses, can it have inﬁnitely many left-inverses? Must it have? 4.26 Assume that H is invertible and that HG is the zero matrix. Show that G is a zero matrix. 4.27 Prove that if H is invertible then the inverse commutes with a matrix GH −1 = H −1 G if and only if H itself commutes with that matrix GH = HG. 4.28 Show that if T is square and if T 4 is the zero matrix then (I − T )−1 = I + T + T 2 + T 3 . Generalize. 4.29 Let D be diagonal. Describe D2 , D3 , . . . , etc. Describe D−1 , D−2 , . . . , etc. Deﬁne D0 appropriately. 4.30 Prove that any matrix row-equivalent to an invertible matrix is also invertible. 4.31 The ﬁrst question below appeared as Exercise 28. (a) Show that the rank of the product of two matrices is less than or equal to the minimum of the rank of each. (b) Show that if T and S are square then T S = I if and only if ST = I. 4.32 Show that the inverse of a permutation matrix is its transpose. 4.33 The ﬁrst two parts of this question appeared as Exercise 25. (a) Show that (GH)trans = H trans Gtrans . (b) A square matrix is symmetric if each i, j entry equals the j, i entry (that is, if the matrix equals its transpose). Show that the matrices HH trans and H trans H are symmetric. (c) Show that the inverse of the transpose is the transpose of the inverse. (d) Show that the inverse of a symmetric matrix is symmetric. 4.34 The items starting this question appeared as Exercise 30. (a) Prove that the composition of the projections πx , πy : R3 → R3 is the zero map despite that neither is the zero map. (b) Prove that the composition of the derivatives d2 /dx2 , d3 /dx3 : P4 → P4 is the zero map despite that neither map is the zero map. (c) Give matrix equations representing each of the prior two items. When two things multiply to give zero despite that neither is zero, each is said to be a zero divisor. Prove that no zero divisor is invertible. 4.35 In real number algebra, there are exactly two numbers, 1 and −1, that are their own multiplicative inverse. Does H 2 = I have exactly two solutions for 2×2 matrices? 4.36 Is the relation ‘is a two-sided inverse of’ transitive? Reﬂexive? Symmetric? 4.37 Prove: if the sum of the elements of a square matrix is k, then the sum of the elements in each row of the inverse matrix is 1/k. [Am. Math. Mon., Nov. 1951]

238

Chapter Three. Maps Between Spaces

V

Change of Basis

Representations, whether of vectors or of maps, vary with the bases. For instance, with respect to the two bases E2 and B= 1 1 , 1 −1

for R2 , the vector e1 has two diﬀerent representations. RepE2 (e1 ) = 1 0 RepB (e1 ) = 1/2 1/2

Similarly, with respect to E2 , E2 and E2 , B, the identity map has two diﬀerent representations. RepE2 ,E2 (id) = 1 0 0 1 RepE2 ,B (id) = 1/2 1/2 1/2 −1/2

With our point of view that the objects of our studies are vectors and maps, in ﬁxing bases we are adopting a scheme of tags or names for these objects, that are convienent for computation. We will now see how to translate among these names — we will see exactly how representations vary as the bases vary.

V.1 Changing Representations of Vectors
In converting RepB (v) to RepD (v) the underlying vector v doesn’t change. Thus, this translation is accomplished by the identity map on the space, described so that the domain space vectors are represented with respect to B and the codomain space vectors are represented with respect to D. Vw.r.t.   id Vw.r.t.
B

D

(The diagram is vertical to ﬁt with the ones in the next subsection.) 1.1 Deﬁnition The change of basis matrix for bases B, D ⊂ V is the representation of the identity map id : V → V with respect to those bases.   . . . . . .   RepB,D (id) = RepD (β1 ) ··· RepD (βn )   . . . . . .

Section V. Change of Basis

239

1.2 Lemma Left-multiplication by the change of basis matrix for B, D converts a representation with respect to B to one with respect to D. Conversly, if leftmultiplication by a matrix changes bases M · RepB (v) = RepD (v) then M is a change of basis matrix.
Proof. For the ﬁrst sentence, for each v, as matrix-vector multiplication repre-

sents a map application, RepB,D (id) · RepB (v) = RepD ( id(v) ) = RepD (v). For the second sentence, with respect to B, D the matrix M represents some linear map, whose action is v → v, and is therefore the identity map. QED 1.3 Example With these bases for R2 , B= because RepD ( id( 2 )) = 1 −1/2 3/2 RepD ( id(
D

2 1 , 1 0

D=

−1 1 , 1 1

1 )) = 0

−1/2 1/2

D

the change of basis matrix is this. RepB,D (id) = −1/2 3/2 −1/2 1/2

We can see this matrix at work by ﬁnding the two representations of e2 RepB ( 0 )= 1 1 −2 RepD ( 0 )= 1 1/2 1/2

and checking that the conversion goes as expected. −1/2 3/2 −1/2 1/2 1 −2 = 1/2 1/2

We ﬁnish this subsection by recognizing that the change of basis matrices are familiar. 1.4 Lemma A matrix changes bases if and only if it is nonsingular.
Proof. For one direction, if left-multiplication by a matrix changes bases then

the matrix represents an invertible function, simply because the function is inverted by changing the bases back. Such a matrix is itself invertible, and so nonsingular. To ﬁnish, we will show that any nonsingular matrix M performs a change of basis operation from any given starting basis B to some ending basis. Because the matrix is nonsingular, it will Gauss-Jordan reduce to the identity, so there are elementatry reduction matrices such that Rr · · · R1 · M = I. Elementary matrices are invertible and their inverses are also elementary, so multiplying from the left ﬁrst by Rr −1 , then by Rr−1 −1 , etc., gives M as a product of

240

Chapter Three. Maps Between Spaces

elementary matrices M = R1 −1 · · · Rr −1 . Thus, we will be done if we show that elementary matrices change a given basis to another basis, for then Rr −1 changes B to some other basis Br , and Rr−1 −1 changes Br to some Br−1 , . . . , and the net eﬀect is that M changes B to B1 . We will prove this about elementary matrices by covering the three types as separate cases. Applying a row-multiplication matrix     c1 c1 .  .  .  .  .  .  Mi (k)  ci  = kci      .  .  .  .  . . cn cn changes a representation with respect to β1 , . . . , βi , . . . , βn to one with respect to β1 , . . . , (1/k)βi , . . . , βn in this way. v = c1 · β1 + · · · + ci · βi + · · · + cn · βn → c1 · β1 + · · · + kci · (1/k)βi + · · · + cn · βn = v Similarly, left-multiplication by a row-swap matrix Pi,j changes a representation with respect to the basis β1 , . . . , βi , . . . , βj , . . . , βn into one with respect to the basis β1 , . . . , βj , . . . , βi , . . . , βn in this way. v = c1 · β1 + · · · + ci · βi + · · · + cj βj + · · · + cn · βn → c1 · β1 + · · · + cj · βj + · · · + ci · βi + · · · + cn · βn = v And, a representation with respect to β1 , . . . , βi , . . . , βj , . . . , βn changes via left-multiplication by a row-combination matrix Ci,j (k) into a representation with respect to β1 , . . . , βi − k βj , . . . , βj , . . . , βn v = c1 · β1 + · · · + ci · βi + cj βj + · · · + cn · βn → c1 · β1 + · · · + ci · (βi − k βj ) + · · · + (kci + cj ) · βj + · · · + cn · βn = v (the deﬁnition of reduction matrices speciﬁes that i = k and k = 0 and so this last one is a basis). QED 1.5 Corollary A matrix is nonsingular if and only if it represents the identity map with respect to some pair of bases. In the next subsection we will see how to translate among representations of maps, that is, how to change RepB,D (h) to RepB,D (h). The above corollary ˆ ˆ is a special case of this, where the domain and range are the same space, and where the map is the identity map.

Section V. Change of Basis Exercises
1.6 In R2 , where D= 2 −2 , 1 4

241

ﬁnd the change of basis matrices from D to E2 and from E2 to D. Multiply the two. 1.7 Find the change of basis matrix for B, D ⊆ R2 . 1 1 (a) B = E2 , D = e2 , e1 (b) B = E2 , D = , 2 4 1 1 −1 2 0 1 , D = E2 (d) B = , ,D= , , 2 4 1 2 4 3 1.8 For the bases in Exercise 7, ﬁnd the change of basis matrix in the other direction, from D to B. 1.9 Find the change of basis matrix for each B, D ⊆ P2 . (a) B = 1, x, x2 , D = x2 , 1, x (b) B = 1, x, x2 , D = 1, 1+x, 1+x+x2 2 2 (c) B = 2, 2x, x , D = 1 + x , 1 − x2 , x + x2 1.10 Decide if each changes bases on R2 . To what basis is E2 changed? 5 0 2 1 −1 4 1 −1 (c) (d) (a) (b) 0 4 3 1 2 −8 1 1 1.11 Find bases such that this matrix represents the identity map with respect to those bases. 3 1 4 2 −1 1 0 0 4 1.12 Conside the vector space of real-valued functions with basis sin(x), cos(x) . Show that 2 sin(x)+cos(x), 3 cos(x) is also a basis for this space. Find the change of basis matrix in each direction. 1.13 Where does this matrix cos(2θ) sin(2θ) sin(2θ) − cos(2θ) (c) B = send the standard basis for R2 ? Any other bases? Hint. Consider the inverse. 1.14 What is the change of basis matrix with respect to B, B? 1.15 Prove that a matrix changes bases if and only if it is invertible. 1.16 Finish the proof of Lemma 1.4. 1.17 Let H be a n×n nonsingular matrix. What basis of Rn does H change to the standard basis? 1.18 (a) In P3 with basis B = 1 + x, 1 − x, x2 + x3 , x2 − x3 we have this represenatation.   0 1 2 3 RepB (1 − x + 3x − x ) =   1 2 B Find a basis D giving this diﬀerent representation for the same polynomial.   1 0 2 3 RepD (1 − x + 3x − x ) =   2 0 D

242

Chapter Three. Maps Between Spaces

(b) State and prove that any nonzero vector representation can be changed to any other. Hint. The proof of Lemma 1.4 is constructive — it not only says the bases change, it shows how they change. ˆ ˆ 1.19 Let V, W be vector spaces, and let B, B be bases for V and D, D be bases for W . Where h : V → W is linear, ﬁnd a formula relating RepB,D (h) to RepB,D (h). ˆ ˆ 1.20 Show that the columns of an n × n change of basis matrix form a basis for Rn . Do all bases appear in that way: can the vectors from any Rn basis make the columns of a change of basis matrix? 1.21 Find a matrix having this eﬀect. 1 3 → 4 −1

That is, ﬁnd a M that left-multiplies the starting vector to yield the ending vector. Is there a matrix having these two eﬀects? 1 1 2 −1 1 1 2 −1 (a) → → (b) → → 3 1 −1 −1 3 1 6 −1 Give a necessary and suﬃcient condition for there to be a matrix such that v1 → w1 and v2 → w2 .

V.2 Changing Map Representations
The ﬁrst subsection shows how to convert the representation of a vector with respect to one basis to the representation of that same vector with respect to another basis. Here we will see how to convert the representation of a map with respect to one pair of bases to the representation of that map with respect to a diﬀerent pair. That is, we want the relationship between the matrices in this arrow diagram. h Vw.r.t. B − − → Ww.r.t. D −− H     id id Vw.r.t.
ˆ B

− − → Ww.r.t. −−
ˆ H

h

ˆ D

To move from the lower-left of this diagram to the lower-right we can either go straight over, or else up to VB then over to WD and then down. Restated in ˆ terms of the matrices, we can calculate H = RepB,D (h) either by simply using ˆ ˆ ˆ ˆ B and D, or else by ﬁrst changing bases with Rep ˆ (id) then multiplying
B,B

by H = RepB,D (h) and then changing bases with RepD,D (id). This equation ˆ summarizes. ˆ H = RepD,D (id) · H · RepB,B (id) (∗) ˆ ˆ (To compare this equation with the sentence before it, remember that the equation is read from right to left because function composition is read right to left and matrix multiplication represent the composition.)

Section V. Change of Basis 2.1 Example The matrix T = cos(π/6) sin(π/6) − sin(π/6) cos(π/6) √ = 3/2 1/2 −1/2 √ 3/2

243

represents, with respect to E2 , E2 , the transformation t : R2 → R2 that rotates vectors π/6 radians counterclockwise.
1 3 √ (−3 + √ 3)/2 (1 + 3 3)/2

−→

tπ/6

We can translate that representation with respect to E2 , E2 to one with respect to 1 0 −1 2 ˆ ˆ B= D= 1 2 0 3 by using the arrow diagram and formula (∗) above. R2 w.r.t.   id R2 w.r.t.
E2 t

− − → R2 −− w.r.t. T   id − − → R2 −− w.r.t.
ˆ T t

E2

ˆ T = RepE2 ,D (id) · T · RepB,E2 (id) ˆ ˆ
ˆ D

ˆ B

Note that RepE2 ,D (id) can be calculated as the matrix inverse of RepD,E2 (id). ˆ ˆ √ −1 −1 2 1 0 3/2 √ −1/2 RepB,D (t) = ˆ ˆ 0 3 1 2 3/2 1/2 √ √ (5 − √3)/6 (3 + 2 3)/3 √ = (1 + 3)/6 3/3 Although the new matrix is messier-appearing, the map that it represents is the ˆ same. For instance, to replicate the eﬀect of t in the picture, start with B, RepB ( ˆ ˆ apply T , √ (5 − √3)/6 (1 + 3)/6 √ (3 + 2 3)/3 √ 3/3 1 1 =
ˆ B

1 )= 3

1 1

ˆ B

ˆ ˆ B,D

√ (11 + 3 3)/6 √ (1 + 3 3)/6 √ (−3 + √ 3)/2 (1 + 3 3)/2

ˆ D

ˆ and check it against D √ √ 11 + 3 3 1+3 3 −1 2 · + · 0 3 6 6 to see that it is the same result as above.

=

244

Chapter Three. Maps Between Spaces

2.2 Example On R3 the map     x y+z t y  −→ x + z  z x+y that is represented with respect to the standard  0 1 RepE3 ,E3 (t) = 1 0 1 1 basis in this way  1 1 0

can also be represented with respect to another basis        1 1 −1 1 then RepB,B (t) =  0 if B = −1 ,  1  , 1 −2 1 0 0

0 −1 0

 0 0 2

in a way that is simpler, in that the action of a diagonal matrix is easy to understand. Naturally, we usually prefer basis changes that make the representation easier to understand. When the representation with respect to equal starting and ending bases is a diagonal matrix we say the map or matrix has been diagonalized. In Chaper Five we shall see which maps and matrices are diagonalizable, and where one is not, we shall see how to get a representation that is nearly diagonal. We ﬁnish this subsection by considering the easier case where representations are with respect to possibly diﬀerent starting and ending bases. Recall that the prior subsection shows that a matrix changes bases if and only if it is nonsingular. That gives us another version of the above arrow diagram and equation (∗). ˆ 2.3 Deﬁnition Same-sized matrices H and H are matrix equivalent if there ˆ = P HQ. are nonsingular matrices P and Q such that H 2.4 Corollary Matrix equivalent matrices represent the same map, with respect to appropriate pairs of bases. Exercise 19 checks that matrix equivalence is an equivalence relation. Thus it partitions the set of matrices into matrix equivalence classes. H matrix equivalent ˆ to H

All matrices:

H ˆ H

...

Section V. Change of Basis

245

We can get some insight into the classes by comparing matrix equivalence with row equivalence (recall that matrices are row equivalent when they can be reˆ duced to each other by row operations). In H = P HQ, the matrices P and Q are nonsingular and thus each can be written as a product of elementary reduction matrices (Lemma 4.8). Left-multiplication by the reduction matrices making up P has the eﬀect of performing row operations. Right-multiplication by the reduction matrices making up Q performs column operations. Therefore, matrix equivalence is a generalization of row equivalence — two matrices are row equivalent if one can be converted to the other by a sequence of row reduction steps, while two matrices are matrix equivalent if one can be converted to the other by a sequence of row reduction steps followed by a sequence of column reduction steps. Thus, if matrices are row equivalent then they are also matrix equivalent (since we can take Q to be the identity matrix and so perform no column operations). The converse, however, does not hold. 2.5 Example These two 1 0 0 0 1 0 1 0

are matrix equivalent because the second can be reduced to the ﬁrst by the column operation of taking −1 times the ﬁrst column and adding to the second. They are not row equivalent because they have diﬀerent reduced echelon forms (in fact, both are already in reduced form). We will close this section by ﬁnding a set of representatives for the matrix equivalence classes.∗ 2.6 Theorem Any m×n matrix of rank k is matrix equivalent to the m×n matrix that is all zeros except that the ﬁrst k diagonal entries are ones.   1 0 ... 0 0 ... 0 0 1 . . . 0 0 . . . 0     . .   .   0 0 . . . 1 0 . . . 0   0 0 . . . 0 0 . . . 0     . .   . 0 0 ... 0 0 ... 0

Sometimes this is described as a block partial-identity form. I Z

Z Z

246

Chapter Three. Maps Between Spaces

Proof. As discussed above, Gauss-Jordan reduce the given matrix and combine

all the reduction matrices used there to make P . Then use the leading entries to do column reduction and ﬁnish by swapping columns to put the leading ones on the diagonal. Combine the reduction matrices used for those column operations into Q. QED 2.7 Example We illustrate the proof  1 2 0 0 2 4 First Gauss-Jordan row-reduce.    1 0 0 1 1 −1 0 0 1 0  0 1 0 0 −2 0 1 2 0 0 1 by ﬁnding the P and Q for this matrix.  1 −1 1 −1 2 −2

2 0 4

1 1 2

  −1 1 −1 = 0 −2 0

2 0 0

0 1 0

 0 −1 0

Then column-reduce, which involves right-multiplication.       1 −2 0 0 1 0 0 0 1 1 2 0 0  0 1 0 0 0 1 0 0  = 0  0 0 1 −1  0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 Finish by swapping columns.  1 0 0 0 0 0 0 1 0  1 0  0 0  0 0 0  0 0 1 0 0 1 0 0   0 1 0  = 0 0 0 1 0 1 0 0 0 0

0 0 0

0 1 0

 0 0 0

 0 0 0

Finally, combine the left-multipliers together together as Q to get the P HQ equation.     1 0 1 −1 0 1 2 1 −1  0 0 0 1 0 0 0 1 −1  0 1 −2 0 1 2 4 2 −2 0 0

as P and the right-multipliers −2 1 0 0   0 1  0  = 0 1 0 1

0 1 0

0 0 0

 0 0 0

2.8 Corollary Two same-sized matrices are matrix equivalent if and only if they have the same rank. That is, the matrix equivalence classes are characterized by rank.
Proof. Two same-sized matrices with the same rank are equivalent to the same block partial-identity matrix. QED

2.9 Example The 2 × 2 matrices have only three possible ranks: zero, one, or two. Thus there are three matrix-equivalence classes.

Section V. Change of Basis

247

00 00

All 2×2 matrices:
10 01

10 00

Three equivalence classes

Each class consists of all of the 2×2 matrices with the same rank. There is only one rank zero matrix, so that class has only one member, but the other two classes each have inﬁnitely many members. In this subsection we have seen how to change the representation of a map with respect to a ﬁrst pair of bases to one with respect to a second pair. That led to a deﬁnition describing when matrices are equivalent in this way. Finally we noted that, with the proper choice of (possibly diﬀerent) starting and ending bases, any map can be represented in block partial-identity form. One of the nice things about this representation is that, in some sense, we can completely understand the map when it is expressed in this way: if the bases are B = β1 , . . . , βn and D = δ1 , . . . , δm then the map sends c1 β1 + · · · + ck βk + ck+1 βk+1 + · · · + cn βn −→ c1 δ1 + · · · + ck δk + 0 + · · · + 0 where k is the map’s rank. Thus, we can understand any linear map as a kind of projection.     c1 c1 .  .  .  .  .  .  ck   ck      ck+1  →  0      .  .  .  .  . . 0 D cn B Of course, “understanding” a map expressed in this way requires that we understand the relationship between B and D. However, despite that diﬃculty, this is a good classiﬁcation of linear maps. Exercises
2.10 Decide if these matrices are matrix equivalent. 1 3 0 2 2 1 (a) , 2 3 0 0 5 −1 (b) (c) 0 1 1 2 3 , 1 3 , 6 4 0 1 2 0 5 3 −6

2.11 Find the canonical representative of the matrix-equivalence class of each matrix.

248
0 1 3 2.12 Suppose that, with respect (a) 2 4 1 2 0 0 (b) B = E2 1 0 1 0 3 3 to

Chapter Three. Maps Between Spaces
2 4 −1 1 1 , 1 −1 2 4

D=

the transformation t : R2 → R2 is represented by this matrix. 1 3

Use change of basis matrices to represent t with respect to each pair. 0 1 −1 2 ˆ ˆ (a) B = , ,D= , 1 1 0 1 ˆ (b) B = 1 1 , 2 0 ˆ ,D= 1 2 , 2 1

ˆ 2.13 What sizes are P and Q in the equation H = P HQ? 2.14 Use Theorem 2.6 to show that a square matrix is nonsingular if and only if it is equivalent to an identity matrix. 2.15 Show that, where A is a nonsingular square matrix, if P and Q are nonsingular square matrices such that P AQ = I then QP = A−1 . 2.16 Why does Theorem 2.6 not show that every matrix is diagonalizable (see Example 2.2)? 2.17 Must matrix equivalent matrices have matrix equivalent transposes? 2.18 What happens in Theorem 2.6 if k = 0? 2.19 Show that matrix-equivalence is an equivalence relation. 2.20 Show that a zero matrix is alone in its matrix equivalence class. Are there other matrices like that? 2.21 What are the matrix equivalence classes of matrices of transformations on R1 ? R3 ? 2.22 How many matrix equivalence classes are there? 2.23 Are matrix equivalence classes closed under scalar multiplication? Addition? 2.24 Let t : Rn → Rn represented by T with respect to En , En . (a) Find RepB,B (t) in this speciﬁc case. T = 1 3 1 −1 B= 1 −1 , 2 −1

(b) Describe RepB,B (t) in the general case where B = β1 , . . . , βn . 2.25 (a) Let V have bases B1 and B2 and suppose that W has the basis D. Where h : V → W , ﬁnd the formula that computes RepB2 ,D (h) from RepB1 ,D (h). (b) Repeat the prior question with one basis for V and two bases for W . 2.26 (a) If two matrices are matrix-equivalent and invertible, must their inverses be matrix-equivalent? (b) If two matrices have matrix-equivalent inverses, must the two be matrixequivalent? (c) If two matrices are square and matrix-equivalent, must their squares be matrix-equivalent? (d) If two matrices are square and have matrix-equivalent squares, must they be matrix-equivalent?

Section V. Change of Basis

249

2.27 Square matrices are similar if they represent the same transformation, but each with respect to the same ending as starting basis. That is, RepB1 ,B1 (t) is similar to RepB2 ,B2 (t). (a) Give a deﬁnition of matrix similarity like that of Deﬁnition 2.3. (b) Prove that similar matrices are matrix equivalent. (c) Show that similarity is an equivalence relation. ˆ ˆ (d) Show that if T is similar to T then T 2 is similar to T 2 , the cubes are similar, etc. Contrast with the prior exercise. (e) Prove that there are matrix equivalent matrices that are not similar.

250

Chapter Three. Maps Between Spaces

VI

Projection

This section is optional; only the last two sections of Chapter Five require this material. We have described the projection π from R3 into its xy plane subspace as a ‘shadow map’. This shows why, but it also shows that some shadows fall upward.
1 2 2

1 2 −1

So perhaps a better description is: the projection of v is the p in the plane with the property that someone standing on p and looking straight up or down sees v. In this section we will generalize this to other projections, both orthogonal (i.e., ‘straight up and down’) and nonorthogonal.

VI.1 Orthogonal Projection Into a Line
We ﬁrst consider orthogonal projection into a line. To orthogonally project a vector v into a line , darken a point on the line if someone on that line and looking straight up or down (from that person’s point of view) sees v.

The picture shows someone who has walked out on the line until the tip of v is straight overhead. That is, where the line is described as the span of some nonzero vector = {c · s c ∈ R}, the person has walked out to ﬁnd the coeﬃcient cp with the property that v − cp · s is orthogonal to cp · s.
v v − cp s cp s

We can solve for this coeﬃcient by noting that because v − cp s is orthogonal to a scalar multiple of s it must be orthogonal to s itself, and then the consequent fact that the dot product (v − cp s) s is zero gives that cp = v s/s s.

Section VI. Projection

251

1.1 Deﬁnition The orthogonal projection of v into the line spanned by a nonzero s is this vector. v s proj[s ] (v) = ·s s s Exercise 19 checks that the outcome of the calculation depends only on the line and not on which vector s happens to be used to describe that line. 1.2 Remark The wording of that deﬁnition says ‘spanned by s ’ instead the more formal ‘the span of the set {s }’. This casual ﬁrst phrase is common. 1.3 Example To orthogonally project the vector 2 into the line y = 2x, we 3 ﬁrst pick a direction vector for the line. For instance, s= will do. Then the calculation is routine.
2 3 1 2 1 2 1 2

1 2

·

1 2

=

8 · 5

1 2

=

8/5 16/5

1.4 Example In R3 , the orthogonal projection of a general vector   x y  z into the y-axis is   x y  z   0 1 0   0 1     0 0 0   · 1 = y  0 0 0 1 0

which matches our intuitive expectation. The picture above with the stick ﬁgure walking out on the line until v’s tip is overhead is one way to think of the orthogonal projection of a vector into a line. We ﬁnish this subsection with two other ways. 1.5 Example A railroad car left on an east-west track without its brake is pushed by a wind blowing toward the northeast at ﬁfteen miles per hour; what speed will the car reach?

252

Chapter Three. Maps Between Spaces

For the wind we use a vector of length 15 that points toward the northeast. v= 15 15 1/2 1/2

The car can only be aﬀected by the part of the wind blowing in the east-west direction — the part of v in the direction of the x-axis is this (the picture has the same perspective as the railroad car picture above).
north

p=
east

15 0

1/2

So the car will reach a velocity of 15

1/2 miles per hour toward the east.

Thus, another way to think of the picture that precedes the deﬁnition is that it shows v as decomposed into two parts, the part with the line (here, the part with the tracks, p), and the part that is orthogonal to the line (shown here lying on the north-south axis). These two are “not interacting” or “independent”, in the sense that the east-west car is not at all aﬀected by the north-south part of the wind (see Exercise 11). So the orthogonal projection of v into the line spanned by s can be thought of as the part of v that lies in the direction of s. Finally, another useful way to think of the orthogonal projection is to have the person stand not on the line, but on the vector that is to be projected to the line. This person has a rope over the line and pulls it tight, naturally making the rope orthogonal to the line.

That is, we can think of the projection p as being the vector in the line that is closest to v (see Exercise 17). 1.6 Example A submarine is tracking a ship moving along the line y = 3x + 2. Torpedo range is one-half mile. Can the sub stay where it is, at the origin on the chart below, or must it move to reach a place where the ship will pass within range?
north

east

Section VI. Projection

253

The formula for projection into a line does not immediately apply because the line doesn’t pass through the origin, and so isn’t the span of any s. To adjust for this, we start by shifting the entire map down two units. Now the line is y = 3x, which is a subspace, and we can project to get the point p of closest approach, the point on the line through the origin closest to v= the sub’s shifted position. 0 −2 1 3 1 3 1 3 0 −2

p=

·

1 3

=

−3/5 −9/5

The distance between v and p is approximately 0.63 miles and so the sub must move to get in range. This subsection has developed a natural projection map: orthogonal projection into a line. As suggested by the examples, it is often called for in applications. The next subsection shows how the deﬁnition of orthogonal projection into a line gives us a way to calculate especially convienent bases for vector spaces, again something that is common in applications. The ﬁnal subsection completely generalizes projection, orthogonal or not, into any subspace at all. Exercises
1.7 Project the ﬁrst vector orthogonally into the line spanned by the second vector. 1 1 1 3 2 3 2 3 2 (a) , (b) , (c) 1 , (d) 1 , 3 1 −2 1 0 4 −1 4 12 1.8 Project the vector orthogonally into the line. 2 −3 −1 (a) −1 , {c 1 c ∈ R} (b) , the line y = 3x −1 4 −3 1.9 Although the development of Deﬁnition 1.1 is guided by the pictures, we are not restricted to spaces that we can draw. In R4 project this vector into this line.     1 −1 2 1 v=  = {c ·   c ∈ R} 1 −1 3 1 1.10 Deﬁnition 1.1 uses two vectors s and v. Consider the transformation of R2 resulting from ﬁxing 3 s= 1 and projecting v into the line that is the span of s. tors. Apply it to these vec-

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Chapter Three. Maps Between Spaces
1 0 (b) 2 4 Show that in general the projection tranformation is this. (a) x1 x2 → (x1 + 3x2 )/10 (3x1 + 9x2 )/10

Express the action of this transformation with a matrix. 1.11 Example 1.5 suggests that projection breaks v into two parts, proj[s ] (v ) and v − proj[s ] (v ), that are “not interacting”. Recall that the two are orthogonal. Show that any two nonzero orthogonal vectors make up a linearly independent set. 1.12 (a) What is the orthogonal projection of v into a line if v is a member of that line? (b) Show that if v is not a member of the line then the set {v, v − proj[s ] (v )} is linearly independent. 1.13 Deﬁnition 1.1 requires that s be nonzero. Why? What is the right deﬁnition of the orthogonal projection of a vector into the (degenerate) line spanned by the zero vector? 1.14 Are all vectors the projection of some other vector into some line? 1.15 Show that the projection of v into the line spanned by s has length equal to the absolute value of the number v s divided by the length of the vector s . 1.16 Find the formula for the distance from a point to a line. 1.17 Find the scalar c such that (cs1 , cs2 ) is a minimum distance from the point (v1 , v2 ) by using calculus (i.e., consider the distance function, set the ﬁrst derivative equal to zero, and solve). Generalize to Rn . 1.18 Prove that the orthogonal projection of a vector into a line is shorter than the vector. 1.19 Show that the deﬁnition of orthogonal projection into a line does not depend on the spanning vector: if s is a nonzero multiple of q then (v s/s s ) · s equals (v q/q q ) · q. 1.20 Consider the function mapping to plane to itself that takes a vector to its projection into the line y = x. These two each show that the map is linear, the ﬁrst one in a way that is bound to the coordinates (that is, it ﬁxes a basis and then computes) and the second in a way that is more conceptual. (a) Produce a matrix that describes the function’s action. (b) Show also that this map can be obtained by ﬁrst rotating everything in the plane π/4 radians clockwise, then projecting into the x-axis, and then rotating π/4 radians counterclockwise. 1.21 For a, b ∈ Rn let v1 be the projection of a into the line spanned by b, let v2 be the projection of v1 into the line spanned by a, let v3 be the projection of v2 into the line spanned by b, etc., back and forth between the spans of a and b. That is, vi+1 is the projection of vi into the span of a if i + 1 is even, and into the span of b if i + 1 is odd. Must that sequence of vectors eventually settle down — must there be a suﬃciently large i such that vi+2 equals vi and vi+3 equals vi+1 ? If so, what is the earliest such i?

Section VI. Projection

255

VI.2 Gram-Schmidt Orthogonalization
This subsection is optional. It requires material from the prior, also optional, subsection. The work done here will only be needed in the ﬁnal two sections of Chapter Five. The prior subsection suggests that projecting into the line spanned by s decomposes a vector v into two parts
v − proj[s] (p)

v

v = proj[s ] (v) + v − proj[s ] (v)
proj[s] (p)

that are orthogonal and so are “not interacting”. We will now develop that suggestion. 2.1 Deﬁnition Vectors v1 , . . . , vk ∈ Rn are mutually orthogonal when any two are orthogonal: if i = j then the dot product vi vj is zero. 2.2 Theorem If the vectors in a set {v1 , . . . , vk } ⊂ Rn are mutually orthogonal and nonzero then that set is linearly independent.
Proof. Consider a linear relationship c1 v1 + c2 v2 + · · · + ck vk = 0. If i ∈ [1..k]

then taking the dot product of vi with both sides of the equation vi (c1 v1 + c2 v2 + · · · + ck vk ) = vi 0 ci · (vi vi ) = 0 shows, since vi is nonzero, that ci is zero.
QED

2.3 Corollary If the vectors in a size k subset of a k dimensional space are mutually orthogonal and nonzero then that set is a basis for the space.
Proof. Any linearly independent size k subset of a k dimensional space is a basis. QED

Of course, the converse of Corollary 2.3 does not hold — not every basis of every subspace of Rn is made of mutually orthogonal vectors. However, we can get the partial converse that for every subspace of Rn there is at least one basis consisting of mutually orthogonal vectors. 2.4 Example The members β1 and β2 of this basis for R2 are not orthogonal.
β2 B= 4 1 , 2 3 β1

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Chapter Three. Maps Between Spaces

However, we can derive from B a new basis for the same space that does have mutually orthogonal members. For the ﬁrst member of the new basis we simply use β1 . 4 κ1 = 2 For the second member of the new basis, we take away from β2 its part in the direction of κ1 ,

κ2 =

1 3

− proj[κ1 ] (

1 )= 3

1 3

2 1

=

−1 2

κ2

which leaves the part, κ2 pictured above, of β2 that is orthogonal to κ1 (it is orthogonal by the deﬁnition of the projection into the span of κ1 ). Note that, by the corollary, {κ1 , κ2 } is a basis for R2 . 2.5 Deﬁnition An orthogonal basis for a vector space is a basis of mutually orthogonal vectors. 2.6 Example To turn this basis for R3       1 0 1 1 , 2 , 0 1 0 3 into an orthogonal basis, we take the ﬁrst vector as it is given.   1 κ1 = 1 1 We get κ2 by starting with the given second vector β2 and subtracting away the part of it in the direction of κ1 .           0 0 0 2/3 −2/3 κ2 = 2 − proj[κ1 ] (2) = 2 − 2/3 =  4/3  0 0 0 2/3 −2/3 Finally, we get κ3 by taking the third given vector and subtracting the part of it in the direction of κ1 , and also the part of it in the direction of κ2 .         1 1 1 −1 κ3 = 0 − proj[κ1 ] (0) − proj[κ2 ] (0) =  0  3 3 3 1

Section VI. Projection Again the corollary gives that       1 −2/3 −1 1 ,  4/3  ,  0  1 −2/3 1 is a basis for the space.

257

The next result veriﬁes that the process used in those examples works with any basis for any subspace of an Rn (we are restricted to Rn only because we have not given a deﬁnition of orthogonality for other vector spaces). 2.7 Theorem (Gram-Schmidt orthogonalization) If β1 , . . . βk is a basis for a subspace of Rn then, where κ1 = β1 κ2 = β2 − proj[κ1 ] (β2 ) κ3 = β3 − proj[κ1 ] (β3 ) − proj[κ2 ] (β3 ) . . . κk = βk − proj[κ1 ] (βk ) − · · · − proj[κk−1 ] (βk ) the κ ’s form an orthogonal basis for the same subspace.
Proof. We will use induction to check that each κi is nonzero, is in the span of

β1 , . . . βi and is orthogonal to all preceding vectors: κ1 κi = · · · = κi−1 κi = 0. With those, and with Corollary 2.3, we will have that κ1 , . . . κk is a basis for the same space as β1 , . . . βk . We shall cover the cases up to i = 3, which give the sense of the argument. Completing the details is Exercise 23. The i = 1 case is trivial — setting κ1 equal to β1 makes it a nonzero vector since β1 is a member of a basis, it is obviously in the desired span, and the ‘orthogonal to all preceding vectors’ condition is vacuously met. For the i = 2 case, expand the deﬁnition of κ2 . κ2 = β2 − proj[κ1 ] (β2 ) = β2 − β2 κ1 β2 κ1 · κ1 = β2 − · β1 κ1 κ1 κ1 κ1

This expansion shows that κ2 is nonzero or else this would be a non-trivial linear dependence among the β’s (it is nontrivial because the coeﬃcient of β2 is 1) and also shows that κ2 is in the desired span. Finally, κ2 is orthogonal to the only preceding vector κ1 κ2 = κ1 (β2 − proj[κ1 ] (β2 )) = 0 because this projection is orthogonal.

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Chapter Three. Maps Between Spaces

The i = 3 case is the same as the i = 2 case except for one detail. As in the i = 2 case, expanding the deﬁnition κ3 = β3 − = β3 − β3 κ1 β3 κ2 · κ1 − · κ2 κ1 κ1 κ2 κ2 β3 κ1 β3 κ2 β2 κ1 · β1 − · β2 − · β1 κ1 κ1 κ2 κ2 κ1 κ1

shows that κ3 is nonzero and is in the span. A calculation shows that κ3 is orthogonal to the preceding vector κ1 . κ1 κ3 = κ1 = κ1 =0 (Here’s the diﬀerence from the i = 2 case — the second line has two kinds of terms. The ﬁrst term is zero because this projection is orthogonal, as in the i = 2 case. The second term is zero because κ1 is orthogonal to κ2 and so is orthogonal to any vector in the line spanned by κ2 .) The check that κ3 is also orthogonal to the other preceding vector κ2 is similar. QED Beyond having the vectors in the basis be orthogonal, we can do more; we can arrange for each vector to have length one by dividing each by its own length (we can normalize the lengths). 2.8 Example Normalizing the length of each vector in the orthogonal basis of Example 2.6 produces this orthonormal basis. √   √   √   1/√3 −1/ 6 −1/ 2 √ 1/ 3 ,  2/ 6  ,  0  √ √ √ 1/ 2 1/ 3 −1/ 6 Besides its intuitive appeal, and its analogy with the standard basis En for Rn , an orthonormal basis also simpliﬁes some computations. See Exercise 17, for example. Exercises
2.9 Perform the Gram-Schmidt process on each of these bases for R2 . 1 2 0 −1 0 −1 (a) , (b) , (c) , 1 1 1 3 1 0 Then turn those orthogonal bases into orthonormal bases. 2.10 Perform the Gram-Schmidt process on each of these bases for R3 .

β3 − proj[κ1 ] (β3 ) − proj[κ2 ] (β3 ) β3 − proj[κ1 ] (β3 ) − κ1 proj[κ2 ] (β3 )

Section VI. Projection

259

2 0 1 0 1 2 −1 , 1 , 3 2 , 0 , 3 (b) 1 0 0 1 −1 2 Then turn those orthogonal bases into orthonormal bases. 2.11 Find an orthonormal basis for this subspace of R3 : the plane x − y + z = 0. 2.12 Find an orthonormal basis for this subspace of R4 .   x y  {  x − y − z + w = 0 and x + z = 0} z w 2.13 Show that any linearly independent subset of Rn can be orthogonalized without changing its span. 2.14 What happens if we apply the Gram-Schmidt process to a basis that is already orthogonal? 2.15 Let κ1 , . . . , κk be a set of mutually orthogonal vectors in Rn . (a) Prove that for any v in the space, the vector v−(proj[κ1 ] (v )+· · ·+proj[vk ] (v )) is orthogonal to each of κ1 , . . . , κk . (b) Illustrate the prior item in R3 by using e1 as κ1 , using e2 as κ2 , and taking v to have components 1, 2, and 3. (c) Show that proj[κ1 ] (v ) + · · · + proj[vk ] (v ) is the vector in the span of the set of κ’s that is closest to v. Hint. To the illustration done for the prior part, add a vector d1 κ1 + d2 κ2 and apply the Pythagorean Theorem to the resulting triangle. 2.16 Find a vector in R3 that is orthogonal to both of these. 1 2 5 2 −1 0 2.17 One advantage of orthogonal bases is that they simplify ﬁnding the representation of a vector with respect to that basis. (a) For this vector and this non-orthogonal basis for R2 2 1 1 v= B= , 3 1 0 ﬁrst represent the vector with respect to the basis. Then project the vector into the span of each basis vector [β1 ] and [β2 ]. (b) With this orthogonal basis for R2 1 1 K= , 1 −1 represent the same vector v with respect to the basis. Then project the vector into the span of each basis vector. Note that the coeﬃcients in the representation and the projection are the same. (c) Let K = κ1 , . . . , κk be an orthogonal basis for some subspace of Rn . Prove that for any v in the subspace, the i-th component of the representation RepK (v ) is the scalar coeﬃcient (v κi )/(κi κi ) from proj[κi ] (v ). (d) Prove that v = proj[κ1 ] (v ) + · · · + proj[κk ] (v ). 2.18 Bessel’s Inequality. Consider these orthonormal sets B1 = {e1 } B2 = {e1 , e2 } B3 = {e1 , e2 , e3 } B4 = {e1 , e2 , e3 , e4 } along with the vector v ∈ R4 whose components are 4, 3, 2, and 1. (a) Find the coeﬃcient c1 for the projection of v into the span of the vector in B1 . Check that v 2 ≥ |c1 |2 . (a)

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Chapter Three. Maps Between Spaces

(b) Find the coeﬃcients c1 and c2 for the projection of v into the spans of the two vectors in B2 . Check that v 2 ≥ |c1 |2 + |c2 |2 . (c) Find c1 , c2 , and c3 associated with the vectors in B3 , and c1 , c2 , c3 , and c4 for the vectors in B4 . Check that v 2 ≥ |c1 |2 + · · · + |c3 |2 and that v 2 ≥ |c1 |2 + · · · + |c4 |2 . Show that this holds in general: where {κ1 , . . . , κk } is an orthonormal set and ci is coeﬃcient of the projection of a vector v from the space then v 2 ≥ |c1 |2 + · · · + |ck |2 . Hint. One way is to look at the inequality 0 ≤ v − (c1 κ1 + · · · + ck κk ) 2 and expand the c’s. 2.19 Prove or disprove: every vector in Rn is in some orthogonal basis. 2.20 Show that the columns of an n×n matrix form an orthonormal set if and only if the inverse of the matrix is its transpose. Produce such a matrix. 2.21 Does the proof of Theorem 2.2 fail to consider the possibility that the set of vectors is empty (i.e., that k = 0)? 2.22 Theorem 2.7 describes a change of basis from any basis B = β1 , . . . , βk to one that is orthogonal K = κ1 , . . . , κk . Consider the change of basis matrix RepB,K (id). (a) Prove that the matrix RepK,B (id) changing bases in the direction opposite to that of the theorem has an upper triangular shape — all of its entries below the main diagonal are zeros. (b) Prove that the inverse of an upper triangular matrix is also upper triangular (if the matrix is invertible, that is). This shows that the matrix RepB,K (id) changing bases in the direction described in the theorem is upper triangular. 2.23 Complete the induction argument in the proof of Theorem 2.7.

VI.3 Projection Into a Subspace
This subsection, like the others in this section, is optional. It also requires material from the optional earlier subsection on Combining Subspaces. The prior subsections project a vector into a line by decomposing it into two parts: the part in the line proj[s ] (v ) and the rest v − proj[s ] (v ). To generalize projection to arbitrary subspaces, we follow this idea. 3.1 Deﬁnition For any direct sum V = M ⊕ N and any v ∈ V , the projection of v into M along N is projM,N (v ) = m where v = m + n with m ∈ M, n ∈ N . This deﬁnition doesn’t involve a sense of ‘orthogonal’ so we can apply it to spaces other than subspaces of an Rn . (Deﬁnitions of orthogonality for other spaces are perfectly possible, but we haven’t seen any in this book.) 3.2 Example The space M2×2 of 2×2 matrices is the direct sum of these two. M ={ a b 0 0 a, b ∈ R} N ={ 0 c 0 d c, d ∈ R}

Section VI. Projection To project A= 3 0 1 4

261

into M along N , we ﬁrst ﬁx bases for the two subspaces. BM = 1 0 0 0 , 0 0 1 0 BN = 0 1 0 0 , 0 0 0 1

The concatenation of these B = BM BN = 1 0 0 0 , 0 0 1 0 , 0 1 0 0 , 0 0 0 1

is a basis for the entire space, because the space is the direct sum, so we can use it to represent A. 3 0 1 4 =3· 1 0 0 0 +1· 0 0 1 0 +0· 0 1 0 0 +4· 0 0 0 1

Now the projection of A into M along N is found by keeping the M part of this sum and dropping the N part. projM,N ( 3 0 1 1 )=3· 4 0 0 0 +1· 0 0 1 0 = 3 0 1 0

3.3 Example Both subscripts on projM,N (v ) are signiﬁcant. The ﬁrst subscript M matters because the result of the projection is an m ∈ M , and changing this subspace would change the possible results. For an example showing that the second subscript matters, ﬁx this plane subspace of R3 and its basis       0 1 x BM = 0 , 2 M = {y  y − 2z = 0} 1 0 z and compare the projections along two diﬀerent subspaces.     0 0 ˆ N = {k 0 k ∈ R} N = {k  1  k ∈ R} 1 −2 ˆ (Veriﬁcation that R3 = M ⊕ N and R3 = M ⊕ N is routine.) We will check that these projections are diﬀerent by checking that they have diﬀerent eﬀects on this vector.   2 v = 2 5 For the ﬁrst one we ﬁnd a basis for N   0 BN = 0 1

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Chapter Three. Maps Between Spaces

and represent v with respect to the concatenation BM BN .         2 1 0 0 2 = 2 · 0 + 1 · 2 + 4 · 0 5 0 1 1 The projection of v into M along N is found by keeping the M part and dropping the N part.       1 0 2 projM,N (v ) = 2 · 0 + 1 · 2 = 2 0 1 1 ˆ For the other subspace N , this basis is natural.   0 BN =  1  ˆ −2 Representing v with respect to the concatenation         1 0 0 2 2 = 2 · 0 + (9/5) · 2 − (8/5) ·  1  0 1 −2 5 and then keeping only the M part gives this.       2 0 1 projM,N (v ) = 2 · 0 + (9/5) · 2 = 18/5 ˆ 9/5 1 0 Therefore projection along diﬀerent subspaces may yield diﬀerent results. These pictures compare the two maps. Both show that the projection is indeed ‘into’ the plane and ‘along’ the line.
N N M M

Notice that the projection along N is not orthogonal — there are members of the plane M that are not orthogonal to the dotted line. But the projection ˆ along N is orthogonal. A natural question is: what is the relationship between the projection operation deﬁned above, and the operation of orthogonal projection into a line? The second picture above suggests the answer — orthogonal projection into a line is a special case of the projection deﬁned above; it is just projection along a subspace perpendicular to the line.

Section VI. Projection

263

N M

In addition to pointing out that projection along a subspace is a generalization, this scheme shows how to deﬁne orthogonal projection into any subspace of Rn , of any dimension. 3.4 Deﬁnition The orthogonal complement of a subspace M of Rn is M ⊥ = {v ∈ Rn v is perpendicular to all vectors in M } (read “M perp”). The orthogonal projection projM (v ) of a vector is its projection into M along M ⊥ . 3.5 Example In R3 , to ﬁnd the orthogonal complement of the plane   x P = {y  3x + 2y − z = 0} z we start with a basis for P .     0 1 B = 0 , 1 2 3 Any v perpendicular to every vector in B is perpendicular to every vector in the span of B (the proof of this assertion is Exercise 19). Therefore, the subspace P ⊥ consists of the vectors that satisfy these two conditions.         1 v1 v1 0 0 v2  = 0 1 v2  = 0 3 v3 2 v3 We can express those conditions more compactly as a linear system.     v1 v 1 0 3  1 0 v2 = P ⊥ = {v2  } 0 1 2 0 v3 v3 We are thus left with ﬁnding the nullspace of the map represented by the matrix, that is, with calculating the solution set of a homogeneous linear system.     v1 −3 v1 + 3v3 = 0 P ⊥ = {v2  } = {k −2 k ∈ R} v2 + 2v3 = 0 v3 1

264

Chapter Three. Maps Between Spaces

3.6 Example Where M is the xy-plane subspace of R3 , what is M ⊥ ? A common ﬁrst reaction is that M ⊥ is the yz-plane, but that’s not right. Some vectors from the yz-plane are not perpendicular to every vector in the xy-plane.
1 1 0 0 3 2

θ = arccos(

1·0+1·3+0·2 √ √ ) ≈ 0.94 rad 2 · 13

Instead M ⊥ is the z-axis, since proceeding as in the prior example and taking the natural basis for the xy-plane gives this.       x x x 1 0 0   0 y = M ⊥ = {y  } = {y  x = 0 and y = 0} 0 1 0 0 z z z The two examples that we’ve seen since Deﬁnition 3.4 illustrate the ﬁrst sentence in that deﬁnition. The next result justiﬁes the second sentence. 3.7 Lemma Let M be a subspace of Rn . The orthogonal complement of M is also a subspace. The space is the direct sum of the two Rn = M ⊕ M ⊥ . And, for any v ∈ Rn , the vector v − projM (v ) is perpendicular to every vector in M .
Proof. First, the orthogonal complement M ⊥ is a subspace of Rn because, as

noted in the prior two examples, it is a nullspace. Next, we can start with any basis BM = µ1 , . . . , µk for M and expand it to a basis for the entire space. Apply the Gram-Schmidt process to get an orthogonal basis K = κ1 , . . . , κn for Rn . This K is the concatenation of two bases κ1 , . . . , κk (with the same number of members as BM ) and κk+1 , . . . , κn . The ﬁrst is a basis for M , so if we show that the second is a basis for M ⊥ then we will have that the entire space is the direct sum of the two subspaces. Exercise 17 from the prior subsection proves this about any orthogonal basis: each vector v in the space is the sum of its orthogonal projections onto the lines spanned by the basis vectors. v = proj[κ1 ] (v ) + · · · + proj[κn ] (v ) (∗)

To check this, represent the vector v = r1 κ1 + · · · + rn κn , apply κi to both sides v κi = (r1 κ1 + · · · + rn κn ) κi = r1 · 0 + · · · + ri · (κi κi ) + · · · + rn · 0, and solve to get ri = (v κi )/(κi κi ), as desired. Since obviously any member of the span of κk+1 , . . . , κn is orthogonal to any vector in M , to show that this is a basis for M ⊥ we need only show the other containment — that any w ∈ M ⊥ is in the span of this basis. The prior paragraph does this. On projections into basis vectors from M , any w ∈ M ⊥ gives proj[κ1 ] (w ) = 0, . . . , proj[κk ] (w ) = 0 and therefore (∗) gives that w is a linear combination of κk+1 , . . . , κn . Thus this is a basis for M ⊥ and Rn is the direct sum of the two.

Section VI. Projection

265

The ﬁnal sentence is proved in much the same way. Write v = proj[κ1 ] (v ) + · · · + proj[κn ] (v ). Then projM (v ) is gotten by keeping only the M part and dropping the M ⊥ part projM (v ) = proj[κk+1 ] (v ) + · · · + proj[κk ] (v ). Therefore v − projM (v ) consists of a linear combination of elements of M ⊥ and so is perpendicular to every vector in M . QED We can ﬁnd the orthogonal projection into a subspace by following the steps of the proof, but the next result gives a convienent formula. 3.8 Theorem Let v be a vector in Rn and let M be a subspace of Rn with basis β1 , . . . , βk . If A is the matrix whose columns are the β’s then projM (v ) = c1 β1 + · · · + ck βk where the coeﬃcients ci are the entries of the vector (Atrans A)Atrans · v. That is, projM (v ) = A(Atrans A)−1 Atrans · v.
Proof. The vector projM (v) is a member of M and so it is a linear combination

of basis vectors c1 · β1 + · · · + ck · βk . Since A’s columns are the β’s, that can be expressed as: there is a c ∈ Rk such that projM (v ) = Ac (this is expressed compactly with matrix multiplication as in Example 3.5 and 3.6). Because v − projM (v ) is perpendicular to each member of the basis, we have this (again, expressed compactly). 0 = Atrans v − Ac = Atrans v − Atrans Ac Solving for c (showing that Atrans A is invertible is an exercise) c = Atrans A
−1

Atrans · v
QED

gives the formula for the projection matrix as projM (v ) = A · c. 3.9 Example To orthogonally project this vector into this subspace     1 x v = −1 P = {y  x + z = 0} 1 z ﬁrst make a matrix whose columns are a basis for the subspace   0 1 A = 1 0  0 −1 and then compute. A Atrans A
−1

Atrans

  0 1 0 1 = 1 0  1/2 0 0 −1   1/2 0 −1/2 1 0  = 0 −1/2 0 1/2

1 0

0 −1 1 0

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Chapter Three. Maps Between Spaces

With the matrix, calculating the orthogonal projection of any vector into P is easy.      1/2 0 −1/2 1 0 1 0  −1 = −1 projP (v) =  0 −1/2 0 1/2 1 0 Exercises
3.10 Project the vectors into M along N . 3 x x + y = 0}, (a) , M ={ −2 y (b) (c) 1 , 2 3 0 , 1 x y x M ={ y z M ={ x y x − y = 0}, x + y = 0}, N ={ N ={ x y 1 0 1 x y −x − 2y = 0} 2x + y = 0} c ∈ R}

N = {c ·

3.11 Find M ⊥ . (a) M = { (c) M = { x + y = 0} (b) M = { x y −2x + 3y = 0}

x x x − y = 0} (d) M = {0 } (e) M = { x = 0} y y x x (f ) M = { y −x + 3y + z = 0} (g) M = { y x = 0 and y + z = 0} z z 3.12 This subsection shows how to project orthogonally in two ways, the method of Example 3.2 and 3.3, and the method of Theorem 3.8. To compare them, consider the plane P speciﬁed by 3x + 2y − z = 0 in R3 . (a) Find a basis for P . (b) Find P ⊥ and a basis for P ⊥ . (c) Represent this vector with respect to the concatenation of the two bases from the prior item. 1 v= 1 2 (d) Find the orthogonal projection of v into P by keeping only the P part from the prior item. (e) Check that against the result from applying Theorem 3.8. 3.13 We have three ways to ﬁnd the orthogonal projection of a vector into a line, the Deﬁnition 1.1 way from the ﬁrst subsection of this section, the Example 3.2 and 3.3 way of representing the vector with respect to a basis for the space and then keeping the M part, and the way of Theorem 3.8. For these cases, do all three ways. 1 x (a) v = , M ={ x + y = 0} −3 y 0 x (b) v = 1 , M = { y x + z = 0 and y = 0} 2 z 3.14 Check that the operation of Deﬁnition 3.1 is well-deﬁned. That is, in Example 3.2 and 3.3, doesn’t the answer depend on the choice of bases?

Section VI. Projection

267

3.15 What is the orthogonal projection into the trivial subspace? 3.16 What is the projection of v into M along N if v ∈ M ? 3.17 Show that if M ⊆ Rn is a subspace with orthonormal basis κ1 , . . . , κn then the orthogonal projection of v into M is this. (v κ1 ) · κ1 + · · · + (v κn ) · κn 3.18 Prove that the map p : V → V is the projection into M along N if and only if the map id − p is the projection into N along M . (Recall the deﬁnition of the diﬀerence of two maps: (id − p) (v) = id(v) − p(v) = v − p(v).) 3.19 Show that if a vector is perpendicular to every vector in a set then it is perpendicular to every vector in the span of that set. 3.20 True or false: the intersection of a subspace and its orthogonal complement is trivial. 3.21 Show that the dimensions of orthogonal complements add to the dimension of the entire space. 3.22 Suppose that v1 , v2 ∈ Rn are such that for all complements M, N ⊆ Rn , the projections of v1 and v2 into M along N are equal. Must v1 equal v2 ? (If so, what if we relax the condition to: all orthogonal projections of the two are equal?) 3.23 Let M, N be subspaces of Rn . The perp operator acts on subspaces; we can ask how it interacts with other such operations. (a) Show that two perps cancel: (M ⊥ )⊥ = M . (b) Prove that M ⊆ N implies that N ⊥ ⊆ M ⊥ . (c) Show that (M + N )⊥ = M ⊥ ∩ N ⊥ . 3.24 The material in this subsection allows us to express a geometric relationship that we have not yet seen between the rangespace and the nullspace of a linear map. (a) Represent f : R3 → R given by v1 v2 v3 → 1v1 + 2v2 + 3v3

with respect to the standard bases and show that 1 2 3 is a member of the perp of the nullspace. Prove that N (f )⊥ is equal to the span of this vector. (b) Generalize that to apply to any f : Rn → R. (c) Represent f : R3 → R2 v1 v2 v3 → 1v1 + 2v2 + 3v3 4v1 + 5v2 + 6v3

with respect to the standard bases and show that 1 2 3 , 4 5 6

are both members of the perp of the nullspace. Prove that N (f )⊥ is the span of these two. (Hint. See the third item of Exercise 23.)

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(d) Generalize that to apply to any f : Rn → Rm . This, and related results, is called the Fundamental Theorem of Linear Algebra in [Strang 93]. 3.25 Deﬁne a projection to be a linear transformation t : V → V with the property that repeating the projection does nothing more than does the projection alone: (t◦ t) (v) = t(v) for all v ∈ V . (a) Show that orthogonal projection into a line has that property. (b) Show that projection along a subspace has that property. (c) Show that for any such t there is a basis B = β1 , . . . , βn for V such that t(βi ) = βi 0 i = 1, 2, . . . , r i = r + 1, r + 2, . . . , n

where r is the rank of t. (d) Conclude that every projection is a projection along a subspace. (e) Also conclude that every projection has a representation RepB,B (t) = I Z Z Z

in block partial-identity form. 3.26 A square matrix is symmetric if each i, j entry equals the j, i entry (i.e., if the matrix equals its transpose). Show that the projection matrix A(Atrans A)−1 Atrans is symmetric. [Strang 80] Hint. Find properties of transposes by looking in the index under ‘transpose’.

Topic: Line of Best Fit

269

Topic: Line of Best Fit
This Topic requires the formulas from the subsections on Orthogonal Projection Into a Line, and Projection Into a Subspace. Scientists are often presented with a system that has no solution and they must ﬁnd an answer anyway. That is, they must ﬁnd a value that is as close as possible to being an answer. For instance, suppose that we have a coin to use in ﬂipping. This coin has some proportion m of heads to total ﬂips, determined by how it is physically constructed, and we want to know if m is near 1/2. We can get experimental data by ﬂipping it many times. This is the result a penny experiment, including some intermediate numbers.
number of ﬂips number of heads 30 16 60 34 90 51

Because of randomness, we do not ﬁnd the exact proportion with this sample — there is no solution to this system. 30m = 16 60m = 34 90m = 51 That is, the vector of experimental data is not in the subspace of solutions.     30 16 34 ∈ {m 60 m ∈ R} 90 51 However, as described above, we want to ﬁnd the m that most nearly works. An orthogonal projection of the data vector into the line subspace gives our best guess.     16 30 34 60     30 30 51 90 7110       · 60 = · 60 12600 30 30 90 90 60 60 90 90 The estimate (m = 7110/12600 ≈ 0.56) is a bit high but not much, so probably the penny is fair enough. The line with the slope m ≈ 0.56 is called the line of best ﬁt for this data.
60

30

30

60

90

ﬂips

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Chapter Three. Maps Between Spaces

Minimizing the distance between the given vector and the vector used as the right-hand side minimizes the total of these vertical lengths, and consequently we say that the line has been obtained through ﬁtting by least-squares

(the vertical scale here has been exaggerated ten times to make the lengths visible). We arranged the equation above so that the line must pass through (0, 0) because we take take it to be (our best guess at) the line whose slope is this coin’s true proportion of heads to ﬂips. We can also handle cases where the line need not pass through the origin. For example, the diﬀerent denominations of U.S. money have diﬀerent average times in circulation (the \$2 bill is left oﬀ as a special case). How long should we expect a \$25 bill to last?
denomination average life (years) 1 1.5 5 2 10 3 20 5 50 9 100 20

The plot (see below) looks roughly linear. It isn’t a perfect line, i.e., the linear system with equations b + 1m = 1.5, . . . , b + 100m = 20 has no solution, but we can again use orthogonal projection to ﬁnd a best approximation. Consider the matrix of coeﬃcients of that linear system and also its vector of constants, the experimentally-determined values.     1.5 1 1 2 1 5      3 1 10   v=  A= 5 1 20      9 1 50  20 1 100 The ending result in the subsection on Projection into a Subspace says that coeﬃcients b and m so that the linear combination of the columns of A is as close as possible to the vector v are the entries of (Atrans A)−1 Atrans · v. Some calculation gives an intercept of b = 1.05 and a slope of m = 0.18.
avg life
15

5 10 30 50 70 90

denom

Plugging x = 25 into the equation of the line shows that such a bill should last between ﬁve and six years.

Topic: Line of Best Fit

271

We close by considering the times for the men’s mile race [Oakley & Baker]. These are the world records that were in force on January ﬁrst of the given years. We want to project when a 3:40 mile will be run. year seconds 1870 268.8 1880 264.5 1890 258.4 1900 255.6 1910 255.6 1920 252.6 1930 250.4

1940 246.4

1950 241.4

1960 234.5

1970 231.1

1980 229.0

1990 226.3

2000 223.1

We can see below that the data is surprisingly linear. With this input     280.0 1 1860 268.8 1 1870      .  . .  .  v= .  A = . .   .  . 226.3 1 1990 223.1 1 2000 the Python program at this Topic’s end gives slope = −0.35 and intercept = 925.53 (rounded to two places; the original data is good to only about a quarter of a second since much of it was hand-timed).
seconds
260

240

220 1870 1890 1910 1930 1950 1970 1990

year

When will a 220 second mile be run? Solving the equation of the line of best ﬁt gives an estimate of the year 2008. This example is amusing, but serves as a caution — obviously the linearity of the data will break down someday (as indeed it does prior to 1860). Exercises
The calculations here are best done on a computer. In addition, some of the problems require more data, available in your library, on the net, or in the Answers to the Exercises. 1 Use least-squares to judge if the coin in this experiment is fair. ﬂips 8 16 24 32 40 heads 4 9 13 17 20 2 For the men’s mile record, rather than give each of the many records and its exact date, we’ve “smoothed” the data somewhat by taking a periodic sample. Do the longer calculation and compare the conclusions. 3 Find the line of best ﬁt for the men’s 1500 meter run. How does the slope compare with that for the men’s mile? (The distances are close; a mile is about 1609 meters.)

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4 Find the line of best ﬁt for the records for women’s mile. 5 Do the lines of best ﬁt for the men’s and women’s miles cross? 6 When the space shuttle Challenger exploded in 1986, one of the criticisms made of NASA’s decision to launch was in the way the analysis of number of O-ring failures versus temperature was made (of course, O-ring failure caused the explosion). Four O-ring failures will cause the rocket to explode. NASA had data from 24 previous ﬂights. temp ◦ F 53 75 57 58 63 70 70 66 67 67 67 failures 3 2 1 1 1 1 1 0 0 0 0 68 69 70 70 72 73 75 76 76 78 79 80 81 0 0 0 0 0 0 0 0 0 0 0 0 0 The temperature that day was forecast to be 31◦ F. (a) NASA based the decision to launch partially on a chart showing only the ﬂights that had at least one O-ring failure. Find the line that best ﬁts these seven ﬂights. On the basis of this data, predict the number of O-ring failures when the temperature is 31, and when the number of failures will exceed four. (b) Find the line that best ﬁts all 24 ﬂights. On the basis of this extra data, predict the number of O-ring failures when the temperature is 31, and when the number of failures will exceed four. Which do you think is the more accurate method of predicting? (An excellent discussion appears in [Dalal, et. al.].) 7 This table lists the average distance from the sun to each of the ﬁrst seven planets, using earth’s average as a unit. Mercury Venus Earth Mars Jupiter Saturn Uranus 0.39 0.72 1.00 1.52 5.20 9.54 19.2 (a) Plot the number of the planet (Mercury is 1, etc.) versus the distance. Note that it does not look like a line, and so ﬁnding the line of best ﬁt is not fruitful. (b) It does, however look like an exponential curve. Therefore, plot the number of the planet versus the logarithm of the distance. Does this look like a line? (c) The asteroid belt between Mars and Jupiter is thought to be what is left of a planet that broke apart. Renumber so that Jupiter is 6, Saturn is 7, and Uranus is 8, and plot against the log again. Does this look better? (d) Use least squares on that data to predict the location of Neptune. (e) Repeat to predict where Pluto is. (f ) Is the formula accurate for Neptune and Pluto? This method was used to help discover Neptune (although the second item is misleading about the history; actually, the discovery of Neptune in position 9 prompted people to look for the “missing planet” in position 5). See [Gardner, 1970] 8 William Bennett has proposed an Index of Leading Cultural Indicators for the US ([Bennett], in 1993). Among the statistics cited are the average daily hours spent watching TV, and the average combined SAT scores. 1960 1965 1970 1975 1980 1985 1990 1992 TV 5:06 5:29 5:56 6:07 6:36 7:07 6:55 7:04 SAT 975 969 948 910 890 906 900 899 Suppose that a cause and eﬀect relationship is proposed between the time spent watching TV and the decline in SAT scores (in this article, Mr. Bennett does not argue that there is a direct connection). (a) Find the line of best ﬁt relating the independent variable of average daily TV hours to the dependent variable of SAT scores.

Topic: Line of Best Fit

273

(b) Find the most recent estimate of the average daily TV hours (Bennett’s cites Neilsen Media Research as the source of these estimates). Estimate the associated SAT score. How close is your estimate to the actual average? (Warning: a change has been made recently in the SAT, so you should investigate whether some adjustment needs to be made to the reported average to make a valid comparison.)

Computer Code
#!/usr/bin/python # least_squares.py calculate the line of best fit for a data set # data file format: each line is two numbers, x and y import math, string n = 0 sum__x = 0 sum_y = 0 sum_x_squared = 0 sum_xy = 0 fn = raw_input("Name of the data file? ") datafile = open(fn,"r") while 1: ln = datafile.readline() if (ln): data = string.split(ln) x = string.atof(data[0]) y = string.atof(data[1]) n = n+1 sum_x = sum_x+x sum_y = sum_y+y sum_x_squared = sum_x_squared+(x*x) sum_xy = sum_xy +(x*y) else: break datafile.close() slope=(sum_of_xy-pow(sum_x,2)/n)/(sum_x_squared-pow(sum_x,2)/n) intercept=(sum_y-(slope*sum_x))/n print "line of best fit: slope=",slope," intercept=",intercept

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Chapter Three. Maps Between Spaces

Topic: Geometry of Linear Maps
The pictures below contrast f1 (x) = ex and f2 (x) = x2 , which are nonlinear, with h1 (x) = 2x and h2 (x) = −x, which are linear. Each of the four pictures shows the domain R1 on the left mapped to the codomain R1 on the right. Arrows trace out where each map sends x = 0, x = 1, x = 2, x = −1, and x = −2. Note how the nonlinear maps distort the domain in transforming it into the range. For instance, f1 (1) is further from f1 (2) than it is from f1 (0) — the map is spreading the domain out unevenly so that an interval near x = 2 is spread apart more than is an interval near x = 0 when they are carried over to the range.

5

5

5

5

0

0

0

0

-5

-5

-5

-5

The linear maps are nicer, more regular, in that for each map all of the domain is spread by the same factor.
5 5 5 5

0

0

0

0

-5

-5

-5

-5

The only linear maps from R1 to R1 are multiplications by a scalar. In higher dimensions more can happen. For instance, this linear transformation of R2 , rotates vectors counterclockwise, and is not just a scalar multiplication.

Topic: Geometry of Linear Maps

275

x cos θ − y sin θ x → x sin θ + y cos θ y − →

The transformation of R3 which projects vectors into the xz-plane is also not just a rescaling.

x y z

→ − →

x 0 z

Nonetheless, even in higher dimensions the situation isn’t too complicated. Below, we use the standard bases to represent each linear map h : Rn → Rm by a matrix H. Recall that any H can be factored H = P BQ, where P and Q are nonsingular and B is a partial-identity matrix. Further, recall that nonsingular matrices factor into elementary matrices P BQ = Tn Tn−1 · · · Tj BTj−1 · · · T1 , which are matrices that are obtained from the identity I with one Gaussian step I −→ Mi (k)
kρi

I −→ Pi,j

ρi ↔ρj

I −→ Ci,j (k)

kρi +ρj

(i = j, k = 0). So if we understand the eﬀect of a linear map described by a partial-identity matrix, and the eﬀect of linear mapss described by the elementary matrices, then we will in some sense understand the eﬀect of any linear map. (The pictures below stick to transformations of R2 for ease of drawing, but the statements hold for maps from any Rn to any Rm .) The geometric eﬀect of the linear transformation represented by a partialidentity matrix is projection.   x y  z
1 0 0 0 1 0 0 0 0

−→

E3 , E3

  x y  0

For the Mi (k) matrices, the geometric action of a transformation represented by such a matrix (with respect to the standard basis) is to stretch vectors by a factor of k along the i-th axis. This map stretches by a factor of 3 along the x-axis.
x 3x → y y − →

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Chapter Three. Maps Between Spaces

Note that if 0 ≤ k < 1 or if k < 0 then the i-th component goes the other way; here, toward the left.
−2x x → y y − →

Either of these is a dilation. The action of a transformation represented by a Pi,j permutation matrix is to interchange the i-th and j-th axes; this is a particular kind of reﬂection.
y x → x y − →

In higher dimensions, permutations involving many axes can be decomposed into a combination of swaps of pairs of axes — see Exercise 5. The remaining case is that of matrices of the form Ci,j (k). Recall that, for instance, that C1,2 (2) performs 2ρ1 + ρ2 .
1 2 0 1

x y

E2 , E2

−→

x 2x + y

In the picture below, the vector u with the ﬁrst component of 1 is aﬀected less than the vector v with the ﬁrst component of 2 — h(u) is only 2 higher than u while h(v) is 4 higher than v.
h(v) h(u)

u v x x → y 2x + y − →

Any vector with a ﬁrst component of 1 would be aﬀected as is u; it would be slid up by 2. And any vector with a ﬁrst component of 2 would be slid up 4, as was v. That is, the transformation represented by Ci,j (k) aﬀects vectors depending on their i-th component. Another way to see this same point is to consider the action of this map on the unit square. In the next picture, vectors with a ﬁrst component of 0, like the origin, are not pushed vertically at all but vectors with a positive ﬁrst component are slid up. Here, all vectors with a ﬁrst component of 1 — the entire right side of the square — is aﬀected to the same extent. More generally, vectors on the same vertical line are slid up the same amount, namely, they are slid up by twice their ﬁrst component. The resulting shape, a rhombus, has the same base and height as the square (and thus the same area) but the right angles are gone.

Topic: Geometry of Linear Maps

277

x x → 2x + y y − →

For contrast the next picture shows the eﬀect of the map represented by C2,1 (1). In this case, vectors are aﬀected according to their second component. The vector x is slid horozontally by twice y. y

x + 2y x → y y − →

Because of this action, this kind of map is called a skew. With that, we have covered the geometric eﬀect of the four types of components in the expansion H = Tn Tn−1 · · · Tj BTj−1 · · · T1 , the partial-identity projection B and the elementary Ti ’s. Since we understand its components, we in some sense understand the action of any H. As an illustration of this assertion, recall that under a linear map, the image of a subspace is a subspace and thus the linear transformation h represented by H maps lines through the origin to lines through the origin. (The dimension of the image space cannot be greater than the dimension of the domain space, so a line can’t map onto, say, a plane.) We will extend that to show that any line, not just those through the origin, is mapped by h to a line. The proof is simply that the partialidentity projection B and the elementary Ti ’s each turn a line input into a line output (verifying the four cases is Exercise 6), and therefore their composition also preserves lines. Thus, by understanding its components we can understand arbitrary square matrices H, in the sense that we can prove things about them. An understanding of the geometric eﬀect of linear transformations on Rn is very important in mathematics. Here is a familiar application from calculus. On the left is a picture of the action of the nonlinear function y(x) = x2 + x. As at that start of this Topic, overall the geometric eﬀect of this map is irregular in that at diﬀerent domain points it has diﬀerent eﬀects (e.g., as the domain point x goes from 2 to −2, the associated range point f (x) at ﬁrst decreases, then pauses instantaneously, and then increases).

5

5

0

0

278

Chapter Three. Maps Between Spaces

But in calculus we don’t focus on the map overall, we focus instead on the local eﬀect of the map. At x = 1 the derivative is y (1) = 3, so that near x = 1 we have ∆y ≈ 3 · ∆x. That is, in a neighborhood of x = 1, in carrying the domain to the codomain this map causes it to grow by a factor of 3 — it is, locally, approximately, a dilation. The picture below shows a small interval in the domain (x − ∆x .. x + ∆x) carried over to an interval in the codomain (y − ∆y .. y + ∆y) that is three times as wide: ∆y ≈ 3 · ∆x.

y = 2 x = 1

(When the above picture is drawn in the traditional cartesian way then the prior sentence about the rate of growth of y(x) is usually stated: the derivative y (1) = 3 gives the slope of the line tangent to the graph at the point (1, 2).) In higher dimensions, the idea is the same but the approximation is not just the R1 -to-R1 scalar multiplication case. Instead, for a function y : Rn → Rm and a point x ∈ Rn , the derivative is deﬁned to be the linear map h : Rn → Rm best approximating how y changes near y(x). So the geometry studied above applies. We will close this Topic by remarking how this point of view makes clear an often-misunderstood, but very important, result about derivatives: the derivative of the composition of two functions is computed by using the Chain Rule for combining their derivatives. Recall that (with suitable conditions on the two functions) d (g ◦ f ) dg df (x) = (f (x)) · (x) dx dx dx so that, for instance, the derivative of sin(x2 + 3x) is cos(x2 + 3x) · (2x + 3). How does this combination arise? From this picture of the action of the composition.

g(f (x))

f (x) x

Topic: Geometry of Linear Maps The ﬁrst map f dilates the neighborhood of x by a factor of df (x) dx

279

and the second map g dilates some more, this time dilating a neighborhood of f (x) by a factor of dg ( f (x) ) dx and as a result, the composition dilates by the product of these two. In higher dimensions the map expressing how a function changes near a point is a linear map, and is expressed as a matrix. (So we understand the basic geometry of higher-dimensional derivatives; they are compositions of dilations, interchanges of axes, shears, and a projection). And, the Chain Rule just multiplies the matrices. Thus, the geometry of linear maps h : Rn → Rm is appealing both for its simplicity and for its usefulness. Exercises
1 Let h : R2 → R2 be the transformation that rotates vectors clockwise by π/4 radians. (a) Find the matrix H representing h with respect to the standard bases. Use Gauss’ method to reduce H to the identity. (b) Translate the row reduction to to a matrix equation Tj Tj−1 · · · T1 H = I (the prior item shows both that H is similar to I, and that no column operations are needed to derive I from H). (c) Solve this matrix equation for H. (d) Sketch the geometric eﬀect matrix, that is, sketch how H is expressed as a combination of dilations, ﬂips, skews, and projections (the identity is a trivial projection). 2 What combination of dilations, ﬂips, skews, and projections produces a rotation counterclockwise by 2π/3 radians? 3 What combination of dilations, ﬂips, skews, and projections produces the map h : R3 → R3 represented with respect to the standard bases by this matrix? 1 3 1 2 6 2 1 0 2

4 Show that any linear transformation of R1 is the map that multiplies by a scalar x → kx. 5 Show that for any permutation (that is, reordering) p of the numbers 1, . . . , n, the map     x1 xp(1)  x2   xp(2)   . → .  .  .  . . xn xp(n) can be accomplished with a composition of maps, each of which only swaps a single pair of coordinates. Hint: it can be done by induction on n. (Remark: in the fourth

280

Chapter Three. Maps Between Spaces

chapter we will show this and we will also show that the parity of the number of swaps used is determined by p. That is, although a particular permutation could be accomplished in two diﬀerent ways with two diﬀerent numbers of swaps, either both ways use an even number of swaps, or both use an odd number.) 6 Show that linear maps preserve the linear structures of a space. (a) Show that for any linear map from Rn to Rm , the image of any line is a line. The image may be a degenerate line, that is, a single point. (b) Show that the image of any linear surface is a linear surface. This generalizes the result that under a linear map the image of a subspace is a subspace. (c) Linear maps preserve other linear ideas. Show that linear maps preserve “betweeness”: if the point B is between A and C then the image of B is between the image of A and the image of C. 7 Use a picture like the one that appears in the discussion of the Chain Rule to answer: if a function f : R → R has an inverse, what’s the relationship between how the function — locally, approximately — dilates space, and how its inverse dilates space (assuming, of course, that it has an inverse)?

Topic: Markov Chains

281

Topic: Markov Chains
Here is a simple game: a player bets on coin tosses, a dollar each time, and the game ends either when the player has no money left or is up to ﬁve dollars. If the player starts with three dollars, what is the chance that the game takes at least ﬁve ﬂips? Twenty-ﬁve ﬂips? At any point, this player has either \$0, or \$1, . . . , or \$5. We say that the player is in the state s0 , s1 , . . . , or s5 . A game consists of moving from state to state. For instance, a player now in state s3 has on the next ﬂip a .5 chance of moving to state s2 and a .5 chance of moving to s4 . The boundary states are a bit diﬀerent; once in state s0 or state s5 , the player never leaves. Let pi (n) be the probability that the player is in state si after n ﬂips. Then, for instance, we have that the probability of being in state s0 after ﬂip n + 1 is p0 (n + 1) = p0 (n) + 0.5 · p1 (n). This matrix equation sumarizes.      p0 (n + 1) p0 (n) 1 .5 0 0 0 0 0 0 .5 0 0 0 p1 (n) p1 (n + 1)      0 .5 0 .5 0 0 p2 (n) p2 (n + 1)  =   0 0 .5 0 .5 0 p3 (n) p3 (n + 1)      0 0 0 .5 0 0 p4 (n) p4 (n + 1) p5 (n + 1) p5 (n) 0 0 0 0 .5 1 With the initial condition that the player starts with three dollars, calculation gives this. n 0  = 0 0    0    1    0  0 n   =1 0 0     .5    0     .5  0 n   =2 0  .25    0     .5    0  .25 n = 3 .125  0    .375     0    .25  .25 n=4 .125  .1875     0    .3125     0 .375 ···  n = 24  .39600  .00276     0    .00447     0 .59676

···

As this computational exploration suggests, the game is not likely to go on for long, with the player quickly ending in either state s0 or state s5 . For instance, after the fourth ﬂip there is a probability of 0.50 that the game is already over. (Because a player who enters either of the boundary states never leaves, they are said to be absorbtive.) This game is an example of a Markov chain, named for A.A. Markov, who worked in the ﬁrst half of the 1900’s. Each vector of p’s is a probability vector and the matrix is a transition matrix. The notable feature of a Markov chain model is that it is historyless in that with a ﬁxed transition matrix, the next state depends only on the current state, not on any prior states. Thus a player, say, who arrives at s2 by starting in state s3 , then going to state s2 , then to s1 , and then to s2 has at this point exactly the same chance of moving next to state s3 as does a player whose history was to start in s3 , then go to s4 , and to s3 , and then to s2 .

282

Chapter Three. Maps Between Spaces

Here is a Markov chain from sociology. A study ([Macdonald & Ridge], p. 202) divided occupations in the United Kingdom into upper level (executives and professionals), middle level (supervisors and skilled manual workers), and lower level (unskilled). To determine the mobility across these levels in a generation, about two thousand men were asked, “At which level are you, and at which level was your father when you were fourteen years old?” This equation summarizes the results.      .60 .29 .16 pU (n) pU (n + 1) .26 .37 .27 pM (n) = pM (n + 1) .14 .34 .57 pL (n) pL (n + 1) For instance, a child of a lower class worker has a .27 probability of growing up to be middle class. Notice that the Markov model assumption about history seems reasonable — we expect that while a parent’s occupation has a direct inﬂuence on the occupation of the child, the grandparent’s occupation has no such direct inﬂuence. With the initial distribution of the respondents’s fathers given below, this table lists the distributions for the next ﬁve generations. n   =0 .12  .32  .56 n   =1 .23  .34  .42 n   =2 .29  .34  .37 n   =3 .31  .34  .35 n   =4 .32  .33  .34 n   =5 .33  .33  .34

One more example, from a very important subject, indeed. The World Series of American baseball is played between the team winning the American League and the team winning the National League (we follow [Brunner] but see also [Woodside]). The series is won by the ﬁrst team to win four games. That means that a series is in one of twenty-four states: 0-0 (no games won yet by either team), 1-0 (one game won for the American League team and no games for the National League team), etc. If we assume that there is a probability p that the American League team wins each game then we have the following transition matrix.  0  p  1 − p   0   0   0  . . . 0 0 0 p 1−p 0 . . . 0 0 0 0 0 0 0 0 p 0 1−p 0 . . . . . .     ... p0-0 (n) p0-0 (n + 1) . . . p1-0 (n) p1-0 (n + 1)     . . . p0-1 (n) p0-1 (n + 1)     . . . p2-0 (n) = p2-0 (n + 1)     . . . p1-1 (n) p1-1 (n + 1)     . . . p0-2 (n) p0-2 (n + 1)     . . . . . .

An especially interesting special case is p = 0.50; this table lists the resulting components of the n = 0 through n = 7 vectors. (The code to generate this table in the computer algebra system Octave follows the exercises.)

Topic: Markov Chains
n=0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n=1 0 0.5 0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n=2 0 0 0 0.25 0.5 0.25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n=3 0 0 0 0 0 0 0.125 0.375 0.375 0.125 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n=4 0 0 0 0 0 0 0 0 0 0 0.0625 0.25 0.375 0.25 0.0625 0 0 0 0 0 0 0 0 0 n=5 0 0 0 0 0 0 0 0 0 0 0.0625 0 0 0 0.0625 0.125 0.3125 0.3125 0.125 0 0 0 0 0 n=6 0 0 0 0 0 0 0 0 0 0 0.0625 0 0 0 0.0625 0.125 0 0 0.125 0.15625 0.3125 0.15625 0 0

283
n=7 0 0 0 0 0 0 0 0 0 0 0.0625 0 0 0 0.0625 0.125 0 0 0.125 0.15625 0 0.15625 0.15625 0.15625

0−0 1−0 0−1 2−0 1−1 0−2 3−0 2−1 1−2 0−3 4−0 3−1 2−2 1−3 0−4 4−1 3−2 2−3 1−4 4−2 3−3 2−4 4−3 3−4

Note that evenly-matched teams are likely to have a long series — there is a probability of 0.625 that the series goes at least six games. One reason for the inclusion of this Topic is that Markov chains are one of the most widely-used applications of matrix operations. Another reason is that it provides an example of the use of matrices where we do not consider the signiﬁcance of the maps represented by the matrices. For more on Markov chains, there are many sources such as [Kemeny & Snell] and [Iosifescu]. Exercises
Use a computer for these problems. You can, for instance, adapt the Octave script given below. 1 These questions refer to the coin-ﬂipping game. (a) Check the computations in the table at the end of the ﬁrst paragraph. (b) Consider the second row of the vector table. Note that this row has alternating 0’s. Must p1 (j) be 0 when j is odd? Prove that it must be, or produce a counterexample. (c) Perform a computational experiment to estimate the chance that the player ends at ﬁve dollars, starting with one dollar, two dollars, and four dollars. 2 We consider throws of a die, and say the system is in state si if the largest number yet appearing on the die was i. (a) Give the transition matrix. (b) Start the system in state s1 , and run it for ﬁve throws. What is the vector at the end?

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Chapter Three. Maps Between Spaces

[Feller], p. 424 3 There has been much interest in whether industries in the United States are moving from the Northeast and North Central regions to the South and West, motivated by the warmer climate, by lower wages, and by less unionization. Here is the transition matrix for large ﬁrms in Electric and Electronic Equipment ([Kelton], p. 43) NE NC S W Z 0.102 0.111 0 0 NE 0.787 0 0 0.034 0.966 NC 0 0 0 0.937 0.063 S 0 0.314 0.612 0.074 0 W 0 Z 0.021 0.954 0.010 0.005 0.009 For example, a ﬁrm in the Northeast region will be in the West region next year with probability 0.111. (The Z entry is a “birth-death” state. For instance, with probability 0.102 a large Electric and Electronic Equipment ﬁrm from the Northeast will move out of this system next year: go out of business, move abroad, or move to another category of ﬁrm. There is a 0.021 probability that a ﬁrm in the National Census of Manufacturers will move into Electronics, or be created, or move in from abroad, into the Northeast. Finally, with probability 0.954 a ﬁrm out of the categories will stay out, according to this research.) (a) Does the Markov model assumption of lack of history seem justiﬁed? (b) Assume that the initial distribution is even, except that the value at Z is 0.9. Compute the vectors for n = 1 through n = 4. (c) Suppose that the initial distribution is this. NE NC S W Z 0.0000 0.6522 0.3478 0.0000 0.0000 Calculate the distributions for n = 1 through n = 4. (d) Find the distribution for n = 50 and n = 51. Has the system settled down to an equilibrium? 4 This model has been suggested for some kinds of learning ([Wickens], p. 41). The learner starts in an undecided state sU . Eventually the learner has to decide to do either response A (that is, end in state sA ) or response B (ending in sB ). However, the learner doesn’t jump right from being undecided to being sure A is the correct thing to do (or B). Instead, the learner spends some time in a “tentative-A” state, or a “tentative-B” state, trying the response out (denoted here tA and tB ). Imagine that once the learner has decided, it is ﬁnal, so once sA or sB is entered it is never left. For the other state changes, imagine a transition is made with probability p in either direction. (a) Construct the transition matrix. (b) Take p = 0.25 and take the initial vector to be 1 at sU . Run this for ﬁve steps. What is the chance of ending up at sA ? (c) Do the same for p = 0.20. (d) Graph p versus the chance of ending at sA . Is there a threshold value for p, above which the learner is almost sure not to take longer than ﬁve steps? 5 A certain town is in a certain country (this is a hypothetical problem). Each year ten percent of the town dwellers move to other parts of the country. Each year one percent of the people from elsewhere move to the town. Assume that there are two states sT , living in town, and sC , living elsewhere. (a) Construct the transistion matrix.

Topic: Markov Chains

285

(b) Starting with an initial distribution sT = 0.3 and sC = 0.7, get the results for the ﬁrst ten years. (c) Do the same for sT = 0.2. (d) Are the two outcomes alike or diﬀerent? 6 For the World Series application, use a computer to generate the seven vectors for p = 0.55 and p = 0.6. (a) What is the chance of the National League team winning it all, even though they have only a probability of 0.45 or 0.40 of winning any one game? (b) Graph the probability p against the chance that the American League team wins it all. Is there a threshold value — a p above which the better team is essentially ensured of winning? (Some sample code is included below.) 7 A Markov matrix has each entry positive and each column sums to 1. (a) Check that the three transistion matrices shown in this Topic meet these two conditions. Must any transition matrix do so? (b) Observe that if Av0 = v1 and Av1 = v2 then A2 is a transition matrix from v0 to v2 . Show that a power of a Markov matrix is also a Markov matrix. (c) Generalize the prior item by proving that the product of two appropriatelysized Markov matrices is a Markov matrix.

Computer Code This script markov.m for the computer algebra system Octave was used to generate the table of World Series outcomes. (The sharp character # marks the rest of a line as a comment.)
# Octave script file to compute chance of World Series outcomes. function w = markov(p,v) q = 1-p; A=[0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 0-0 p,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 1-0 q,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 0-1_ 0,p,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 2-0 0,q,p,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 1-1 0,0,q,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 0-2__ 0,0,0,p,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 3-0 0,0,0,q,p,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 2-1 0,0,0,0,q,p, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 1-2_ 0,0,0,0,0,q, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 0-3 0,0,0,0,0,0, p,0,0,0,1,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 4-0 0,0,0,0,0,0, q,p,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 3-1__ 0,0,0,0,0,0, 0,q,p,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 2-2 0,0,0,0,0,0, 0,0,q,p,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 1-3 0,0,0,0,0,0, 0,0,0,q,0,0, 0,0,1,0,0,0, 0,0,0,0,0,0; # 0-4_ 0,0,0,0,0,0, 0,0,0,0,0,p, 0,0,0,1,0,0, 0,0,0,0,0,0; # 4-1 0,0,0,0,0,0, 0,0,0,0,0,q, p,0,0,0,0,0, 0,0,0,0,0,0; # 3-2 0,0,0,0,0,0, 0,0,0,0,0,0, q,p,0,0,0,0, 0,0,0,0,0,0; # 2-3__ 0,0,0,0,0,0, 0,0,0,0,0,0, 0,q,0,0,0,0, 1,0,0,0,0,0; # 1-4 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,p,0, 0,1,0,0,0,0; # 4-2 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,q,p, 0,0,0,0,0,0; # 3-3_ 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,q, 0,0,0,1,0,0; # 2-4

286

Chapter Three. Maps Between Spaces

0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,p,0,1,0; # 4-3 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,q,0,0,1]; # 3-4 w = A * v; endfunction

> > > > v0=[1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0] p=.5 v1=markov(p,v0) v2=markov(p,v1) ...

Translating to another computer algebra system should be easy — all have commands similar to these.

Topic: Orthonormal Matrices

287

Topic: Orthonormal Matrices
In The Elements, Euclid considers two ﬁgures to be the same if they have the same size and shape. That is, the triangles below are not equal because they are not the same set of points. But they are congruent — essentially indistinguishable for Euclid’s purposes — because we can imagine picking the plane up, sliding it over and rotating it a bit, although not warping or stretching it, and then putting it back down, to superimpose the ﬁrst ﬁgure on the second. (Euclid never explicitly states this principle but he uses it often [Casey].)
P2 P1 Q1 Q3 Q2 P3

In modern terminology, “picking the plane up . . . ” means considering a map from the plane to itself. Euclid has limited consideration to only certain transformations of the plane, ones that may possibly slide or turn the plane but not bend or stretch it. Accordingly, we deﬁne a map f : R2 → R2 to be distancepreserving or a rigid motion or an isometry, if for all points P1 , P2 ∈ R2 , the distance from f (P1 ) to f (P2 ) equals the distance from P1 to P2 . We also deﬁne a plane ﬁgure to be a set of points in the plane and we say that two ﬁgures are congruent if there is a distance-preserving map from the plane to itself that carries one ﬁgure onto the other. Many statements from Euclidean geometry follow easily from these deﬁnitions. Some are: (i) collinearity is invariant under any distance-preserving map (that is, if P1 , P2 , and P3 are collinear then so are f (P1 ), f (P2 ), and f (P3 )), (ii) betweeness is invariant under any distance-preserving map (if P2 is between P1 and P3 then so is f (P2 ) between f (P1 ) and f (P3 )), (iii) the property of being a triangle is invariant under any distance-preserving map (if a ﬁgure is a triangle then the image of that ﬁgure is also a triangle), (iv) and the property of being a circle is invariant under any distance-preserving map. In 1872, F. Klein suggested that Euclidean geometry can be characterized as the study of properties that are invariant under these maps. (This forms part of Klein’s Erlanger Program, which proposes the organizing principle that each kind of geometry — Euclidean, projective, etc. — can be described as the study of the properties that are invariant under some group of transformations. The word ‘group’ here means more than just ‘collection’, but that lies outside of our scope.) We can use linear algebra to characterize the distance-preserving maps of the plane. First, there are distance-preserving transformations of the plane that are not linear. The obvious example is this translation. x y → x 1 + y 0 = x+1 y

288

Chapter Three. Maps Between Spaces

However, this example turns out to be the only example, in the sense that if f is distance-preserving and sends 0 to v0 then the map v → f (v)−v0 is linear. That will follow immediately from this statement: a map t that is distance-preserving and sends 0 to itself is linear. To prove this equivalent statement, let t(e1 ) = a b t(e2 ) = c d

for some a, b, c, d ∈ R. Then to show that t is linear, we can show that it can be represented by a matrix, that is, that t acts in this way for all x, y ∈ R. v= x y −→
t

ax + cy bx + dy

(∗)

Recall that if we ﬁx three non-collinear points then any point in the plane can be described by giving its distance from those three. So any point v in the domain is determined by its distance from the three ﬁxed points 0, e1 , and e2 . Similarly, any point t(v) in the codomain is determined by its distance from the three ﬁxed points t(0), t(e1 ), and t(e2 ) (these three are not collinear because, as mentioned above, collinearity is invariant and 0, e1 , and e2 are not collinear). In fact, because t is distance-preserving, we can say more: for the point v in the plane that is determined by being the distance d0 from 0, the distance d1 from e1 , and the distance d2 from e2 , its image t(v) must be the unique point in the codomain that is determined by being d0 from t(0), d1 from t(e1 ), and d2 from t(e2 ). Because of the uniqueness, checking that the action in (∗) works in the d0 , d1 , and d2 cases dist( x x ax + cy , 0) = dist(t( ), t(0)) = dist( , 0) y y bx + dy

(t is assumed to send 0 to itself) dist( and dist( x x ax + cy c , e2 ) = dist(t( ), t(e2 )) = dist( , ) y y bx + dy d x x ax + cy a , e1 ) = dist(t( ), t(e1 )) = dist( , ) y y bx + dy b

suﬃces to show that (∗) describes t. Those checks are routine. Thus, any distance-preserving f : R2 → R2 can be written f (v) = t(v) + v0 for some constant vector v0 and linear map t that is distance-preserving. Not every linear map is distance-preserving, for example, v → 2v does not preserve distances. But there is a neat characterization: a linear transformation t of the plane is distance-preserving if and only if both t(e1 ) = t(e2 ) = 1 and t(e1 ) is orthogonal to t(e2 ). The ‘only if’ half of that statement is easy — because t is distance-preserving it must preserve the lengths of vectors, and because t is distance-preserving the Pythagorean theorem shows that it must preserve

Topic: Orthonormal Matrices

289

orthogonality. For the ‘if’ half, it suﬃces to check that the map preserves lengths of vectors, because then for all p and q the distance between the two is preserved t(p − q ) = t(p) − t(q ) = p − q . For that check, let v= x y t(e1 ) = a b t(e2 ) = c d

and, with the ‘if’ assumptions that a2 + b2 = c2 + d2 = 1 and ac + bd = 0 we have this. t(v )
2

= (ax + cy)2 + (bx + dy)2 = a2 x2 + 2acxy + c2 y 2 + b2 x2 + 2bdxy + d2 y 2 = x2 (a2 + b2 ) + y 2 (c2 + d2 ) + 2xy(ac + bd) = x2 + y 2 = v
2

One thing that is neat about this characterization is that we can easily recognize matrices that represent such a map with respect to the standard bases. Those matrices have that when the columns are written as vectors then they are of length one and are mutually orthogonal. Such a matrix is called an orthonormal matrix or orthogonal matrix (the second term is commonly used to mean not just that the columns are orthogonal, but also that they have length one). We can use this insight to delimit the geometric actions possible in distancepreserving maps. Because t(v ) = v , any v is mapped by t to lie somewhere on the circle about the origin that has radius equal to the length of v. In particular, e1 and e2 are mapped to the unit circle. What’s more, once we ﬁx the unit vector e1 as mapped to the vector with components a and b then there are only two places where e2 can be mapped if that image is to be perpendicular to the ﬁrst vector: one where e2 maintains its position a quarter circle clockwise from e1
−b a a b

RepE2 ,E2 (t) =

a b

−b a

and one where is is mapped a quarter circle counterclockwise.
a b

RepE2 ,E2 (t) =
b −a

a b

b −a

290

Chapter Three. Maps Between Spaces

We can geometrically describe these two cases. Let θ be the angle between the x-axis and the image of e1 , measured counterclockwise. The ﬁrst matrix above represents, with respect to the standard bases, a rotation of the plane by θ radians.
−b a a b

x y

−→

t

x cos θ − y sin θ x sin θ + y cos θ

The second matrix above represents a reﬂection of the plane through the line bisecting the angle between e1 and t(e1 ).
a b

x y
b −a

−→

t

x cos θ + y sin θ x sin θ − y cos θ

(This picture shows e1 reﬂected up into the ﬁrst quadrant and e2 reﬂected down into the fourth quadrant.) Note again: the angle between e1 and e2 runs counterclockwise, and in the ﬁrst map above the angle from t(e1 ) to t(e2 ) is also counterclockwise, so the orientation of the angle is preserved. But in the second map the orientation is reversed. A distance-preserving map is direct if it preserves orientations and opposite if it reverses orientation. So, we have characterized the Euclidean study of congruence: it considers, for plane ﬁgures, the properties that are invariant under combinations of (i) a rotation followed by a translation, or (ii) a reﬂection followed by a translation (a reﬂection followed by a non-trivial translation is a glide reﬂection). Another idea, besides congruence of ﬁgures, encountered in elementary geometry is that ﬁgures are similar if they are congruent after a change of scale. These two triangles are similar since the second is the same shape as the ﬁrst, but 3/2-ths the size.
P2 P1 Q1 Q2

P3

Q3

From the above work, we have that ﬁgures are similar if there is an orthonormal matrix T such that the points q on one are derived from the points p by q = (kT )v + p0 for some nonzero real number k and constant vector p0 .

Topic: Orthonormal Matrices

291

Although many of these ideas were ﬁrst explored by Euclid, mathematics is timeless and they are very much in use today. One application of the maps studied above is in computer graphics. We can, for example, animate this top view of a cube by putting together ﬁlm frames of it rotating; that’s a rigid motion.

Frame 1

Frame 2

Frame 3

We could also make the cube appear to be moving away from us by producing ﬁlm frames of it shrinking, which gives us ﬁgures that are similar.

Frame 1:

Frame 2:

Frame 3:

Computer graphics incorporates techniques from linear algebra in many other ways (see Exercise 4). So the analysis above of distance-preserving maps is useful as well as interesting. A beautiful book that explores some of this area is [Weyl]. More on groups, of transformations and otherwise, can be found in any book on Modern Algebra, for instance [Birkhoﬀ & MacLane]. More on Klein and the Erlanger Program is in [Yaglom]. Exercises
1 Decide if √ each of these is an orthonormal matrix. √ 1/ √ 2 −1/√2 (a) −1/ 2 −1/ 2 √ √ 1/ √ 3 −1/√3 (b) −1/ 3 −1/ 3 √ √ √ 1/ 3 − 2/ 3 √ √ √ (c) − 2/ 3 −1/ 3 2 Write down the formula for each of these distance-preserving maps. (a) the map that rotates π/6 radians, and then translates by e2 (b) the map that reﬂects about the line y = 2x (c) the map that reﬂects about y = −2x and translates over 1 and up 1 3 (a) The proof that a map that is distance-preserving and sends the zero vector to itself incidentally shows that such a map is one-to-one and onto (the point in the domain determined by d0 , d1 , and d2 corresponds to the point in the codomain determined by those three). Therefore any distance-preserving map has an inverse. Show that the inverse is also distance-preserving. (b) Prove that congruence is an equivalence relation between plane ﬁgures. 4 In practice the matrix for the distance-preserving linear transformation and the translation are often combined into one. Check that these two computations yield the same ﬁrst two components. a c e x a c x e b d f y + b d y f 0 0 1 1

292

Chapter Three. Maps Between Spaces

(These are homogeneous coordinates; see the Topic on Projective Geometry). 5 (a) Verify that the properties described in the second paragraph of this Topic as invariant under distance-preserving maps are indeed so. (b) Give two more properties that are of interest in Euclidean geometry from your experience in studying that subject that are also invariant under distancepreserving maps. (c) Give a property that is not of interest in Euclidean geometry and is not invariant under distance-preserving maps.

Chapter Four

Determinants
In the ﬁrst chapter of this book we considered linear systems and we picked out the special case of systems with the same number of equations as unknowns, those of the form T x = b where T is a square matrix. We noted a distinction between two classes of T ’s. While such systems may have a unique solution or no solutions or inﬁnitely many solutions, if a particular T is associated with a unique solution in any system, such as the homogeneous system b = 0, then T is associated with a unique solution for every b. We call such a matrix of coeﬃcients ‘nonsingular’. The other kind of T , where every linear system for which it is the matrix of coeﬃcients has either no solution or inﬁnitely many solutions, we call ‘singular’. Through the second and third chapters the value of this distinction has been a theme. For instance, we now know that nonsingularity of an n×n matrix T is equivalent to each of these: • a system T x = b has a solution, and that solution is unique; • Gauss-Jordan reduction of T yields an identity matrix; • the rows of T form a linearly independent set; • the columns of T form a basis for Rn ; • any map that T represents is an isomorphism; • an inverse matrix T −1 exists. So when we look at a particular square matrix, the question of whether it is nonsingular is one of the ﬁrst things that we ask. This chapter develops a formula to determine this. (Since we will restrict the discussion to square matrices, in this chapter we will usually simply say ‘matrix’ in place of ‘square matrix’.) More precisely, we will develop inﬁnitely many formulas, one for 1×1 matrices, one for 2×2 matrices, etc. Of course, these formulas are related — that is, we will develop a family of formulas, a scheme that describes the formula for each size. 293

294

Chapter Four. Determinants

I

Definition
a is nonsingular iﬀ a = 0

For 1×1 matrices, determining nonsingularity is trivial.

The 2×2 formula came out in the course of developing the inverse. a c The 3×3  a d g b d is nonsingular iﬀ ad − bc = 0

formula can be produced similarly (see Exercise 9).  b c e f  is nonsingular iﬀ aei + bf g + cdh − hf a − idb − gec = 0 h i

With these cases in mind, we posit a family of formulas, a, ad − bc, etc. For each n the formula gives rise to a determinant function detn×n : Mn×n → R such that an n×n matrix T is nonsingular if and only if detn×n (T ) = 0. (We usually omit the subscript because if T is n×n then ‘det(T )’ could only mean ‘detn×n (T )’.)

I.1 Exploration
This subsection is optional. It brieﬂy describes how an investigator might come to a good general deﬁnition, which is given in the next subsection. The three cases above don’t show an evident pattern to use for the general n×n formula. We may spot that the 1×1 term a has one letter, that the 2×2 terms ad and bc have two letters, and that the 3×3 terms aei, etc., have three letters. We may also observe that in those terms there is a letter from each row and column of the matrix, e.g., the letters in the cdh term   c  d h come one from each row and one from each column. But these observations perhaps seem more puzzling than enlightening. For instance, we might wonder why some of the terms are added while others are subtracted. A good problem solving strategy is to see what properties a solution must have and then search for something with those properties. So we shall start by asking what properties we require of the formulas. At this point, our primary way to decide whether a matrix is singular is to do Gaussian reduction and then check whether the diagonal of resulting echelon form matrix has any zeroes (that is, to check whether the product down the diagonal is zero). So, we may expect that the proof that a formula

Section I. Definition

295

determines singularity will involve applying Gauss’ method to the matrix, to show that in the end the product down the diagonal is zero if and only if the determinant formula gives zero. This suggests our initial plan: we will look for a family of functions with the property of being unaﬀected by row operations and with the property that a determinant of an echelon form matrix is the product of its diagonal entries. Under this plan, a proof that the functions ˆ determine singularity would go, “Where T → · · · → T is the Gaussian reduction, ˆ (because the determinant is the determinant of T equals the determinant of T unchanged by row operations), which is the product down the diagonal, which is zero if and only if the matrix is singular”. In the rest of this subsection we will test this plan on the 2×2 and 3×3 determinants that we know. We will end up modifying the “unaﬀected by row operations” part, but not by much. The ﬁrst step in checking the plan is to test whether the 2 × 2 and 3 × 3 formulas are unaﬀected by the row operation of pivoting: if
kρi +ρj ˆ T −→ T

ˆ then is det(T ) = det(T )? This check of the 2×2 determinant after the kρ1 + ρ2 operation det( a ka + c b ) = a(kb + d) − (ka + c)b = ad − bc kb + d

shows that it is indeed unchanged, and the other 2×2 pivot kρ2 + ρ1 gives the same result. The 3×3 pivot kρ3 + ρ2 leaves the determinant unchanged   a b c det(kg + d kh + e ki + f ) = a(kh + e)i + b(ki + f )g + c(kg + d)h g h i − h(ki + f )a − i(kg + d)b − g(kh + e)c = aei + bf g + cdh − hf a − idb − gec as do the other 3×3 pivot operations. So there seems to be promise in the plan. Of course, perhaps the 4 × 4 determinant formula is aﬀected by pivoting. We are exploring a possibility here and we do not yet have all the facts. Nonetheless, so far, so good. ˆ The next step is to compare det(T ) with det(T ) for the operation
ρi ↔ρj ˆ T −→ T

of swapping two rows. The 2×2 row swap ρ1 ↔ ρ2 det( c a d ) = cb − ad b

does not yield ad − bc. This ρ1 ↔ ρ3 swap inside of a 3×3 matrix   g h i det(d e f ) = gec + hf a + idb − bf g − cdh − aei a b c

296

Chapter Four. Determinants

also does not give the same determinant as before the swap — again there is a sign change. Trying a diﬀerent 3×3 swap ρ1 ↔ ρ2   d e f det(a b c ) = dbi + ecg + f ah − hcd − iae − gbf g h i also gives a change of sign. Thus, row swaps appear to change the sign of a determinant. This modiﬁes our plan, but does not wreck it. We intend to decide nonsingularity by considering only whether the determinant is zero, not by considering its sign. Therefore, instead of expecting determinants to be entirely unaﬀected by row operations, will look for them to change sign on a swap. ˆ To ﬁnish, we compare det(T ) to det(T ) for the operation
kρi ˆ T −→ T

of multiplying a row by a scalar k = 0. One of the 2×2 cases is det( a kc b ) = a(kd) − (kc)b = k · (ad − bc) kd

and the other case has  a b e det( d kg kh

the same result. Here is one 3×3 case  c f ) = ae(ki) + bf (kg) + cd(kh) ki −(kh)f a − (ki)db − (kg)ec = k · (aei + bf g + cdh − hf a − idb − gec)

and the other two are similar. These lead us to suspect that multiplying a row by k multiplies the determinant by k. This ﬁts with our modiﬁed plan because we are asking only that the zeroness of the determinant be unchanged and we are not focusing on the determinant’s sign or magnitude. In summary, to develop the scheme for the formulas to compute determinants, we look for determinant functions that remain unchanged under the pivoting operation, that change sign on a row swap, and that rescale on the rescaling of a row. In the next two subsections we will ﬁnd that for each n such a function exists and is unique. For the next subsection, note that, as above, scalars come out of each row without aﬀecting other rows. For instance, in this equality     3 3 9 1 1 3 1 ) = 3 · det(2 1 1 ) det(2 1 5 10 −5 5 10 −5 the 3 isn’t factored out of all three rows, only out of the top row. The determinant acts on each row of independently of the other rows. When we want to use this property of determinants, we shall write the determinant as a function of the rows: ‘det(ρ1 , ρ2 , . . . ρn )’, instead of as ‘det(T )’ or ‘det(t1,1 , . . . , tn,n )’. The deﬁnition of the determinant that starts the next subsection is written in this way.

Section I. Definition Exercises

297

1.1 Evaluate the determinant of each. 4 0 1 2 0 1 3 1 1 3 1 1 (c) 0 0 (a) (b) −1 1 1 3 −1 −1 0 1 1.2 Evaluate the determinant of each. 2 3 4 2 1 1 2 0 5 −2 (c) 5 6 7 (a) (b) 0 −1 3 8 9 1 1 −3 4 1.3 Verify that the determinant of an upper-triangular 3×3 matrix is the product down the diagonal. a b c det( 0 e f ) = aei 0 0 i Do lower-triangular matrices work the same way? 1.4 Use the determinant to decide if each is singular or nonsingular. 2 1 0 1 4 2 (a) (b) (c) 3 1 1 −1 2 1 1.5 Singular or nonsingular? Use the determinant to decide. 2 1 1 1 0 1 2 1 0 (a) 3 2 2 (b) 2 1 1 (c) 3 −2 0 0 1 4 4 1 3 1 0 0 1.6 Each pair of matrices diﬀer by one row operation. Use this operation to compare det(A) with det(B). 1 2 1 2 (a) A = B= 2 3 0 −1 3 1 0 3 1 0 (b) A = 0 0 1 B = 0 1 2 0 1 2 0 0 1 1 −1 3 1 −1 3 2 −6 B = 1 1 −3 (c) A = 2 1 0 4 1 0 4 1.7 Show this. 1 1 1 b c ) = (b − a)(c − a)(c − b) det( a a2 b2 c2 1.8 Which real numbers x make this matrix singular? 12 − x 4 8 8−x 1.9 Do the Gaussian reduction to check the formula for 3×3 matrices stated in the preamble to this section. a b c d e f is nonsingular iﬀ aei + bf g + cdh − hf a − idb − gec = 0 g h i 1.10 Show that the equation of a line in R2 thru (x1 , y1 ) and (x2 , y2 ) is expressed by this determinant. x y 1 det( x1 y1 1 ) = 0 x1 = x2 x2 y2 1

298

Chapter Four. Determinants

1.11 Many people know this mnemonic for the determinant of a 3×3 matrix: ﬁrst repeat the ﬁrst two columns and then sum the products on the forward diagonals and subtract the products on the backward diagonals. That is, ﬁrst write h1,1 h2,1 h3,1 h1,2 h2,2 h3,2 h1,3 h2,3 h3,3 h1,1 h2,1 h3,1 h1,2 h2,2 h3,2

and then calculate this. h1,1 h2,2 h3,3 + h1,2 h2,3 h3,1 + h1,3 h2,1 h3,2 −h3,1 h2,2 h1,3 − h3,2 h2,3 h1,1 − h3,3 h2,1 h1,2 (a) Check that this agrees with the formula given in the preamble to this section. (b) Does it extend to other-sized determinants? 1.12 The cross product of the vectors x= x1 x2 x3 y= y1 y2 y3 e3 x3 ) y3

is the vector computed as this determinant. e1 x × y = det( x1 y1 e2 x2 y2

Note that the ﬁrst row is composed of vectors, the vectors from the standard basis for R3 . Show that the cross product of two vectors is perpendicular to each vector. 1.13 Prove that each statement holds for 2×2 matrices. (a) The determinant of a product is the product of the determinants det(ST ) = det(S) · det(T ). (b) If T is invertible then the determinant of the inverse is the inverse of the determinant det(T −1 ) = ( det(T ) )−1 . Matrices T and T are similar if there is a nonsingular matrix P such that T = P T P −1 . (This deﬁnition is in Chapter Five.) Show that similar 2×2 matrices have the same determinant. 1.14 Prove that the area of this region in the plane
x2 y2

x1 y1

is equal to the value of this determinant. det( Compare with this. det( x2 y2 x1 ) y1 x1 y1 x2 ) y2

1.15 Prove that for 2×2 matrices, the determinant of a matrix equals the determinant of its transpose. Does that also hold for 3×3 matrices? 1.16 Is the determinant function linear — is det(x·T +y ·S) = x·det(T )+y ·det(S)? 1.17 Show that if A is 3×3 then det(c · A) = c3 · det(A) for any scalar c.

Section I. Definition
1.18 Which real numbers θ make cos θ sin θ − sin θ cos θ

299

singular? Explain geometrically. ? 1.19 If a third order determinant has elements 1, 2, . . . , 9, what is the maximum value it may have? [Am. Math. Mon., Apr. 1955]

I.2 Properties of Determinants
As described above, we want a formula to determine whether an n×n matrix is nonsingular. We will not begin by stating such a formula. Instead, we will begin by considering the function that such a formula calculates. We will deﬁne the function by its properties, then prove that the function with these properties exist and is unique and also describe formulas that compute this function. (Because we will show that the function exists and is unique, from the start we will say ‘det(T )’ instead of ‘if there is a determinant function then det(T )’ and ‘the determinant’ instead of ‘any determinant’.) 2.1 Deﬁnition A n×n determinant is a function det : Mn×n → R such that (1) det(ρ1 , . . . , k · ρi + ρj , . . . , ρn ) = det(ρ1 , . . . , ρj , . . . , ρn ) for i = j (2) det(ρ1 , . . . , ρj , . . . , ρi , . . . , ρn ) = − det(ρ1 , . . . , ρi , . . . , ρj , . . . , ρn ) for i = j (3) det(ρ1 , . . . , kρi , . . . , ρn ) = k · det(ρ1 , . . . , ρi , . . . , ρn ) for k = 0 (4) det(I) = 1 where I is an identity matrix (the ρ ’s are the rows of the matrix). We often write |T | for det(T ). 2.2 Remark Property (2) is redundant since T −→
ρi +ρj −ρj +ρi ρi +ρj −ρi

−→

ˆ −→ −→ T

swaps rows i and j. It is listed only for convenience. The ﬁrst result shows that a function satisfying these conditions gives a criteria for nonsingularity. (Its last sentence is that, in the context of the ﬁrst three conditions, (4) is equivalent to the condition that the determinant of an echelon form matrix is the product down the diagonal.) 2.3 Lemma A matrix with two identical rows has a determinant of zero. A matrix with a zero row has a determinant of zero. A matrix is nonsingular if and only if its determinant is nonzero. The determinant of an echelon form matrix is the product down its diagonal.

300

Chapter Four. Determinants

Proof. To verify the ﬁrst sentence, swap the two equal rows. The sign of the

determinant changes, but the matrix is unchanged and so its determinant is unchanged. Thus the determinant is zero. The second sentence is clearly true if the matrix is 1×1. If it has at least two rows then apply property (1) of the deﬁnition with the zero row as row j and with k = 1. det(. . . , ρi , . . . , 0, . . . ) = det(. . . , ρi , . . . , ρi + 0, . . . ) The ﬁrst sentence of this lemma gives that the determinant is zero. ˆ For the third sentence, where T → · · · → T is the Gauss-Jordan reduction, by the deﬁnition the determinant of T is zero if and only if the determinant of ˆ T is zero (although they could diﬀer in sign or magnitude). A nonsingular T Gauss-Jordan reduces to an identity matrix and so has a nonzero determinant. ˆ A singular T reduces to a T with a zero row; by the second sentence of this lemma its determinant is zero. Finally, for the fourth sentence, if an echelon form matrix is singular then it has a zero on its diagonal, that is, the product down its diagonal is zero. The third sentence says that if a matrix is singular then its determinant is zero. So if the echelon form matrix is singular then its determinant equals the product down its diagonal. If an echelon form matrix is nonsingular then none of its diagonal entries is zero so we can use property (3) of the deﬁnition to factor them out (again, the vertical bars | · · · | indicate the determinant operation). t1,1 0 0 t1,2 t2,2 .. . tn,n t1,n t2,n 1 0 0 t1,2 /t1,1 1 .. . 1 t1,n /t1,1 t2,n /t2,2

= t1,1 · t2,2 · · · tn,n ·

Next, the Jordan half of Gauss-Jordan elimination, using property (1) of the deﬁnition, leaves the identity matrix. 1 0 0 0 1 .. . 1 0 0

= t1,1 · t2,2 · · · tn,n ·

= t1,1 · t2,2 · · · tn,n · 1

Therefore, if an echelon form matrix is nonsingular then its determinant is the product down its diagonal. QED That result gives us a way to compute the value of a determinant function on a matrix. Do Gaussian reduction, keeping track of any changes of sign caused by row swaps and any scalars that are factored out, and then ﬁnish by multiplying down the diagonal of the echelon form result. This procedure takes the same time as Gauss’ method and so is suﬃciently fast to be practical on the size matrices that we see in this book.

Section I. Definition 2.4 Example Doing 2×2 determinants 2 −1 2 4 = 0 3 4 = 10 5

301

with Gauss’ method won’t give a big savings because the 2 × 2 determinant formula is so easy. However, a 3×3 determinant is usually easier to calculate with Gauss’ method than with the formula given earlier. 2 4 0 2 4 −3 2 6 3 = 0 0 5 2 0 −3 2 6 −9 = − 0 0 5 2 −3 0 6 5 = −54 −9

2.5 Example Determinants of matrices any bigger than 3×3 are almost always most quickly done with this Gauss’ method procedure. 1 0 0 0 0 1 0 1 1 1 0 0 1 3 0 4 = 0 5 0 1 0 1 0 0 1 1 0 −1 3 4 =− 5 −3 1 0 0 1 0 0 0 0 1 1 −1 0 3 4 = −(−5) = 5 −3 5

The prior example illustrates an important point. Although we have not yet found a 4×4 determinant formula, if one exists then we know what value it gives to the matrix — if there is a function with properties (1)-(4) then on the above matrix the function must return 5. 2.6 Lemma For each n, if there is an n × n determinant function then it is unique.
Proof. For any n × n matrix we can perform Gauss’ method on the matrix,

keeping track of how the sign alternates on row swaps, and then multiply down the diagonal of the echelon form result. By the deﬁnition and the lemma, all n×n determinant functions must return this value on this matrix. Thus all n×n determinant functions are equal, that is, there is only one input argument/output value relationship satisfying the four conditions. QED The ‘if there is an n×n determinant function’ emphasizes that, although we can use Gauss’ method to compute the only value that a determinant function could possibly return, we haven’t yet shown that such a determinant function exists for all n. In the rest of the section we will produce determinant functions. Exercises
For these, assume that an n×n determinant function exists for all n. 2.7 Use Gauss’ method to ﬁnd each determinant. 1 0 0 1 3 1 2 2 1 1 0 (a) 3 1 0 (b) −1 0 1 0 0 1 4 1 1 1 0 2.8 Use Gauss’ method to ﬁnd each.

302

Chapter Four. Determinants

1 1 0 (b) 3 0 2 5 2 2 2.9 For which values of k does this system have a unique solution? x + z−w=2 y − 2z =3 x + kz =4 z−w=2 (a) 2 −1 −1 −1 2.10 Express each of these in terms of |H|. h3,1 h3,2 h3,3 (a) h2,1 h2,2 h2,3 h1,1 h1,2 h1,3 −h1,1 −h1,2 −h1,3 (b) −2h2,1 −2h2,2 −2h2,3 −3h3,1 −3h3,2 −3h3,3 h1,1 + h3,1 h1,2 + h3,2 h1,3 + h3,3 h2,1 h2,2 h2,3 (c) 5h3,1 5h3,2 5h3,3 2.11 Find the determinant of a diagonal matrix. 2.12 Describe the solution set of a homogeneous linear system if the determinant of the matrix of coeﬃcients is nonzero. 2.13 Show that this determinant is zero. y+z x+z x+y x y z 1 1 1 2.14 (a) Find the 1×1, 2×2, and 3×3 matrices with i, j entry given by (−1)i+j . (b) Find the determinant of the square matrix with i, j entry (−1)i+j . 2.15 (a) Find the 1×1, 2×2, and 3×3 matrices with i, j entry given by i + j. (b) Find the determinant of the square matrix with i, j entry i + j. 2.16 Show that determinant functions are not linear by giving a case where |A + B| = |A| + |B|. 2.17 The second condition in the deﬁnition, that row swaps change the sign of a determinant, is somewhat annoying. It means we have to keep track of the number of swaps, to compute how the sign alternates. Can we get rid of it? Can we replace it with the condition that row swaps leave the determinant unchanged? (If so then we would need new 1 × 1, 2 × 2, and 3 × 3 formulas, but that would be a minor matter.) 2.18 Prove that the determinant of any triangular matrix, upper or lower, is the product down its diagonal. 2.19 Refer to the deﬁnition of elementary matrices in the Mechanics of Matrix Multiplication subsection. (a) What is the determinant of each kind of elementary matrix? (b) Prove that if E is any elementary matrix then |ES| = |E||S| for any appropriately sized S. (c) (This question doesn’t involve determinants.) Prove that if T is singular then a product T S is also singular. (d) Show that |T S| = |T ||S|. (e) Show that if T is nonsingular then |T −1 | = |T |−1 .

Section I. Definition

303

2.20 Prove that the determinant of a product is the product of the determinants |T S| = |T | |S| in this way. Fix the n × n matrix S and consider the function d : Mn×n → R given by T → |T S|/|S|. (a) Check that d satisﬁes property (1) in the deﬁnition of a determinant function. (b) Check property (2). (c) Check property (3). (d) Check property (4). (e) Conclude the determinant of a product is the product of the determinants. 2.21 A submatrix of a given matrix A is one that can be obtained by deleting some of the rows and columns of A. Thus, the ﬁrst matrix here is a submatrix of the second. 3 4 1 3 1 0 9 −2 2 5 2 −1 5 Prove that for any square matrix, the rank of the matrix is r if and only if r is the largest integer such that there is an r×r submatrix with a nonzero determinant. 2.22 Prove that a matrix with rational entries has a rational determinant. ? 2.23 Find the element of likeness in (a) simplifying a fraction, (b) powdering the nose, (c) building new steps on the church, (d) keeping emeritus professors on campus, (e) putting B, C, D in the determinant 1 a3 B C [Am. Math. Mon., Feb. 1953] a 1 a3 D a2 a 1 a3 a3 a2 . a 1

I.3 The Permutation Expansion
The prior subsection deﬁnes a function to be a determinant if it satisﬁes four conditions and shows that there is at most one n×n determinant function for each n. What is left is to show that for each n such a function exists. How could such a function not exist? After all, we have done computations that start with a square matrix, follow the conditions, and end with a number. The diﬃculty is that, as far as we know, the computation might not give a well-deﬁned result. To illustrate this possibility, suppose that we were to change the second condition in the deﬁnition of determinant to be that the value of a determinant does not change on a row swap. By Remark 2.2 we know that this conﬂicts with the ﬁrst and third conditions. Here is an instance of the conﬂict: here are two Gauss’ method reductions of the same matrix, the ﬁrst without any row swap 1 3 2 4
−3ρ1 +ρ2

−→

1 0

2 −2

304 and the second with a swap. 1 3 2 4
ρ1 ↔ρ2

Chapter Four. Determinants

−→

3 1

4 2

−(1/3)ρ1 +ρ2

−→

3 0

4 2/3

Following Deﬁnition 2.1 gives that both calculations yield the determinant −2 since in the second one we keep track of the fact that the row swap changes the sign of the result of multiplying down the diagonal. But if we follow the supposition and change the second condition then the two calculations yield diﬀerent values, −2 and 2. That is, under the supposition the outcome would not be well-deﬁned — no function exists that satisﬁes the changed second condition along with the other three. Of course, observing that Deﬁnition 2.1 does the right thing in this one instance is not enough; what we will do in the rest of this section is to show that there is never a conﬂict. The natural way to try this would be to deﬁne the determinant function with: “The value of the function is the result of doing Gauss’ method, keeping track of row swaps, and ﬁnishing by multiplying down the diagonal”. (Since Gauss’ method allows for some variation, such as a choice of which row to use when swapping, we would have to ﬁx an explicit algorithm.) Then we would be done if we veriﬁed that this way of computing the determinant ˆ satisﬁes the four properties. For instance, if T and T are related by a row swap then we would need to show that this algorithm returns determinants that are negatives of each other. However, how to verify this is not evident. So the development below will not proceed in this way. Instead, in this subsection we will deﬁne a diﬀerent way to compute the value of a determinant, a formula, and we will use this way to prove that the conditions are satisﬁed. The formula that we shall use is based on an insight gotten from property (3) of the deﬁnition of determinants. This property shows that determinants are not linear. 3.1 Example For this matrix det(2A) = 2 · det(A). A= 2 −1 1 3

Instead, the scalar comes out of each of the two rows. 4 −2 2 2 =2· 6 −2 1 2 =4· 6 −1 1 3

Since scalars come out a row at a time, we might guess that determinants are linear a row at a time. 3.2 Deﬁnition Let V be a vector space. A map f : V n → R is multilinear if (1) f (ρ1 , . . . , v + w, . . . , ρn ) = f (ρ1 , . . . , v, . . . , ρn ) + f (ρ1 , . . . , w, . . . , ρn ) (2) f (ρ1 , . . . , kv, . . . , ρn ) = k · f (ρ1 , . . . , v, . . . , ρn ) for v, w ∈ V and k ∈ R.

Section I. Definition 3.3 Lemma Determinants are multilinear.

305

Proof. The deﬁnition of determinants gives property (2) (Lemma 2.3 following

that deﬁnition covers the k = 0 case) so we need only check property (1). det(ρ1 , . . . , v + w, . . . , ρn ) = det(ρ1 , . . . , v, . . . , ρn ) + det(ρ1 , . . . , w, . . . , ρn ) If the set {ρ1 , . . . , ρi−1 , ρi+1 , . . . , ρn } is linearly dependent then all three matrices are singular and so all three determinants are zero and the equality is trivial. Therefore assume that the set is linearly independent. This set of n-wide row vectors has n − 1 members, so we can make a basis by adding one more vector ρ1 , . . . , ρi−1 , β, ρi+1 , . . . , ρn . Express v and w with respect to this basis v = v1 ρ1 + · · · + vi−1 ρi−1 + vi β + vi+1 ρi+1 + · · · + vn ρn w = w1 ρ1 + · · · + wi−1 ρi−1 + wi β + wi+1 ρi+1 + · · · + wn ρn giving this. v + w = (v1 + w1 )ρ1 + · · · + (vi + wi )β + · · · + (vn + wn )ρn By the deﬁnition of determinant, the value of det(ρ1 , . . . , v + w, . . . , ρn ) is unchanged by the pivot operation of adding −(v1 + w1 )ρ1 to v + w. v + w − (v1 + w1 )ρ1 = (v2 + w2 )ρ2 + · · · + (vi + wi )β + · · · + (vn + wn )ρn Then, to the result, we can add −(v2 + w2 )ρ2 , etc. Thus det(ρ1 , . . . , v + w, . . . , ρn ) = det(ρ1 , . . . , (vi + wi ) · β, . . . , ρn ) = (vi + wi ) · det(ρ1 , . . . , β, . . . , ρn ) = vi · det(ρ1 , . . . , β, . . . , ρn ) + wi · det(ρ1 , . . . , β, . . . , ρn ) (using (2) for the second equality). To ﬁnish, bring vi and wi back inside in front of β and use pivoting again, this time to reconstruct the expressions of v and w in terms of the basis, e.g., start with the pivot operations of adding v1 ρ1 to vi β and w1 ρ1 to wi ρ1 , etc. QED Multilinearity allows us to expand a determinant into a sum of determinants, each of which involves a simple matrix. 3.4 Example We can use multilinearity to split this determinant into two, ﬁrst breaking up the ﬁrst row 2 4 1 2 = 3 4 0 0 + 3 4 1 3

and then separating each of those two, breaking along the second rows. = 2 4 0 2 + 0 0 0 0 + 3 4 1 0 + 0 0 1 3

306

Chapter Four. Determinants

We are left with four determinants, such that in each row of each matrix there is a single entry from the original matrix. 3.5 Example In the same way, a 3 × 3 determinant separates into a sum of many simpler determinants. We start by splitting along the ﬁrst row, producing three determinants (the zero in the 1, 3 position is underlined to set it oﬀ visually from the zeroes that appear in the splitting). 2 1 4 3 2 1 2 0 0 0 1 0 0 0 −1 0 = 4 3 0 + 4 3 0 + 4 3 5 2 1 5 2 1 5 2 1 −1 0 5

Each of these three will itself split in three along the second row. Each of the resulting nine splits in three along the third row, resulting in twenty seven determinants 2 = 4 2 0 0 0 0 2 0 + 4 0 0 0 0 1 0 2 0 + 4 0 0 0 0 0 0 2 0 + 0 5 2 0 3 0 0 0 0 0 + ··· + 0 0 0 0 0 −1 0 5

such that each row contains a single entry from the starting matrix. So an n×n determinant expands into a sum of nn determinants where each row of each summands contains a single entry from the starting matrix. However, many of these summand determinants are zero. 3.6 Example In each of these three matrices from the above expansion, two of the rows have their entry from the starting matrix in the same column, e.g., in the ﬁrst matrix, the 2 and the 4 both come from the ﬁrst column. 2 4 0 0 0 1 0 0 0 0 0 0 0 −1 3 0 0 5 0 0 0 1 0 0 0 0 5

Any such matrix is singular, because in each, one row is a multiple of the other (or is a zero row). Thus, any such determinant is zero, by Lemma 2.3. Therefore, the above expansion of the 3×3 determinant into the sum of the twenty seven determinants simpliﬁes to the sum of these six. 2 4 2 1 3 1 −1 2 0 = 0 5 0 0 + 4 0 0 + 4 0 0 3 0 1 0 0 0 0 1 0 2 0 0 0 + 0 0 0 0 1 0 5 0 0 0 + 0 2 5 1 0 0 0 0 0 −1 0 0

−1 0 0 0 + 0 3 0 2 0

Section I. Definition We can bring out the scalars. 1 = (2)(3)(5) 0 0 0 + (1)(4)(5) 1 0 0 + (−1)(4)(1) 1 0 0 1 0 1 0 0 0 0 1 0 1 0 + (2)(0)(1) 0 1 0 0 0 0 + (1)(0)(2) 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0

307

0 1 0 + (−1)(3)(2) 0 1 0

To ﬁnish, we evaluate those six determinants by row-swapping them to the identity matrix, keeping track of the resulting sign changes. = 30 · (+1) + 0 · (−1) + 20 · (−1) + 0 · (+1) − 4 · (+1) − 6 · (−1) = 12 That example illustrates the key idea. We’ve applied multilinearity to a 3×3 determinant to get 33 separate determinants, each with one distinguished entry per row. We can drop most of these new determinants because the matrices are singular, with one row a multiple of another. We are left with the oneentry-per-row determinants also having only one entry per column (one entry from the original determinant, that is). And, since we can factor scalars out, we can further reduce to only considering determinants of one-entry-per-row-andcolumn matrices where the entries are ones. These are permutation matrices. Thus, the determinant can be computed in this three-step way (Step 1) for each permutation matrix, multiply together the entries from the original matrix where that permutation matrix has ones, (Step 2) multiply that by the determinant of the permutation matrix and (Step 3) do that for all permutation matrices and sum the results together. To state this as a formula, we introduce a notation for permutation matrices. Let ιj be the row vector that is all zeroes except for a one in its j-th entry, so that the four-wide ι2 is 0 1 0 0 . We can construct permutation matrices by permuting — that is, scrambling — the numbers 1, 2, . . . , n, and using them as indices on the ι’s. For instance, to get a 4×4 permutation matrix matrix, we can scramble the numbers from 1 to 4 into this sequence 3, 2, 1, 4 and take the corresponding row vector ι’s.     ι3 0 0 1 0 ι2  0 1 0 0  =  ι1  1 0 0 0 ι4 0 0 0 1 3.7 Deﬁnition An n-permutation is a sequence consisting of an arrangement of the numbers 1, 2, . . . , n.

308

Chapter Four. Determinants

3.8 Example The 2-permutations are φ1 = 1, 2 and φ2 = 2, 1 . These are the associated permutation matrices. Pφ 1 = ι1 ι2 = 1 0 0 1 Pφ2 = ι2 ι1 = 0 1 1 0

We sometimes write permutations as functions, e.g., φ2 (1) = 2, and φ2 (2) = 1. Then the rows of Pφ2 are ιφ2 (1) = ι2 and ιφ2 (2) = ι1 . The 3-permutations are φ1 = 1, 2, 3 , φ2 = 1, 3, 2 , φ3 = 2, 1, 3 , φ4 = 2, 3, 1 , φ5 = 3, 1, 2 , and φ6 = 3, 2, 1 . Here are two of the associated permutation matrices.         ι1 1 0 0 ι3 0 0 1 Pφ2 = ι3  = 0 0 1 Pφ5 = ι1  = 1 0 0 ι2 0 1 0 ι2 0 1 0 For instance, the rows of Pφ5 are ιφ5 (1) = ι3 , ιφ5 (2) = ι1 , and ιφ5 (3) = ι2 . 3.9 Deﬁnition The permutation expansion for determinants is t1,1 t2,1 tn,1 t1,2 t2,2 . . . tn,2 ... ... ... t1,n t2,n tn,n

= t1,φ1 (1) t2,φ1 (2) · · · tn,φ1 (n) |Pφ1 | + t1,φ2 (1) t2,φ2 (2) · · · tn,φ2 (n) |Pφ2 | . . . + t1,φk (1) t2,φk (2) · · · tn,φk (n) |Pφk |

where φ1 , . . . , φk are all of the n-permutations. This formula is often written in summation notation |T | = t1,φ(1) t2,φ(2) permutations φ · · · tn,φ(n) |Pφ |

read aloud as “the sum, over all permutations φ, of terms having the form t1,φ(1) t2,φ(2) · · · tn,φ(n) |Pφ |”. This phrase is just a restating of the three-step process (Step 1) for each permutation matrix, compute t1,φ(1) t2,φ(2) · · · tn,φ(n) (Step 2) multiply that by |Pφ | and (Step 3) sum all such terms together. 3.10 Example The familiar formula for the determinant of a 2×2 matrix can be derived in this way. t1,1 t2,1 t1,2 = t1,1 t2,2 · |Pφ1 | + t1,2 t2,1 · |Pφ2 | t2,2 = t1,1 t2,2 · 1 0 0 0 + t1,2 t2,1 · 1 1 1 0

= t1,1 t2,2 − t1,2 t2,1

Section I. Definition

309

(the second permutation matrix takes one row swap to pass to the identity). Similarly, the formula for the determinant of a 3×3 matrix is this. t1,1 t2,1 t3,1 t1,2 t2,2 t3,2 t1,3 t2,3 = t1,1 t2,2 t3,3 |Pφ1 | + t1,1 t2,3 t3,2 |Pφ2 | + t1,2 t2,1 t3,3 |Pφ3 | t3,3 + t1,2 t2,3 t3,1 |Pφ4 | + t1,3 t2,1 t3,2 |Pφ5 | + t1,3 t2,2 t3,1 |Pφ6 | = t1,1 t2,2 t3,3 − t1,1 t2,3 t3,2 − t1,2 t2,1 t3,3 + t1,2 t2,3 t3,1 + t1,3 t2,1 t3,2 − t1,3 t2,2 t3,1 Computing a determinant by permutation expansion usually takes longer than Gauss’ method. However, here we are not trying to do the computation eﬃciently, we are instead trying to give a determinant formula that we can prove to be well-deﬁned. While the permutation expansion is impractical for computations, it is useful in proofs. In particular, we can use it for the result that we are after. 3.11 Theorem For each n there is a n×n determinant function. The proof is deferred to the following subsection. Also there is the proof of the next result (they share some features). 3.12 Theorem The determinant of a matrix equals the determinant of its transpose. The consequence of this theorem is that, while we have so far stated results in terms of rows (e.g., determinants are multilinear in their rows, row swaps change the signum, etc.), all of the results also hold in terms of columns. The ﬁnal result gives examples. 3.13 Corollary A matrix with two equal columns is singular. Column swaps change the sign of a determinant. Determinants are multilinear in their columns.
Proof. For the ﬁrst statement, transposing the matrix results in a matrix with

the same determinant, and with two equal rows, and hence a determinant of zero. The other two are proved in the same way. QED We ﬁnish with a summary (although the ﬁnal subsection contains the unﬁnished business of proving the two theorems). Determinant functions exist, are unique, and we know how to compute them. As for what determinants are about, perhaps these lines [Kemp] help make it memorable. Determinant none, Solution: lots or none. Determinant some, Solution: just one.

310 Exercises

Chapter Four. Determinants

These summarize the notation used in this book for the 2- and 3- permutations. i φ1 (i) φ2 (i) 1 1 2 2 2 1 i φ1 (i) φ2 (i) φ3 (i) φ4 (i) φ5 (i) φ6 (i) 1 1 1 2 2 3 3 2 2 3 1 3 1 2 3 3 2 3 1 2 1

3.14 Compute the determinant by using the permutation expansion. 2 2 1 1 2 3 −1 0 (b) 3 (a) 4 5 6 −2 0 5 7 8 9 3.15 Compute these both with Gauss’ method and with the permutation expansion formula. 0 1 4 2 1 (a) (b) 0 2 3 3 1 1 5 1 3.16 Use the permutation expansion formula to derive the formula for 3×3 determinants. 3.17 List all of the 4-permutations. 3.18 A permutation, regarded as a function from the set {1, .., n} to itself, is oneto-one and onto. Therefore, each permutation has an inverse. (a) Find the inverse of each 2-permutation. (b) Find the inverse of each 3-permutation. 3.19 Prove that f is multilinear if and only if for all v, w ∈ V and k1 , k2 ∈ R, this holds. f (ρ1 , . . . , k1 v1 + k2 v2 , . . . , ρn ) = k1 f (ρ1 , . . . , v1 , . . . , ρn ) + k2 f (ρ1 , . . . , v2 , . . . , ρn ) 3.20 Find the only nonzero term in the permutation expansion of this matrix. 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0

Compute that determinant by ﬁnding the signum of the associated permutation. 3.21 How would determinants change if we changed property (4) of the deﬁnition to read that |I| = 2? 3.22 Verify the second and third statements in Corollary 3.13. 3.23 Show that if an n×n matrix has a nonzero determinant then any column vector v ∈ Rn can be expressed as a linear combination of the columns of the matrix. 3.24 True or false: a matrix whose entries are only zeros or ones has a determinant equal to zero, one, or negative one. [Strang 80] 3.25 (a) Show that there are 120 terms in the permutation expansion formula of a 5×5 matrix. (b) How many are sure to be zero if the 1, 2 entry is zero? 3.26 How many n-permutations are there?

Section I. Definition
3.27 A matrix A is skew-symmetric if Atrans = −A, as in this matrix. A= 0 −3 3 0

311

Show that n×n skew-symmetric matrices with nonzero determinants exist only for even n. 3.28 What is the smallest number of zeros, and the placement of those zeros, needed to ensure that a 4×4 matrix has a determinant of zero? 3.29 If we have n data points (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ) and want to ﬁnd a polynomial p(x) = an−1 xn−1 + an−2 xn−2 + · · · + a1 x + a0 passing through those points then we can plug in the points to get an n equation/n unknown linear system. The matrix of coeﬃcients for that system is called the Vandermonde matrix. Prove that the determinant of the transpose of that matrix of coeﬃcients 1 x1 x1 2 x1 n−1 1 x2 x2 2 . . . x2 n−1 ... ... ... ... 1 xn xn 2 xn n−1

equals the product, over all indices i, j ∈ {1, . . . , n} with i < j, of terms of the form xj − xi . (This shows that the determinant is zero, and the linear system has no solution, if and only if the xi ’s in the data are not distinct.) 3.30 A matrix can be divided into blocks, as here, 1 3 0 2 4 0 0 0 −2

which shows four blocks, the square 2×2 and 1×1 ones in the upper left and lower right, and the zero blocks in the upper right and lower left. Show that if a matrix can be partitioned as T = J Z1 Z2 K

where J and K are square, and Z1 and Z2 are all zeroes, then |T | = |J| · |K|. 3.31 Prove that for any n×n matrix T there are at most n distinct reals r such that the matrix T − rI has determinant zero (we shall use this result in Chapter Five). ? 3.32 The nine positive digits can be arranged into 3×3 arrays in 9! ways. Find the sum of the determinants of these arrays. [Math. Mag., Jan. 1963, Q307] 3.33 Show that x−2 x+1 x−4 [Math. Mag., Jan. 1963, Q237] ? 3.34 Let S be the sum of the integer elements of a magic square of order three and let D be the value of the square considered as a determinant. Show that D/S is an integer. [Am. Math. Mon., Jan. 1949] x−3 x−1 x−7 x−4 x − 3 = 0. x − 10

312

Chapter Four. Determinants

? 3.35 Show that the determinant of the n2 elements in the upper left corner of the Pascal triangle 1 1 1 1 . . 1 2 3 . . 1 3 . . 1 . . . . has the value unity. [Am. Math. Mon., Jun. 1931]

I.4 Determinants Exist
This subsection is optional. It consists of proofs of two results from the prior subsection. These proofs involve the properties of permutations, which will not be used later, except in the optional Jordan Canonical Form subsection. The prior subsection attacks the problem of showing that for any size there is a determinant function on the set of square matrices of that size by using multilinearity to develop the permutation expansion. t1,1 t2,1 tn,1 t1,2 t2,2 . . . tn,2 ... ... ... t1,n t2,n tn,n

= t1,φ1 (1) t2,φ1 (2) · · · tn,φ1 (n) |Pφ1 | + t1,φ2 (1) t2,φ2 (2) · · · tn,φ2 (n) |Pφ2 | . . . + t1,φk (1) t2,φk (2) · · · tn,φk (n) |Pφk | = t1,φ(1) t2,φ(2) permutations φ · · · tn,φ(n) |Pφ |

This reduces the problem to showing that there is a determinant function on the set of permutation matrices of that size. Of course, a permutation matrix can be row-swapped to the identity matrix and to calculate its determinant we can keep track of the number of row swaps. However, the problem is still not solved. We still have not shown that the result is well-deﬁned. For instance, the determinant of   0 1 0 0 1 0 0 0  Pφ =  0 0 1 0 0 0 0 1 could be computed with one swap  1 ρ1 ↔ρ2 0 Pφ −→  0 0 0 1 0 0 0 0 1 0  0 0  0 1

Section I. Definition or with three.  0 ρ3 ↔ρ1 1 Pφ −→  0 0 0 0 1 0 1 0 0 0  0 0 ρ2 ↔ρ3  −→ 0 1  0 0  1 0 0 1 0 0 1 0 0 0  0 0 ρ1 ↔ρ3  −→ 0 1  1 0  0 0 0 1 0 0 0 0 1 0  0 0  0 1

313

Both reductions have an odd number of swaps so we ﬁgure that |Pφ | = −1 but how do we know that there isn’t some way to do it with an even number of swaps? Corollary 4.6 below proves that there is no permutation matrix that can be row-swapped to an identity matrix in two ways, one with an even number of swaps and the other with an odd number of swaps. 4.1 Deﬁnition Two rows of a permutation matrix   . . . ιk    . . .  ιj    . . . such that k > j are in an inversion of their natural order. 4.2 Example This permutation matrix    0 ι3 ι2  = 0 ι1 1

0 1 0

 1 0 0

has three inversions: ι3 precedes ι1 , ι3 precedes ι2 , and ι2 precedes ι1 . 4.3 Lemma A row-swap in a permutation matrix changes the number of inversions from even to odd, or from odd to even.
Proof. Consider a swap of rows j and k, where k > j. If the two rows are adjacent     . . . . .  .     ιφ(j)  ρk ↔ρj ιφ(k)    −→   Pφ =    ιφ(j)   ιφ(k)   . . . . . .

then the swap changes the total number of inversions by one — either removing or producing one inversion, depending on whether φ(j) > φ(k) or not, since inversions involving rows not in this pair are not aﬀected. Consequently, the total number of inversions changes from odd to even or from even to odd.

314

Chapter Four. Determinants

If the rows are not adjacent then they can be swapped via a sequence of adjacent swaps, ﬁrst bringing row k up  . . .   . . . 

   ιφ(j)    ιφ(j+1)    ιφ(j+2)  ρk ↔ρk−1  −→   .   .   .   ιφ(k)    . . . and then bringing row j down.

ρk−1 ↔ρk−2

−→

...

ρj+1 ↔ρj

−→

   ιφ(k)     ιφ(j)     ιφ(j+1)     .   .   .  ιφ(k−1)    . . .

. . .

ρj+1 ↔ρj+2 ρj+2 ↔ρj+3

−→

−→

...

ρk−1 ↔ρk

−→

   ιφ(k)    ιφ(j+1)    ιφ(j+2)     .   .   .   ιφ(j)    . . .

Each of these adjacent swaps changes the number of inversions from odd to even or from even to odd. There are an odd number (k − j) + (k − j − 1) of them. The total change in the number of inversions is from even to odd or from odd to even. QED 4.4 Deﬁnition The signum of a permutation sgn(φ) is +1 if the number of inversions in Pφ is even, and is −1 if the number of inversions is odd. 4.5 Example With the subscripts from Example 3.8 for the 3-permutations, sgn(φ1 ) = 1 while sgn(φ2 ) = −1. 4.6 Corollary If a permutation matrix has an odd number of inversions then swapping it to the identity takes an odd number of swaps. If it has an even number of inversions then swapping to the identity takes an even number of swaps.
Proof. The identity matrix has zero inversions. To change an odd number to

zero requires an odd number of swaps, and to change an even number to zero requires an even number of swaps. QED We still have not shown that the permutation expansion is well-deﬁned because we have not considered row operations on permutation matrices other than

Section I. Definition

315

row swaps. We will ﬁnesse this problem: we will deﬁne a function d : Mn×n → R by altering the permutation expansion formula, replacing |Pφ | with sgn(φ) d(T ) =
permutations φ

t1,φ(1) t2,φ(2) . . . tn,φ(n) sgn(φ)

(this gives the same value as the permutation expansion because the prior result shows that det(Pφ ) = sgn(φ)). This formula’s advantage is that the number of inversions is clearly well-deﬁned — just count them. Therefore, we will show that a determinant function exists for all sizes by showing that d is it, that is, that d satisﬁes the four conditions. 4.7 Lemma The function d is a determinant. Hence determinants exist for every n.
Proof. We’ll must check that it has the four properties from the deﬁnition.

Property (4) is easy; in d(I) =
perms φ

ι1,φ(1) ι2,φ(2) · · · ιn,φ(n) sgn(φ)

all of the summands are zero except for the product down the diagonal, which is one. kρi ˆ ˆ For property (3) consider d(T ) where T −→T . ˆ ˆ ˆ t1,φ(1) · · · ti,φ(i) · · · tn,φ(n) sgn(φ) =
perms φ φ

t1,φ(1) · · · kti,φ(i) · · · tn,φ(n) sgn(φ)

Factor the k out of each term to get the desired equality. =k·
φ ρi ↔ρj

t1,φ(1) · · · ti,φ(i) · · · tn,φ(n) sgn(φ) = k · d(T )

ˆ For (2), let T −→ T . ˆ d(T ) = ˆ ˆ ˆ ˆ t1,φ(1) · · · ti,φ(i) · · · tj,φ(j) · · · tn,φ(n) sgn(φ)
perms φ

To convert to unhatted t’s, for each φ consider the permutation σ that equals φ except that the i-th and j-th numbers are interchanged, σ(i) = φ(j) and σ(j) = ˆ ˆ ˆ ˆ φ(i). Replacing the φ in t1,φ(1) · · · ti,φ(i) · · · tj,φ(j) · · · tn,φ(n) with this σ gives t1,σ(1) · · · tj,σ(j) · · · ti,σ(i) · · · tn,σ(n) . Now sgn(φ) = − sgn(σ) (by Lemma 4.3) and so we get =
σ

t1,σ(1) · · · tj,σ(j) · · · ti,σ(i) · · · tn,σ(n) · − sgn(σ) t1,σ(1) · · · tj,σ(j) · · · ti,σ(i) · · · tn,σ(n) · sgn(σ)
σ

=−

316

Chapter Four. Determinants

where the sum is over all permutations σ derived from another permutation φ by a swap of the i-th and j-th numbers. But any permutation can be derived from some other permutation by such a swap, in one and only one way, so this summation is in fact a sum over all permutations, taken once and only once. ˆ Thus d(T ) = −d(T ). ˆ To do property (1) let T −→ T and consider ˆ d(T ) =
perms φ kρi +ρj

ˆ ˆ ˆ ˆ t1,φ(1) · · · ti,φ(i) · · · tj,φ(j) · · · tn,φ(n) sgn(φ) t1,φ(1) · · · ti,φ(i) · · · (kti,φ(j) + tj,φ(j) ) · · · tn,φ(n) sgn(φ)
φ

=

(notice: that’s kti,φ(j) , not ktj,φ(j) ). Distribute, commute, and factor. =
φ

t1,φ(1) · · · ti,φ(i) · · · kti,φ(j) · · · tn,φ(n) sgn(φ) + t1,φ(1) · · · ti,φ(i) · · · tj,φ(j) · · · tn,φ(n) sgn(φ) t1,φ(1) · · · ti,φ(i) · · · kti,φ(j) · · · tn,φ(n) sgn(φ)
φ

=

+
φ

t1,φ(1) · · · ti,φ(i) · · · tj,φ(j) · · · tn,φ(n) sgn(φ) t1,φ(1) · · · ti,φ(i) · · · ti,φ(j) · · · tn,φ(n) sgn(φ)

= k·
φ

+ d(T ) We ﬁnish by showing that the terms t1,φ(1) · · · ti,φ(i) · · · ti,φ(j) . . . tn,φ(n) sgn(φ) add to zero. This sum represents d(S) where S is a matrix equal to T except that row j of S is a copy of row i of T (because the factor is ti,φ(j) , not tj,φ(j) ). Thus, S has two equal rows, rows i and j. Since we have already shown that d changes sign on row swaps, as in Lemma 2.3 we conclude that d(S) = 0. QED We have now shown that determinant functions exist for each size. We already know that for each size there is at most one determinant. Therefore, the permutation expansion computes the one and only determinant value of a square matrix. We end this subsection by proving the other result remaining from the prior subsection, that the determinant of a matrix equals the determinant of its transpose. 4.8 Example Writing out the permutation expansion of the general 3×3 matrix and of its transpose, and comparing corresponding terms a d g b e h c 0 f = · · · + cdh · 1 i 0 0 0 1 1 0 + ··· 0

Section I. Definition (terms with the same letters) a b c d e f 0 g h = · · · + dhc · 0 1 i 1 0 0 0 1 + ··· 0

317

shows that the corresponding permutation matrices are transposes. That is, there is a relationship between these corresponding permutations. Exercise 15 shows that they are inverses. 4.9 Theorem The determinant of a matrix equals the determinant of its transpose.
Proof. Call the matrix T and denote the entries of T trans with s’s so that

ti,j = sj,i . Substitution gives this |T | =
perms φ

t1,φ(1) . . . tn,φ(n) sgn(φ) =
φ

sφ(1),1 . . . sφ(n),n sgn(φ)

and we can ﬁnish the argument by manipulating the expression on the right to be recognizable as the determinant of the transpose. We have written all permutation expansions (as in the middle expression above) with the row indices ascending. To rewrite the expression on the right in this way, note that because φ is a permutation, the row indices in the term on the right φ(1), . . . , φ(n) are just the numbers 1, . . . , n, rearranged. We can thus commute to have these ascend, giving s1,φ−1 (1) · · · sn,φ−1 (n) (if the column index is j and the row index is φ(j) then, where the row index is i, the column index is φ−1 (i)). Substituting on the right gives =
φ−1

s1,φ−1 (1) · · · sn,φ−1 (n) sgn(φ−1 )

(Exercise 14 shows that sgn(φ−1 ) = sgn(φ)). Since every permutation is the inverse of another, a sum over all φ−1 is a sum over all permutations φ =
perms σ

s1,σ( 1) . . . sn,σ(n) sgn(σ) = T trans
QED

as required. Exercises
These summarize the notation used in this book for the 2i 1 2 i 1 2 φ1 (i) 1 2 φ1 (i) 1 2 φ2 (i) 2 1 φ2 (i) 1 3 φ3 (i) 2 1 φ4 (i) 2 3 φ5 (i) 3 1 φ6 (i) 3 2

and 3- permutations. 3 3 2 3 1 2 1

318

Chapter Four. Determinants

4.10 Give the permutation expansion of a general 2×2 matrix and its transpose. 4.11 This problem appears also in the prior subsection. (a) Find the inverse of each 2-permutation. (b) Find the inverse of each 3-permutation. 4.12 (a) Find the signum of each 2-permutation. (b) Find the signum of each 3-permutation. 4.13 What is the signum of the n-permutation φ = n, n − 1, . . . , 2, 1 ? [Strang 80] 4.14 Prove these. (a) Every permutation has an inverse. (b) sgn(φ−1 ) = sgn(φ) (c) Every permutation is the inverse of another. 4.15 Prove that the matrix of the permutation inverse is the transpose of the matrix of the permutation Pφ−1 = Pφ trans , for any permutation φ. 4.16 Show that a permutation matrix with m inversions can be row swapped to the identity in m steps. Contrast this with Corollary 4.6. 4.17 For any permutation φ let g(φ) be the integer deﬁned in this way. g(φ) =
i<j

[φ(j) − φ(i)]

(This is the product, over all indices i and j with i < j, of terms of the given form.) (a) Compute the value of g on all 2-permutations. (b) Compute the value of g on all 3-permutations. (c) Prove this. g(φ) sgn(φ) = |g(φ)| Many authors give this formula as the deﬁnition of the signum function.

Section II. Geometry of Determinants

319

II

Geometry of Determinants

The prior section develops the determinant algebraically, by considering what formulas satisfy certain properties. This section complements that with a geometric approach. One advantage of this approach is that, while we have so far only considered whether or not a determinant is zero, here we shall give a meaning to the value of that determinant. (The prior section handles determinants as functions of the rows, but in this section columns are more convenient. The ﬁnal result of the prior section says that we can make the switch.)

II.1 Determinants as Size Functions
This parallelogram picture

x2 y2

x1 y1

is familiar from the construction of the sum of the two vectors. One way to compute the area that it encloses is to draw this rectangle and subtract the area of each subregion.
A y2 y1 C E x2 F x1 B D

area of parallelogram = area of rectangle − area of A − area of B − · · · − area of F = (x1 + x2 )(y1 + y2 ) − x2 y1 − x1 y1 /2 − x2 y2 /2 − x2 y2 /2 − x1 y1 /2 − x2 y1 = x1 y2 − x2 y1

The fact that the area equals the value of the determinant x1 y1 x2 = x1 y2 − x2 y1 y2

is no coincidence. The properties in the deﬁnition of determinants make reasonable postulates for a function that measures the size of the region enclosed by the vectors in the matrix. For instance, this shows the eﬀect of multiplying one of the box-deﬁning vectors by a scalar (the scalar used is k = 1.4).
w v w kv

320

Chapter Four. Determinants

The region formed by kv and w is bigger, by a factor of k, than the shaded region enclosed by v and w. That is, size(kv, w) = k · size(v, w) and in general we expect of the size measure that size(. . . , kv, . . . ) = k · size(. . . , v, . . . ). Of course, this postulate is already familiar as one of the properties in the deﬁntion of determinants. Another property of determinants is that they are unaﬀected by pivoting. Here are before-pivoting and after-pivoting boxes (the scalar used is k = 0.35).
w v kv + w v

Although the region on the right, the box formed by v and kv + w, is more slanted than the shaded region, the two have the same base and the same height and hence the same area. This illustrates that size(v, kv + w) = size(v, w). Generalized, size(. . . , v, . . . , w, . . . ) = size(. . . , v, . . . , kv + w, . . . ), which is a restatement of the determinant postulate. Of course, this picture
e2 e1

shows that size(e1 , e2 ) = 1, and we naturally extend that to any number of dimensions size(e1 , . . . , en ) = 1, which is a restatement of the property that the determinant of the identity matrix is one. With that, because property (2) of determinants is redundant (as remarked right after the deﬁnition), we have that all of the properties of determinants are reasonable to expect of a function that gives the size of boxes. We can now cite the work done in the prior section to show that the determinant exists and is unique to be assured that these postulates are consistent and suﬃcient (we do not need any more postulates). That is, we’ve got an intuitive justiﬁcation to interpret det(v1 , . . . , vn ) as the size of the box formed by the vectors. (Comment. An even more basic approach, which also leads to the deﬁnition below, is in [Weston].) 1.1 Example The volume of this parallelepiped, which can be found by the usual formula from high school geometry, is 12.

−1 0 1 2 0 2

0 3 1

2 0 0 3 2 1

−1 0 = 12 1

Section II. Geometry of Determinants

321

1.2 Remark Although property (2) of the deﬁnition of determinants is redundant, it raises an important point. Consider these two.
v u v u

4 2

1 = 10 3

1 3

4 = −10 2

The only diﬀerence between them is in the order in which the vectors are taken. If we take u ﬁrst and then go to v, follow the counterclockwise arc shown, then the sign is positive. Following a clockwise arc gives a negative sign. The sign returned by the size function reﬂects the ‘orientation’ or ‘sense’ of the box. (We see the same thing if we picture the eﬀect of scalar multiplication by a negative scalar.) Although it is both interesting and important, the idea of orientation turns out to be tricky. It is not needed for the development below, and so we will pass it by. (See Exercise 27.) 1.3 Deﬁnition The box (or parallelepiped ) formed by v1 , . . . , vn (where each vector is from Rn ) includes all of the set {t1 v1 + · · · + tn vn t1 , . . . , tn ∈ [0..1]}. The volume of a box is the absolute value of the determinant of the matrix with those vectors as columns. 1.4 Example Volume, because it is an absolute value, does not depend on the order in which the vectors are given. The volume of the parallelepiped in Exercise 1.1, can also be computed as the absolute value of this determinant. 0 3 1 2 0 2 0 3 = −12 1

The deﬁnition of volume gives a geometric interpretation to something in the space, boxes made from vectors. The next result relates the geometry to the functions that operate on spaces. 1.5 Theorem A transformation t : Rn → Rn changes the size of all boxes by the same factor, namely the size of the image of a box |t(S)| is |T | times the size of the box |S|, where T is the matrix representing t with respect to the standard basis. That is, for all n×n matrices, the determinant of a product is the product of the determinants |T S| = |T | · |S|. The two sentences state the same idea, ﬁrst in map terms and then in matrix terms. Although we tend to prefer a map point of view, the second sentence, the matrix version, is more convienent for the proof and is also the way that we shall use this result later. (Alternate proofs are given as Exercise 23 and Exercise 28.)

322

Chapter Four. Determinants

Proof. The two statements are equivalent because |t(S)| = |T S|, as both give the size of the box that is the image of the unit box En under the composition t ◦ s (where s is the map represented by S with respect to the standard basis). First consider the case that |T | = 0. A matrix has a zero determinant if and only if it is not invertible. Observe that if T S is invertible, so that there is an M such that (T S)M = I, then the associative property of matrix multiplication T (SM ) = I shows that T is also invertible (with inverse SM ). Therefore, if T is not invertible then neither is T S — if |T | = 0 then |T S| = 0, and the result holds in this case. Now consider the case that |T | = 0, that T is nonsingular. Recall that any nonsingular matrix can be factored into a product of elementary matrices, so that T S = E1 E2 · · · Er S. In the rest of this argument, we will verify that if E is an elementary matrix then |ES| = |E| · |S|. The result will follow because then |T S| = |E1 · · · Er S| = |E1 | · · · |Er | · |S| = |E1 · · · Er | · |S| = |T | · |S|. If the elementary matrix E is Mi (k) then Mi (k)S equals S except that row i has been multiplied by k. The third property of determinant functions then gives that |Mi (k)S| = k · |S|. But |Mi (k)| = k, again by the third property because Mi (k) is derived from the identity by multiplication of row i by k, and so |ES| = |E| · |S| holds for E = Mi (k). The E = Pi,j = −1 and E = Ci,j (k) checks are similar. QED

1.6 Example Application of the map t represented with respect to the standard bases by 1 1 −2 0 will double sizes of boxes, e.g., from this
v w

2 1

1 =3 2

to this

t(v)

3 −4
t(w)

3 =6 −2

1.7 Corollary If a matrix is invertible then the determinant of its inverse is the inverse of its determinant |T −1 | = 1/|T |.
Proof. 1 = |I| = |T T −1 | = |T | · |T −1 | QED

Recall that determinants are not additive homomorphisms, det(A + B) need not equal det(A) + det(B). The above theorem says, in contrast, that determinants are multiplicative homomorphisms: det(AB) does equal det(A) · det(B).

Section II. Geometry of Determinants Exercises
1.8 Find the volume of the region formed. 1 −1 (a) , 3 4 8 3 2 1 , −2 , −3 (b) 8 4 0         0 −1 2 1 2 2  3  1 (c)   ,   ,   ,   0 0 2 0 7 5 2 1 1.9 Is 4 1 2 inside of the box formed by these three? 3 3 1 1.10 Find the volume of this region. 2 6 1 1 0 5

323

1.11 Suppose that |A| = 3. By what factor do these change volumes? (a) A (b) A2 (c) A−2 1.12 By what factor does each transformation change the size of boxes? x x−y x 2x x 3x − y (a) → (b) → (c) y → x + y + z y 3y y −2x + y z y − 2z 1.13 What is the area of the image of the rectangle [2..4] × [2..5] under the action of this matrix? 2 3 4 −1 1.14 If t : R3 → R3 changes volumes by a factor of 7 and s : R3 → R3 changes volumes by a factor of 3/2 then by what factor will their composition changes volumes? 1.15 In what way does the deﬁnition of a box diﬀer from the deﬁntion of a span? 1.16 Why doesn’t this picture contradict Theorem 1.5? −→ area is 2 determinant is 2 area is 5 1.17 Does |T S| = |ST |? |T (SP )| = |(T S)P |? 1.18 (a) Suppose that |A| = 3 and that |B| = 2. Find |A2 · B trans · B −2 · Atrans |. (b) Assume that |A| = 0. Prove that |6A3 + 5A2 + 2A| = 0. 1.19 Let T be the matrix representing (with respect to the standard bases) the map that rotates plane vectors counterclockwise thru θ radians. By what factor does T change sizes? 1.20 Must a transformation t : R2 → R2 that preserves areas also preserve lengths?
2 1 0 1

324

Chapter Four. Determinants

1.21 What is the volume of a parallelepiped in R3 bounded by a linearly dependent set? 1.22 Find the area of the triangle in R3 with endpoints (1, 2, 1), (3, −1, 4), and (2, 2, 2). (Area, not volume. The triangle deﬁnes a plane — what is the area of the triangle in that plane?) 1.23 An alternate proof of Theorem 1.5 uses the deﬁnition of determinant functions. (a) Note that the vectors forming S make a linearly dependent set if and only if |S| = 0, and check that the result holds in this case. (b) For the |S| = 0 case, to show that |T S|/|S| = |T | for all transformations, consider the function d : Mn×n → R given by T → |T S|/|S|. Show that d has the ﬁrst property of a determinant. (c) Show that d has the remaining three properties of a determinant function. (d) Conclude that |T S| = |T | · |S|. 1.24 Give a non-identity matrix with the property that Atrans = A−1 . Show that if Atrans = A−1 then |A| = ±1. Does the converse hold? 1.25 The algebraic property of determinants that factoring a scalar out of a single row will multiply the determinant by that scalar shows that where H is 3×3, the determinant of cH is c3 times the determinant of H. Explain this geometrically, that is, using Theorem 1.5, 1.26 Matrices H and G are said to be similar if there is a nonsingular matrix P such that H = P −1 GP (we will study this relation in Chapter Five). Show that similar matrices have the same determinant. 1.27 We usually represent vectors in R2 with respect to the standard basis so vectors in the ﬁrst quadrant have both coordinates positive.
v

RepE2 (v) =

+3 +2

Moving counterclockwise around the origin, we cycle thru four regions:
· · · −→ + + −→ − + −→ − − −→ + − −→ · · · .

Using this basis B= 0 −1 , 1 0
β1

β2

gives the same counterclockwise cycle. We say these two bases have the same orientation. (a) Why do they give the same cycle? (b) What other conﬁgurations of unit vectors on the axes give the same cycle? (c) Find the determinants of the matrices formed from those (ordered) bases. (d) What other counterclockwise cycles are possible, and what are the associated determinants? (e) What happens in R1 ? (f ) What happens in R3 ? A fascinating general-audience discussion of orientations is in [Gardner]. 1.28 This question uses material from the optional Determinant Functions Exist subsection. Prove Theorem 1.5 by using the permutation expansion formula for the determinant.

Section II. Geometry of Determinants
1.29

325

(a) Show that this gives the equation of a line in R2 thru (x2 , y2 ) and (x3 , y3 ). x y 1 x2 y2 1 x3 y3 = 0 1

(b) [Petersen] Prove that the area of a triangle with vertices (x1 , y1 ), (x2 , y2 ), and (x3 , y3 ) is x x2 x3 1 1 y1 y2 y3 . 2 1 1 1 (c) [Math. Mag., Jan. 1973] Prove that the area of a triangle with vertices at (x1 , y1 ), (x2 , y2 ), and (x3 , y3 ) whose coordinates are integers has an area of N or N/2 for some positive integer N .

326

Chapter Four. Determinants

III

Other Formulas

(This section is optional. Later sections do not depend on this material.) Determinants are a fount of interesting and amusing formulas. Here is one that is often seen in calculus classes and used to compute determinants by hand.

III.1 Laplace’s Expansion
1.1 Example In this permutation expansion t1,1 t2,1 t3,1 t1,2 t2,2 t3,2 t1,3 1 t2,3 = t1,1 t2,2 t3,3 0 t3,3 0 0 1 0 1 0 0 + t1,1 t2,3 t3,2 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0

0 + t1,2 t2,1 t3,3 1 0 0 + t1,3 t2,1 t3,2 1 0

0 0 0 + t1,2 t2,3 t3,1 0 1 1 1 0 0 + t1,3 t2,2 t3,1 0 0 1

we can, for instance, factor out the entries from the ﬁrst row   1 0 0 1 0 0 = t1,1 · t2,2 t3,3 0 1 0 + t2,3 t3,2 0 0 1  0 1 0 0 0 1   0 1 0 0 1 0 + t1,2 · t2,1 t3,3 1 0 0 + t2,3 t3,1 0 0 1  1 0 0 0 0 1   0 0 1 0 0 1 + t1,3 · t2,1 t3,2 1 0 0 + t2,2 t3,1 0 1 0  1 0 0 0 1 0 and swap rows in the permutation matrices to get this.  1 0 0 1 0 = t1,1 · t2,2 t3,3 0 1 0 + t2,3 t3,2 0 0 0 0 1 0 1  1 0 0 1 − t1,2 · t2,1 t3,3 0 1 0 + t2,3 t3,1 0 0 0 1 0  1 0 0 1 + t1,3 · t2,1 t3,2 0 1 0 + t2,2 t3,1 0 0 0 1 0  0 1  0 0 0 1 0 0 1  0 1  0  0 1  0

Section III. Other Formulas

327

The point of the swapping (one swap to each of the permutation matrices on the second line and two swaps to each on the third line) is that the three lines simplify to three terms. = t1,1 · t2,2 t3,2 t t2,3 − t1,2 · 2,1 t3,1 t3,3 t t2,3 + t1,3 · 2,1 t3,1 t3,3 t2,2 t3,2

The formula given in Theorem 1.5, which generalizes this example, is a recurrence — the determinant is expressed as a combination of determinants. This formula isn’t circular because, as here, the determinant is expressed in terms of determinants of matrices of smaller size. 1.2 Deﬁnition For any n×n matrix T , the (n − 1)×(n − 1) matrix formed by deleting row i and column j of T is the i, j minor of T . The i, j cofactor Ti,j of T is (−1)i+j times the determinant of the i, j minor of T . 1.3 Example The 1, 2 cofactor of the matrix from Example 1.1 is the negative of the second 2×2 determinant. T1,2 = −1 · 1.4 Example Where 1 T = 4 7 these are the 1, 2 and 2, 2 cofactors. T1,2 = (−1)1+2 · 4 7 6 =6 9 T2,2 = (−1)2+2 · 1 7 3 = −12 9  2 5 8  3 6 9 t2,1 t3,1 t2,3 t3,3

1.5 Theorem (Laplace Expansion of Determinants) Where T is an n×n matrix, the determinant can be found by expanding by cofactors on row i or column j. |T | = ti,1 · Ti,1 + ti,2 · Ti,2 + · · · + ti,n · Ti,n = t1,j · T1,j + t2,j · T2,j + · · · + tn,j · Tn,j
Proof. Exercise 27. QED

1.6 Example We can compute the determinant 1 |T | = 4 7 2 5 8 3 6 9

by expanding along the ﬁrst row, as in Example 1.1. |T | = 1 · (+1) 5 8 6 4 + 2 · (−1) 9 7 6 4 + 3 · (+1) 9 7 5 = −3 + 12 − 9 = 0 8

328

Chapter Four. Determinants

Alternatively, we can expand down the second column. |T | = 2 · (−1) 4 7 1 6 + 5 · (+1) 7 9 1 3 + 8 · (−1) 4 9 3 = 12 − 60 + 48 = 0 6

1.7 Example A row or column with many zeroes suggests a Laplace expansion. 1 2 3 5 1 −1 0 2 1 = 0 · (+1) 3 0 1 1 + 1 · (−1) 3 −1 1 5 + 0 · (+1) 2 −1 5 = 16 1

We ﬁnish by applying this result to derive a new formula for the inverse of a matrix. With Theorem 1.5, the determinant of an n × n matrix T can be calculated by taking linear combinations of entries from a row and their associated cofactors. ti,1 · Ti,1 + ti,2 · Ti,2 + · · · + ti,n · Ti,n = |T | (∗)

Recall that a matrix with two identical rows has a zero determinant. Thus, for any matrix T , weighing the cofactors by entries from the “wrong” row — row k with k = i — gives zero ti,1 · Tk,1 + ti,2 · Tk,2 + · · · + ti,n · Tk,n = 0 (∗∗)

because it represents the expansion along the row k of a matrix with row i equal to row k. This equation summarizes (∗) and (∗∗).      t1,1 t1,2 . . . t1,n T1,1 T2,1 . . . Tn,1 |T | 0 . . . 0  t2,1 t2,2 . . . t2,n   T1,2 T2,2 . . . Tn,2   0 |T | . . . 0          = . . . . . .      . . . tn,1 tn,2 ... tn,n T1,n T2,n ... Tn,n 0 0 ... |T | Note that the order of the subscripts in the matrix of cofactors is opposite to the order of subscripts in the other matrix; e.g., along the ﬁrst row of the matrix of cofactors the subscripts are 1, 1 then 2, 1, etc. 1.8 Deﬁnition The matrix adjoint to the  T1,1 T2,1  T1,2 T2,2  adj(T ) =  . .  . T1,n T2,n where Tj,i is the j, i cofactor. 1.9 Theorem Where T is a square matrix, T · adj(T ) = adj(T ) · T = |T | · I.
Proof. Equations (∗) and (∗∗). QED

square matrix T is  . . . Tn,1 . . . Tn,2     . . . Tn,n

Section III. Other Formulas 1.10 Example If  1 T = 2 1 then the adjoint adj(T ) is   T1,1 T1,2 T1,3 T2,1 T2,2 T2,3 0 1 0  4 −1 1

329

1   0  T3,1   2 T3,2 =−  1 T3,3   2 1

−1 1 −1 1 1 0

0 − 0 1 1 − 1 1

4 1 4 1 0 0 −

0 1 1 2 1 2

 4 −1    1  4   = −3 −1  −1  0  1

0 −3 0

 −4 9 1

and taking the product with T gives the diagonal matrix |T | · I.      −3 0 0 1 0 −4 1 0 4 2 1 −1 −3 −3 9  =  0 −3 0  0 0 −3 −1 0 1 1 0 1 1.11 Corollary If |T | = 0 then T −1 = (1/|T |) · adj(T ). 1.12 Example The inverse of the matrix from Example 1.10 is (1/−3)·adj(T ).     1/−3 0/−3 −4/−3 −1/3 0 4/3 1 −3  T −1 = −3/−3 −3/−3 9/−3  =  1 −1/−3 0/−3 1/−3 1/3 0 −1/3 The formulas from this section are often used for by-hand calculation and are sometimes useful with special types of matrices. However, they are not the best choice for computation with arbitrary matrices because they require more arithmetic than, for instance, the Gauss-Jordan method. Exercises
1.13 Find the cofactor. T = 1 −1 0 0 1 2 2 3 −1

(a) T2,3 (b) T3,2 (c) T1,3 1.14 Find the determinant by expanding 3 1 −1 0 2 3 1 2 0

(a) on the ﬁrst row (b) on the second row (c) on the third column. 1.15 Find the adjoint of the matrix in Example 1.6. 1.16 Find the matrix adjoint to each.

330
2 −1 1 1.17 Find the 1.18 Find the (a)

Chapter Four. Determinants
1 4 3 1 4 3 −1 1 1 0 2 (b) (c) (d) −1 0 3 2 4 5 0 1 8 9 0 1 inverse of each matrix in the prior question with Theorem 1.9. matrix adjoint to this one.   2 1 0 0 1 2 1 0  0 1 2 1  0 0 1 2

1.19 Expand across the ﬁrst row to derive the formula for the determinant of a 2×2 matrix. 1.20 Expand across the ﬁrst row to derive the formula for the determinant of a 3×3 matrix. 1.21 (a) Give a formula for the adjoint of a 2×2 matrix. (b) Use it to derive the formula for the inverse. 1.22 Can we compute a determinant by expanding down the diagonal? 1.23 Give a formula for the adjoint of a diagonal matrix. 1.24 Prove that the transpose of the adjoint is the adjoint of the transpose. 1.25 Prove or disprove: adj(adj(T )) = T . 1.26 A square matrix is upper triangular if each i, j entry is zero in the part above the diagonal, that is, when i > j. (a) Must the adjoint of an upper triangular matrix be upper triangular? Lower triangular? (b) Prove that the inverse of a upper triangular matrix is upper triangular, if an inverse exists. 1.27 This question requires material from the optional Determinants Exist subsection. Prove Theorem 1.5 by using the permutation expansion. 1.28 Prove that the determinant of a matrix equals the determinant of its transpose using Laplace’s expansion and induction on the size of the matrix. ? 1.29 Show that 1 −1 1 −1 1 −1 . . . 1 1 0 1 0 1 ... 1 1 0 1 0 ... Fn = 0 0 0 1 1 0 1 ... . . . . . . ... where Fn is the n-th term of 1, 1, 2, 3, 5, . . . , x, y, x + y, . . . , the Fibonacci sequence, and the determinant is of order n − 1. [Am. Math. Mon., Jun. 1949]

Topic: Cramer’s Rule

331

Topic: Cramer’s Rule
We have introduced determinant functions algebraically by looking for a formula to decide whether a matrix is nonsingular. After that introduction we saw a geometric interpretation, that the determinant function gives the size of the box with sides formed by the columns of the matrix. This Topic makes a connection between the two views. First, a linear system x1 + 2x2 = 6 3x1 + x2 = 8 is equivalent to a linear relationship among vectors. x1 · 1 2 + x2 · 3 1 = 6 8
1 3 2 1 .

The picture below shows a parallelogram with sides formed from nested inside a parallelogram with sides formed from x1 1 and x2 3
6 8 x1 · 1 3

and

2 1

1 3 x2 · 2 1 2 1

So even without determinants we can state the algebraic issue that opened this book, ﬁnding the solution of a linear system, in geometric terms: by what factors x1 and x2 must we dilate the vectors to expand the small parallegram to ﬁll the larger one? However, by employing the geometric signiﬁcance of determinants we can get something that is not just a restatement, but also gives us a new insight and sometimes allows us to compute answers quickly. Compare the sizes of these shaded boxes.
6 8

x1 ·

1 3

1 3

2 1

2 1

2 1

The second is formed from x1 1 and 2 , and one of the properties of the size 1 3 function — the determinant — is that its size is therefore x1 times the size of the

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ﬁrst box. Since the third box is formed from x1 1 + x2 2 = 6 and 2 , and 3 1 8 1 the determinant is unchanged by adding x2 times the second column to the ﬁrst column, the size of the third box equals that of the second. We have this. 6 8 x ·1 2 = 1 x1 · 3 1 1 2 = x1 · 3 1 2 1

Solving gives the value of one of the variables. 6 2 8 1 −10 = x1 = =2 −5 1 2 3 1 The theorem that generalizes this example, Cramer’s Rule, is: if |A| = 0 then the system Ax = b has the unique solution xi = |Bi |/|A| where the matrix Bi is formed from A by replacing column i with the vector b. Exercise 3 asks for a proof. For instance, to solve this system for x2      x1 1 0 4 2 2 1 −1 x2  =  1  1 0 1 x3 −1 we do this computation. 1 2 4 2 1 −1 1 −1 1 −18 = x2 = −3 1 0 4 2 1 −1 1 0 1 Cramer’s Rule allows us to solve many two equations/two unknowns systems by eye. It is also sometimes used for three equations/three unknowns systems. But computing large determinants takes a long time, so solving large systems by Cramer’s Rule is not practical. Exercises
1 Use Cramer’s Rule to solve each for each of the variables. x− y= 4 −2x + y = −2 (a) (b) −x + 2y = −7 x − 2y = −2 2 Use Cramer’s Rule to solve this system for z. 2x + y + z = 1 3x +z=4 x−y−z=2 3 Prove Cramer’s Rule.

Topic: Cramer’s Rule

333

4 Suppose that a linear system has as many equations as unknowns, that all of its coeﬃcients and constants are integers, and that its matrix of coeﬃcients has determinant 1. Prove that the entries in the solution are all integers. (Remark. This is often used to invent linear systems for exercises. If an instructor makes the linear system with this property then the solution is not some disagreeable fraction.) 5 Use Cramer’s Rule to give a formula for the solution of a two equations/two unknowns linear system. 6 Can Cramer’s Rule tell the diﬀerence between a system with no solutions and one with inﬁnitely many? 7 The ﬁrst picture in this Topic (the one that doesn’t use determinants) shows a unique solution case. Produce a similar picture for the case of inﬁntely many solutions, and the case of no solutions.

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Topic: Speed of Calculating Determinants
The permutation expansion formula for computing determinants is useful for proving theorems, but the method of using row operations is a much better for ﬁnding the determinants of a large matrix. We can make this statement precise by considering, as computer algorithm designers do, the number of arithmetic operations that each method uses. The speed of an algorithm is measured by ﬁnding how the time taken by the computer grows as the size of its input data set grows. For instance, how much longer will the algorithm take if we increase the size of the input data by a factor of ten, from a 1000 row matrix to a 10, 000 row matrix or from 10, 000 to 100, 000? Does the time taken grow by a factor of ten, or by a factor of a hundred, or by a factor of a thousand? That is, is the time taken by the algorithm proportional to the size of the data set, or to the square of that size, or to the cube of that size, etc.? Recall the permutation expansion formula for determinants. t1,1 t2,1 tn,1 t1,2 t2,2 . . . tn,2 ... ... ... t1,n t2,n tn,n = t1,φ1 (1) · t2,φ1 (2) · · · tn,φ1 (n) |Pφ1 | + t1,φ2 (1) · t2,φ2 (2) · · · tn,φ2 (n) |Pφ2 | . . . + t1,φk (1) · t2,φk (2) · · · tn,φk (n) |Pφk | There are n! = n · (n − 1) · (n − 2) · · · 2 · 1 diﬀerent n-permutations. For numbers n of any size at all, this is a large value; for instance, even if n is only 10 then the expansion has 10! = 3, 628, 800 terms, all of which are obtained by multiplying n entries together. This is a very large number of multiplications (for instance, [Knuth] suggests 10! steps as a rough boundary for the limit of practical calculation). The factorial function grows faster than the square function. It grows faster than the cube function, the fourth power function, or any polynomial function. (One way to see that the factorial function grows faster than the square is to note that multiplying the ﬁrst two factors in n! gives n · (n − 1), which for large n is approximately n2 , and then multiplying in more factors will make it even larger. The same argument works for the cube function, etc.) So a computer that is programmed to use the permutation expansion formula, and thus to perform a number of operations that is greater than or equal to the factorial of the number of rows, would take very long times as its input data set grows. In contrast, the time taken by the row reduction method does not grow so fast. This fragment of row-reduction code is in the computer language FORTRAN. The matrix is stored in the N ×N array A. For each ROW between 1 and N parts of the program not shown here have already found the pivot entry

=

t1,φ(1) t2,φ(2) permutations φ

· · · tn,φ(n) |Pφ |

Topic: Speed of Calculating Determinants A(ROW, COL). Now the program does a row pivot. −PIVINV · ρROW + ρi

335

(This code fragment is for illustration only and is incomplete. Still, analysis of a ﬁnished version that includes all of the tests and subcases is messier but gives essentially the same conclusion.)
PIVINV=1.0/A(ROW,COL) DO 10 I=ROW+1, N DO 20 J=I, N A(I,J)=A(I,J)-PIVINV*A(ROW,J) 20 CONTINUE 10 CONTINUE

The outermost loop (not shown) runs through N − 1 rows. For each row, the nested I and J loops shown perform arithmetic on the entries in A that are below and to the right of the pivot entry. Assume that the pivot is found in the expected place, that is, that COL = ROW . Then there are (N − ROW )2 entries below and to the right of the pivot. On average, ROW will be N/2. Thus, we estimate that the arithmetic will be performed about (N/2)2 times, that is, will run in a time proportional to the square of the number of equations. Taking into account the outer loop that is not shown, we get the estimate that the running time of the algorithm is proportional to the cube of the number of equations. Finding the fastest algorithm to compute the determinant is a topic of current research. Algorithms are known that run in time between the second and third power. Speed estimates like these help us to understand how quickly or slowly an algorithm will run. Algorithms that run in time proportional to the size of the data set are fast, algorithms that run in time proportional to the square of the size of the data set are less fast, but typically quite usable, and algorithms that run in time proportional to the cube of the size of the data set are still reasonable in speed for not-too-big input data. However, algorithms that run in time (greater than or equal to) the factorial of the size of the data set are not practical for input of any appreciable size. There are other methods besides the two discussed here that are also used for computation of determinants. Those lie outside of our scope. Nonetheless, this contrast of the two methods for computing determinants makes the point that although in principle they give the same answer, in practice the idea is to select the one that is fast. Exercises
Most of these problems presume access to a computer. 1 Computer systems generate random numbers (of course, these are only pseudorandom, in that they are generated by an algorithm, but they pass a number of reasonable statistical tests for randomness). (a) Fill a 5×5 array with random numbers (say, in the range [0..1)). See if it is singular. Repeat that experiment a few times. Are singular matrices frequent or rare (in this sense)?

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Topic: Projective Geometry

337

Topic: Projective Geometry
There are geometries other than the familiar Euclidean one. One such geometry arose in art, where it was observed that what a viewer sees is not necessarily what is there. This is Leonardo da Vinci’s The Last Supper.

What is there in the room, for instance where the ceiling meets the left and right walls, are lines that are parallel. However, what a viewer sees is lines that, if extended, would intersect. The intersection point is called the vanishing point. This aspect of perspective is also familiar as the image of a long stretch of railroad tracks that appear to converge at the horizon. To depict the room, da Vinci has adopted a model of how we see, of how we project the three dimensional scene to a two dimensional image. This model is only a ﬁrst approximation — it does not take into account that our retina is curved and our lens bends the light, that we have binocular vision, or that our brain’s processing greatly aﬀects what we see — but nonetheless it is interesting, both artistically and mathematically. The projection is not orthogonal, it is a central projection from a single point, to the plane of the canvas.

A B C

(It is not an orthogonal projection since the line from the viewer to C is not orthogonal to the image plane.) As the picture suggests, the operation of central projection preserves some geometric properties — lines project to lines. However, it fails to preserve some others — equal length segments can project to segments of unequal length; the length of AB is greater than the length of

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Chapter Four. Determinants

BC because the segment projected to AB is closer to the viewer and closer things look bigger. The study of the eﬀects of central projections is projective geometry. We will see how linear algebra can be used in this study. There are three cases of central projection. The ﬁrst is the projection done by a movie projector.

projector P

source S

image I

We can think that each source point is “pushed” from the domain plane outward to the image point in the codomain plane. This case of projection has a somewhat diﬀerent character than the second case, that of the artist “pulling” the source back to the canvas.

painter P

image I

source S

In the ﬁrst case S is in the middle while in the second case I is in the middle. One more conﬁguration is possible, with P in the middle. An example of this is when we use a pinhole to shine the image of a solar eclipse onto a piece of paper.

source S

pinhole P

image I

We shall take each of the three to be a central projection by P of S to I.

Topic: Projective Geometry

339

Consider again the eﬀect of railroad tracks that appear to converge to a point. We model this with parallel lines in a domain plane S and a projection via a P to a codomain plane I. (The gray lines are parallel to S and I.)

S P

I

All three projection cases appear here. The ﬁrst picture below shows P acting like a movie projector by pushing points from part of S out to image points on the lower half of I. The middle picture shows P acting like the artist by pulling points from another part of S back to image points in the middle of I. In the third picture, P acts like the pinhole, projecting points from S to the upper part of I. This picture is the trickiest — the points that are projected near to the vanishing point are the ones that are far out on the bottom left of S. Points in S that are near to the vertical gray line are sent high up on I.

S P P

S P

S

I

I

I

There are two awkward things about this situation. The ﬁrst is that neither of the two points in the domain nearest to the vertical gray line (see below) has an image because a projection from those two is along the gray line that is parallel to the codomain plane (we sometimes say that these two are projected “to inﬁnity”). The second awkward thing is that the vanishing point in I isn’t the image of any point from S because a projection to this point would be along the gray line that is parallel to the domain plane (we sometimes say that the vanishing point is the image of a projection “from inﬁnity”).

S P

I

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Chapter Four. Determinants

For a better model, put the projector P at the origin. Imagine that P is covered by a glass hemispheric dome. As P looks outward, anything in the line of vision is projected to the same spot on the dome. This includes things on the line between P and the dome, as in the case of projection by the movie projector. It includes things on the line further from P than the dome, as in the case of projection by the painter. It also includes things on the line that lie behind P , as in the case of projection by a pinhole.
= {k · 1 2 3 k ∈ R}

From this perspective P , all of the spots on the line are seen as the same point. Accordingly, for any nonzero vector v ∈ R3 , we deﬁne the associated point v in the projective plane to be the set {kv k ∈ R and k = 0} of nonzero vectors lying on the same line through the origin as v. To describe a projective point we can give any representative member of the line, so that the projective point shown above can be represented in any of these three ways.       1/3 −2 1 2/3 −4 2 1 −6 3 Each of these is a homogeneous coordinate vector for v. This picture, and the above deﬁnition that arises from it, clariﬁes the description of central projection but there is something awkward about the dome model: what if the viewer looks down? If we draw P ’s line of sight so that the part coming toward us, out of the page, goes down below the dome then we can trace the line of sight backward, up past P and toward the part of the hemisphere that is behind the page. So in the dome model, looking down gives a projective point that is behind the viewer. Therefore, if the viewer in the picture above drops the line of sight toward the bottom of the dome then the projective point drops also and as the line of sight continues down past the equator, the projective point suddenly shifts from the front of the dome to the back of the dome. This discontinuity in the drawing means that we often have to treat equatorial points as a separate case. That is, while the railroad track discussion of central projection has three cases, the dome model has two. We can do better than this. Consider a sphere centered at the origin. Any line through the origin intersects the sphere in two spots, which are said to be antipodal. Because we associate each line through the origin with a point in the projective plane, we can draw such a point as a pair of antipodal spots on the sphere. Below, the two antipodal spots are shown connected by a dashed line

Topic: Projective Geometry

341

to emphasize that they are not two diﬀerent points, the pair of spots together make one projective point.

While drawing a point as a pair of antipodal spots is not as natural as the onespot-per-point dome mode, on the other hand the awkwardness of the dome model is gone, in that if as a line of view slides from north to south, no sudden changes happen on the picture. This model of central projection is uniform — the three cases are reduced to one. So far we have described points in projective geometry. What about lines? What a viewer at the origin sees as a line is shown below as a great circle, the intersection of the model sphere with a plane through the origin.

(One of the projective points on this line is shown to bring out a subtlety. Because two antipodal spots together make up a single projective point, the great circle’s behind-the-paper part is the same set of projective points as its in-front-of-the-paper part.) Just as we did with each projective point, we will also describe a projective line with a triple of reals. For instance, the members of this plane through the origin in R3   x {y  x + y − z = 0} z project to a line that we can described with the triple 1 1 −1 (we use row vectors to typographically distinguish lines from points). In general, for any nonzero three-wide row vector L we deﬁne the associated line in the projective plane, to be the set L = {k L k ∈ R and k = 0} of nonzero multiples of L. The reason that this description of a line as a triple is convienent is that in the projective plane, a point v and a line L are incident — the point lies on the line, the line passes throught the point — if and only if a dot product of their representatives v1 L1 + v2 L2 + v3 L3 is zero (Exercise 4 shows that this is independent of the choice of representatives v and L). For instance, the projective point described above by the column vector with components 1, 2, and 3 lies in the projective line described by 1 1 −1 , simply because any

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vector in R3 whose components are in ratio 1 : 2 : 3 lies in the plane through the origin whose equation is of the form 1k · x + 1k · y − 1k · z = 0 for any nonzero k. That is, the incidence formula is inherited from the three-space lines and planes of which v and L are projections. Thus, we can do analytic projective geometry. For instance, the projective line L = 1 1 −1 has the equation 1v1 + 1v2 − 1v3 = 0, because points incident on the line are characterized by having the property that their representatives satisfy this equation. One diﬀerence from familiar Euclidean anlaytic geometry is that in projective geometry we talk about the equation of a point. For a ﬁxed point like   1 v = 2 3 the property that characterizes lines through this point (that is, lines incident on this point) is that the components of any representatives satisfy 1L1 + 2L2 + 3L3 = 0 and so this is the equation of v. This symmetry of the statements about lines and points brings up the Duality Principle of projective geometry: in any true statement, interchanging ‘point’ with ‘line’ results in another true statement. For example, just as two distinct points determine one and only one line, in the projective plane, two distinct lines determine one and only one point. Here is a picture showing two lines that cross in antipodal spots and thus cross at one projective point.

(∗)

Contrast this with Euclidean geometry, where two distinct lines may have a unique intersection or may be parallel. In this way, projective geometry is simpler, more uniform, than Euclidean geometry. That simplicity is relevant because there is a relationship between the two spaces: the projective plane can be viewed as an extension of the Euclidean plane. Take the sphere model of the projective plane to be the unit sphere in R3 and take Euclidean space to be the plane z = 1. This gives us a way of viewing some points in projective space as corresponding to points in Euclidean space, because all of the points on the plane are projections of antipodal spots from the sphere.

(∗∗)

Topic: Projective Geometry

343

Note though that projective points on the equator don’t project up to the plane. Instead, these project ‘out to inﬁnity’. We can thus think of projective space as consisting of the Euclidean plane with some extra points adjoined — the Euclidean plane is embedded in the projective plane. These extra points, the equatorial points, are the ideal points or points at inﬁnity and the equator is the ideal line or line at inﬁnity (note that it is not a Euclidean line, it is a projective line). The advantage of the extension to the projective plane is that some of the awkwardness of Euclidean geometry disappears. For instance, the projective lines shown above in (∗) cross at antipodal spots, a single projective point, on the sphere’s equator. If we put those lines into (∗∗) then they correspond to Euclidean lines that are parallel. That is, in moving from the Euclidean plane to the projective plane, we move from having two cases, that lines either intersect or are parallel, to having only one case, that lines intersect (possibly at a point at inﬁnity). The projective case is nicer in many ways than the Euclidean case but has the problem that we don’t have the same experience or intuitions with it. That’s one advantage of doing analytic geometry, where the equations can lead us to the right conclusions. Analytic projective geometry uses linear algebra. For instance, for three points of the projective plane t, u, and v, setting up the equations for those points by ﬁxing vectors representing each, shows that the three are collinear — incident in a single line — if and only if the resulting threeequation system has inﬁnitely many row vector solutions representing that line. That, in turn, holds if and only if this determinant is zero. t1 t2 t3 u1 u2 u3 v1 v2 v3

Thus, three points in the projective plane are collinear if and only if any three representative column vectors are linearly dependent. Similarly (and illustrating the Duality Principle), three lines in the projective plane are incident on a single point if and only if any three row vectors representing them are linearly dependent. The following result is more evidence of the ‘niceness’ of the geometry of the projective plane, compared to the Euclidean case. These two triangles are said to be in perspective from P because their corresponding vertices are collinear.
O T1 U1 T2 V2 U2

V1

Consider the pairs of corresponding sides: the sides T1 U1 and T2 U2 , the sides T1 V1 and T2 V2 , and the sides U1 V1 and U2 V2 . Desargue’s Theorem is that

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when the three pairs of corresponding sides are extended to lines, they intersect (shown here as the point T U , the point T V , and the point U V ), and further, those three intersection points are collinear.
UV TV TU

We will prove this theorem, using projective geometry. (These are drawn as Euclidean ﬁgures because it is the more familiar image. To consider them as projective ﬁgures, we can imagine that, although the line segments shown are parts of great circles and so are curved, the model has such a large radius compared to the size of the ﬁgures that the sides appear in this sketch to be straight.) For this proof, we need a preliminary lemma [Coxeter]: if W , X, Y , Z are four points in the projective plane (no three of which are collinear) then there are homogeneous coordinate vectors w, x, y, and z for the projective points, and a basis B for R3 , satisfying this.         0 0 1 1 RepB (w) = 0 RepB (x) = 1 RepB (y) = 0 RepB (z) = 1 0 1 1 0 The proof is straightforward. Because W, X, Y are not on the same projective line, any homogeneous coordinate vectors w0 , x0 , y0 do not line on the same plane through the origin in R3 and so form a spanning set for R3 . Thus any homogeneous coordinate vector for Z can be written as a combination z0 = a · w0 + b · x0 + c · y0 . Then, we can take w = a · w0 , x = b · x0 , y = c · y0 , and z = z0 , where the basis is B = w, x, y . Now, to prove of Desargue’s Theorem, use the lemma to ﬁx homogeneous coordinate vectors and a basis.         1 0 0 1 RepB (t1 ) = 0 RepB (u1 ) = 1 RepB (v1 ) = 0 RepB (o) = 1 0 0 1 1 Because the projective point T2 is incident on the projective line OT1 , any homogeneous coordinate vector for T2 lies in the plane through the origin in R3 that is spanned by homogeneous coordinate vectors of O and T1 :     1 1 RepB (t2 ) = a 1 + b 0 1 0

Topic: Projective Geometry

345

for some scalars a and b. That is, the homogenous coordinate vectors of members T2 of the line OT1 are of the form on the left below, and the forms for U2 and V2 are similar.       t2 1 1 RepB (t2 ) =  1  RepB (u2 ) = u2  RepB (v2 ) =  1  1 1 v2 The projective line T1 U1 is the image of a plane through the origin in R3 . A quick way to get its equation is to note that any vector in it is linearly dependent on the vectors for T1 and U1 and so this determinant is zero. 1 0 0 0 1 0 x y =0 z =⇒ z=0

The equation of the plane in R3 whose image is the projective line T2 U2 is this. t2 1 1 1 u2 1 x y =0 z =⇒ (1 − u2 ) · x + (1 − t2 ) · y + (t2 u2 − 1) · z = 0

Finding the intersection of the two is routine.   t2 − 1 T1 U1 ∩ T2 U2 = 1 − u2  0 (This is, of course, the homogeneous coordinate vector of a projective point.) The other two intersections are similar.     1 − t2 0 T1 V1 ∩ T2 V2 =  0  U1 V1 ∩ U2 V2 = u2 − 1 1 − v2 v2 − 1 The proof is ﬁnished by noting that these projective points are on one projective line because the sum of the three homogeneous coordinate vectors is zero. Every projective theorem has a translation to a Euclidean version, although the Euclidean result is often messier to state and prove. Desargue’s theorem illustrates this. In the translation to Euclidean space, the case where O lies on the ideal line must be treated separately for then the lines T1 T2 , U1 U2 , and V1 V2 are parallel. The parenthetical remark following the statement of Desargue’s Theorem suggests thinking of the Euclidean pictures as ﬁgures from projective geometry for a model of very large radius. That is, just as a small area of the earth appears ﬂat to people living there, the projective plane is also ‘locally Euclidean’. Although its local properties are the familiar Euclidean ones, there is a global property of the projective plane that is quite diﬀerent. The picture below shows

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Chapter Four. Determinants

a projective point. At that point is drawn an xy-axis. There is something interesting about the way this axis appears at the antipodal ends of the sphere. In the northern hemisphere, where the axis are drawn in black, a right hand put down with ﬁngers on the x-axis will have the thumb point along the y-axis. But the antipodal axis has just the opposite: a right hand placed with its ﬁngers on the x-axis will have the thumb point in the wrong way, instead, it is a left hand that works. Brieﬂy, the projective plane is not orientable: in this geometry, left and right handedness are not ﬁxed properties of ﬁgures.

The sequence of pictures below dramatizes this non-orientability. They sketch a trip around this space in the direction of the y part of the xy-axis. (Warning: the trip shown is not halfway around, it is a full circuit. True, if we made this into a movie then we could watch the northern hemisphere spots in the drawing above gradually rotate about halfway around the sphere to the last picture below. And we could watch the southern hemisphere spots in the picture above slide through the south pole and up through the equator to the last picture. But: the spots at either end of the dashed line are the same projective point. We don’t need to continue on much further; we are pretty much back to the projective point where we started by the last picture.)

=⇒

=⇒

At the end of the circuit, the x part of the xy-axes sticks out in the other direction. Thus, in the projective plane we cannot describe a ﬁgure as right- or left-handed (another way to make this point is that we cannot describe a spiral as clockwise or counterclockwise). This exhibition of the existence of a non-orientable space raises the question of whether our universe is orientable: is is possible for an astronaut to leave right-handed and return left-handed? An excellent nontechnical reference is [Gardner]. An classic science ﬁction story about orientation reversal is [Clarke]. So projective geometry is mathematically interesting, in addition to the natural way in which it arises in art. It is more than just a technical device to shorten some proofs. For an overview, see [Courant & Robbins]. The approach we’ve taken here, the analytic approach, leads to quick theorems and — most importantly for us — illustrates the power of linear algebra (see [Hanes], [Ryan], and [Eggar]). But another approach, the synthetic approach of deriving the

Topic: Projective Geometry

347

results from an axiom system, is both extraordinarily beautiful and is also the historical route of development. Two ﬁne sources for this approach are [Coxeter] or [Seidenberg]. An interesting and easy application is [Davies] Exercises
1 What is the equation of this point? 1 0 0 2 (a) Find the line incident on these points in the projective plane. 1 2 3 1 2 , 4 5 6 5 6

(b) Find the point incident on both of these projective lines. 3 , 4 3 Find the formula for the line incident on two projective points. Find the formula for the point incident on two projective lines. 4 Prove that the deﬁnition of incidence is independent of the choice of the representatives of p and L. That is, if p1 , p2 , p3 , and q1 , q2 , q3 are two triples of homogeneous coordinates for p, and L1 , L2 , L3 , and M1 , M2 , M3 are two triples of homogeneous coordinates for L, prove that p1 L1 + p2 L2 + p3 L3 = 0 if and only if q1 M1 + q2 M2 + q3 M3 = 0. 5 Give a drawing to show that central projection does not preserve circles, that a circle may project to an ellipse. Can a (non-circular) ellipse project to a circle? 6 Give the formula for the correspondence between the non-equatorial part of the antipodal modal of the projective plane, and the plane z = 1. 7 (Pappus’s Theorem) Assume that T0 , U0 , and V0 are collinear and that T1 , U1 , and V1 are collinear. Consider these three points: (i) the intersection V2 of the lines T0 U1 and T1 U0 , (ii) the intersection U2 of the lines T0 V1 and T1 V0 , and (iii) the intersection T2 of U0 V1 and U1 V0 . (a) Draw a (Euclidean) picture. (b) Apply the lemma used in Desargue’s Theorem to get simple homogeneous coordinate vectors for the T ’s and V0 . (c) Find the resulting homogeneous coordinate vectors for U ’s (these must each involve a parameter as, e.g., U0 could be anywhere on the T0 V0 line). (d) Find the resulting homogeneous coordinate vectors for V1 . (Hint: it involves two parameters.) (e) Find the resulting homogeneous coordinate vectors for V2 . (It also involves two parameters.) (f ) Show that the product of the three parameters is 1. (g) Verify that V2 is on the T2 U2 line..

Chapter Five

Similarity
While studying matrix equivalence, we have shown that for any homomorphism there are bases B and D such that the representation matrix has a block partialidentity form. Identity Zero RepB,D (h) = Zero Zero This representation describes the map as sending c1 β1 + · · · + cn βn to c1 δ1 + · · · + ck δk + 0 + · · · + 0, where n is the dimension of the domain and k is the dimension of the range. So, under this representation the action of the map is easy to understand because most of the matrix entries are zero. This chapter considers the special case where the domain and the codomain are equal, that is, where the homomorphism is a transformation. In this case we naturally ask to ﬁnd a single basis B so that RepB,B (t) is as simple as possible (we will take ‘simple’ to mean that it has many zeroes). A matrix having the above block partial-identity form is not always possible here. But we will develop a form that comes close, a representation that is nearly diagonal.

I

Complex Vector Spaces

This chapter requires that we factor polynomials. Of course, many polynomials do not factor over the real numbers; for instance, x2 + 1 does not factor into the product of two linear polynomials with real coeﬃcients. For that reason, we shall from now on take our scalars from the complex numbers. That is, we are shifting from studying vector spaces over the real numbers to vector spaces over the complex numbers — in this chapter vector and matrix entries are complex. Any real number is a complex number and a glance through this chapter shows that most of the examples use only real numbers. Nonetheless, the critical theorems require that the scalars be complex numbers, so the ﬁrst section below is a quick review of complex numbers. 349

350

Chapter Five. Similarity

In this book we are moving to the more general context of taking scalars to be complex only for the pragmatic reason that we must do so in order to develop the representation. We will not go into using other sets of scalars in more detail because it could distract from our goal. However, the idea of taking scalars from a structure other than the real numbers is an interesting one. Delightful presentations taking this approach are in [Halmos] and [Hoﬀman & Kunze].

I.1 Factoring and Complex Numbers; A Review
This subsection is a review only and we take the main results as known. For proofs, see [Birkhoﬀ & MacLane] or [Ebbinghaus]. Just as integers have a division operation — e.g., ‘4 goes 5 times into 21 with remainder 1’ — so do polynomials. 1.1 Theorem (Division Theorem for Polynomials) Let c(x) be a polynomial. If m(x) is a non-zero polynomial then there are quotient and remainder polynomials q(x) and r(x) such that c(x) = m(x) · q(x) + r(x) where the degree of r(x) is strictly less than the degree of m(x). In this book constant polynomials, including the zero polynomial, are said to have degree 0. (This is not the standard deﬁnition, but it is convienent here.) The point of the integer division statement ‘4 goes 5 times into 21 with remainder 1’ is that the remainder is less than 4 — while 4 goes 5 times, it does not go 6 times. In the same way, the point of the polynomial division statement is its ﬁnal clause. 1.2 Example If c(x) = 2x3 − 3x2 + 4x and m(x) = x2 + 1 then q(x) = 2x − 3 and r(x) = 2x + 3. Note that r(x) has a lower degree than m(x). 1.3 Corollary The remainder when c(x) is divided by x − λ is the constant polynomial r(x) = c(λ).
Proof. The remainder must be a constant polynomial because it is of degree less

than the divisor x − λ, To determine the constant, take m(x) from the theorem to be x − λ and substitute λ for x to get c(λ) = (λ − λ) · q(λ) + r(x). QED If a divisor m(x) goes into a dividend c(x) evenly, meaning that r(x) is the zero polynomial, then m(x) is a factor of c(x). Any root of the factor (any λ ∈ R such that m(λ) = 0) is a root of c(x) since c(λ) = m(λ) · q(λ) = 0. The prior corollary immediately yields the following converse. 1.4 Corollary If λ is a root of the polynomial c(x) then x − λ divides c(x) evenly, that is, x − λ is a factor of c(x).

Section I. Complex Vector Spaces

351

Finding the roots and factors of a high-degree polynomial can be hard. But for second-degree polynomials we have the quadratic formula: the roots of ax2 + bx + c are √ √ −b + b2 − 4ac −b − b2 − 4ac λ1 = λ2 = 2a 2a (if the discriminant b2 − 4ac is negative then the polynomial has no real number roots). A polynomial that cannot be factored into two lower-degree polynomials with real number coeﬃcients is irreducible over the reals. 1.5 Theorem Any constant or linear polynomial is irreducible over the reals. A quadratic polynomial is irreducible over the reals if and only if its discriminant is negative. No cubic or higher-degree polynomial is irreducible over the reals. 1.6 Corollary Any polynomial with real coeﬃcients can be factored into linear and irreducible quadratic polynomials. That factorization is unique; any two factorizations have the same powers of the same factors. Note the analogy with the prime factorization of integers. In both cases, the uniqueness clause is very useful. 1.7 Example Because of uniqueness we know, without multiplying them out, that (x + 3)2 (x2 + 1)3 does not equal (x + 3)4 (x2 + x + 1)2 . 1.8 Example By uniqueness, if c(x) = m(x) · q(x) then where c(x) = (x − 3)2 (x + 2)3 and m(x) = (x − 3)(x + 2)2 , we know that q(x) = (x − 3)(x + 2). While x2 + 1 has no real roots and so doesn’t factor over the real numbers, if we imagine a root — traditionally denoted i so that i2 + 1 = 0 — then x2 + 1 factors into a product of linears (x − i)(x + i). So we adjoin this root i to the reals and close the new system with respect to addition, multiplication, etc. (i.e., we also add 3 + i, and 2i, and 3 + 2i, etc., putting in all linear combinations of 1 and i). We then get a new structure, the complex numbers, denoted C. In C we can factor (obviously, at least some) quadratics that would be irreducible if we were to stick to the real numbers. Surprisingly, in C we can not only factor x2 + 1 and its close relatives, we can factor any quadratic. √ √ −b + b2 − 4ac −b − b2 − 4ac ax2 + bx + c = a · x − · x− 2a 2a 1.9 Example The second degree polynomial x2 +x+1 factors over the complex numbers into the product of two ﬁrst degree polynomials. √ √ √ √ −1 + −3 −1 − −3 1 3 1 3 x− x− = x − (− + i) x − (− − i) 2 2 2 2 2 2 1.10 Corollary (Fundamental Theorem of Algebra) Polynomials with complex coeﬃcients factor into linear polynomials with complex coeﬃcients. The factorization is unique.

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Chapter Five. Similarity

I.2 Complex Representations
Recall the deﬁnitions of the complex number addition (a + bi) + (c + di) = (a + c) + (b + d)i and multiplication. (a + bi)(c + di) = ac + adi + bci + bd(−1) = (ac − bd) + (ad + bc)i 2.1 Example For instance, (1 − 2i) + (5 + 4i) = 6 + 2i and (2 − 3i)(4 − 0.5i) = 6.5 − 13i. Handling scalar operations with those rules, all of the operations that we’ve covered for real vector spaces carry over unchanged. 2.2 Example Matrix multiplication is the same, although the scalar arithmetic involves more bookkeeping. 1 + 1i i = = 2 − 0i −2 + 3i 1 + 0i 3i 1 − 0i −i (1 + 1i) · (1 − 0i) + (2 − 0i) · (−i) (i) · (1 − 0i) + (−2 + 3i) · (−i)

(1 + 1i) · (1 + 0i) + (2 − 0i) · (3i) (i) · (1 + 0i) + (−2 + 3i) · (3i) 1 + 7i −9 − 5i 1 − 1i 3 + 3i

Everything else from prior chapters that we can, we shall also carry over unchanged. For instance, we shall call this     1 + 0i 0 + 0i 0 + 0i 0 + 0i      . ,..., .   .   .  . . 0 + 0i 1 + 0i the standard basis for Cn as a vector space over C and again denote it En .

Section II. Similarity

353

II

Similarity

II.1 Deﬁnition and Examples
ˆ We’ve deﬁned H and H to be matrix-equivalent if there are nonsingular maˆ trices P and Q such that H = P HQ. That deﬁnition is motivated by this diagram h Vw.r.t. B − − → Ww.r.t. D −− H     id id Vw.r.t.
ˆ B

− − → Ww.r.t. −−
ˆ H

h

ˆ D

ˆ showing that H and H both represent h but with respect to diﬀerent pairs of bases. We now specialize that setup to the case where the codomain equals the domain, and where the codomain’s basis equals the domain’s basis. Vw.r.t.   id Vw.r.t.
B

− − → Vw.r.t. −−   id − − → Vw.r.t. −−
t

t

B

D

D

To move from the lower left to the lower right we can either go straight over, or up, over, and then down. In matrix terms, RepD,D (t) = RepB,D (id) RepB,B (t) RepB,D (id)
−1

(recall that a representation of composition like this one reads right to left). 1.1 Deﬁnition The matrices T and S are similar if there is a nonsingular P such that T = P SP −1 . Since nonsingular matrices are square, the similar matrices T and S must be square and of the same size. 1.2 Example With these two, P = 2 1 1 1 S= 2 1 −3 −1

calculation gives that S is similar to this matrix. T = 0 1 −1 1

354

Chapter Five. Similarity

1.3 Example The only matrix similar to the zero matrix is itself: P ZP −1 = P Z = Z. The only matrix similar to the identity matrix is itself: P IP −1 = P P −1 = I. Since matrix similarity is a special case of matrix equivalence, if two matrices are similar then they are equivalent. What about the converse: must matrix equivalent square matrices be similar? The answer is no. The prior example shows that the similarity classes are diﬀerent from the matrix equivalence classes, because the matrix equivalence class of the identity consists of all nonsingular matrices of that size. Thus, for instance, these two are matrix equivalent but not similar. T = 1 0 0 1 S= 1 0 2 3

So some matrix equivalence classes split into two or more similarity classes — similarity gives a ﬁner partition than does equivalence. This picture shows some matrix equivalence classes subdivided into similarity classes.
A B

...

To understand the similarity relation we shall study the similarity classes. We approach this question in the same way that we’ve studied both the row equivalence and matrix equivalence relations, by ﬁnding a canonical form for representatives∗ of the similarity classes, called Jordan form. With this canonical form, we can decide if two matrices are similar by checking whether they reduce to the same representative. We’ve also seen with both row equivalence and matrix equivalence that a canonical form gives us insight into the ways in which members of the same class are alike (e.g., two identically-sized matrices are matrix equivalent if and only if they have the same rank). Exercises
1.4 For S= 1 −2 3 −6 T = 0 −11/2 0 −5 P = 4 −3 2 2

check that T = P SP −1 . 1.5 Example 1.3 shows that the only matrix similar to a zero matrix is itself and that the only matrix similar to the identity is itself. (a) Show that the 1×1 matrix (2), also, is similar only to itself. (b) Is a matrix of the form cI for some scalar c similar only to itself? (c) Is a diagonal matrix similar only to itself? 1.6 Show that these matrices are not similar. 1 0 4 1 0 1 1 1 3 0 1 1 2 1 7 3 1 2

Section II. Similarity

355

1.7 Consider the transformation t : P2 → P2 described by x2 → x + 1, x → x2 − 1, and 1 → 3. (a) Find T = RepB,B (t) where B = x2 , x, 1 . (b) Find S = RepD,D (t) where D = 1, 1 + x, 1 + x + x2 . (c) Find the matrix P such that T = P SP −1 . 1.8 Exhibit an nontrivial similarity relationship in this way: let t : C2 → C2 act by 1 2 → 3 0 −1 1 → −1 2

and pick two bases, and represent t with respect to then T = RepB,B (t) and S = RepD,D (t). Then compute the P and P −1 to change bases from B to D and back again. 1.9 Explain Example 1.3 in terms of maps. 1.10 Are there two matrices A and B that are similar while A2 and B 2 are not similar? [Halmos] 1.11 Prove that if two matrices are similar and one is invertible then so is the other. 1.12 Show that similarity is an equivalence relation. 1.13 Consider a matrix representing, with respect to some B, B, reﬂection across the x-axis in R2 . Consider also a matrix representing, with respect to some D, D, reﬂection across the y-axis. Must they be similar? 1.14 Prove that similarity preserves determinants and rank. Does the converse hold? 1.15 Is there a matrix equivalence class with only one matrix similarity class inside? One with inﬁnitely many similarity classes? 1.16 Can two diﬀerent diagonal matrices be in the same similarity class? 1.17 Prove that if two matrices are similar then their k-th powers are similar when k > 0. What if k ≤ 0? 1.18 Let p(x) be the polynomial cn xn + · · · + c1 x + c0 . Show that if T is similar to S then p(T ) = cn T n + · · · + c1 T + c0 I is similar to p(S) = cn S n + · · · + c1 S + c0 I. 1.19 List all of the matrix equivalence classes of 1×1 matrices. Also list the similarity classes, and describe which similarity classes are contained inside of each matrix equivalence class. 1.20 Does similarity preserve sums? 1.21 Show that if T − λI and N are similar matrices then T and N + λI are also similar.

II.2 Diagonalizability
The prior subsection deﬁnes the relation of similarity and shows that, although similar matrices are necessarily matrix equivalent, the converse does not hold. Some matrix-equivalence classes break into two or more similarity classes (the nonsingular n×n matrices, for instance). This means that the canonical form for matrix equivalence, a block partial-identity, cannot be used as a canonical form for matrix similarity because the partial-identities cannot be in more than

356

Chapter Five. Similarity

one similarity class, so there are similarity classes without one. This picture illustrates. As earlier in this book, class representatives are shown with stars.

...

We are developing a canonical form for representatives of the similarity classes. We naturally try to build on our previous work, meaning ﬁrst that the partial identity matrices should represent the similarity classes into which they fall, and beyond that, that the representatives should be as simple as possible. The simplest extension of the partial-identity form is a diagonal form. 2.1 Deﬁnition A transformation is diagonalizable if it has a diagonal representation with respect to the same basis for the codomain as for the domain. A diagonalizable matrix is one that is similar to a diagonal matrix: T is diagonalizable if there is a nonsingular P such that P T P −1 is diagonal. 2.2 Example The matrix 4 1 is diagonalizable. 2 0 0 3 = −1 1 2 −1 4 1 −2 1 −1 2 1 −1
−1

−2 1

2.3 Example Not every matrix is diagonalizable. The square of N= 0 1 0 0

is the zero matrix. Thus, for any map n that N represents (with respect to the same basis for the domain as for the codomain), the composition n ◦ n is the zero map. This implies that no such map n can be diagonally represented (with respect to any B, B) because no power of a nonzero diagonal matrix is zero. That is, there is no diagonal matrix in N ’s similarity class. That example shows that a diagonal form will not do for a canonical form — we cannot ﬁnd a diagonal matrix in each matrix similarity class. However, the canonical form that we are developing has the property that if a matrix can be diagonalized then the diagonal matrix is the canonical representative of the similarity class. The next result characterizes which maps can be diagonalized. 2.4 Corollary A transformation t is diagonalizable if and only if there is a basis B = β1 , . . . , βn and scalars λ1 , . . . , λn such that t(βi ) = λi βi for each i.

Section II. Similarity

357

Proof. This follows from the deﬁnition by considering a diagonal representation

matrix. . . .  RepB,B (t) = RepB (t(β1 ))  . . .    . . λ1 .  . RepB (t(βn )) =  . .  . 0 . . ..  0 .  .  . λn

···

.

This representation is equivalent to the existence of a basis satisfying the stated conditions simply by the deﬁnition of matrix representation. QED 2.5 Example To diagonalize T = 3 0 2 1

we take it as the representation of a transformation with respect to the standard basis T = RepE2 ,E2 (t) and we look for a basis B = β1 , β2 such that RepB,B (t) = λ1 0 0 λ2

that is, such that t(β1 ) = λ1 β1 and t(β2 ) = λ2 β2 . 3 0 2 β = λ1 · β 1 1 1 3 0 2 β = λ2 · β2 1 2

We are looking for scalars x such that this equation 3 0 2 1 b1 b2 =x· b1 b2

has solutions b1 and b2 , which are not both zero. Rewrite that as a linear system. (3 − x) · b1 + 2 · b2 = 0 (1 − x) · b2 = 0 (∗)

In the bottom equation the two numbers multiply to give zero only if at least one of them is zero so there are two possibilities, b2 = 0 and x = 1. In the b2 = 0 possibility, the ﬁrst equation gives that either b1 = 0 or x = 3. Since the case of both b1 = 0 and b2 = 0 is disallowed, we are left looking at the possibility of x = 3. With it, the ﬁrst equation in (∗) is 0 · b1 + 2 · b2 = 0 and so associated with 3 are vectors with a second component of zero and a ﬁrst component that is free. 3 2 b1 b =3· 1 0 1 0 0 That is, one solution to (∗) is λ1 = 3, and we have a ﬁrst basis vector. β1 = 1 0

358

Chapter Five. Similarity

In the x = 1 possibility, the ﬁrst equation in (∗) is 2 · b1 + 2 · b2 = 0, and so associated with 1 are vectors whose second component is the negative of their ﬁrst component. 3 2 b1 b1 =1· 0 1 −b1 −b1 Thus, another solution is λ2 = 1 and a second basis vector is this. β2 = 1 −1

To ﬁnish, drawing the similarity diagram R2 w.r.t.   id R2 w.r.t.
E2

− − → R2 −− w.r.t. T   id − − → R2 −− w.r.t.
D t

t

E2

B

B

and noting that the matrix RepB,E2 (id) is easy leads to this diagonalization. 3 0 0 1 = 1 0 1 −1
−1

3 0

2 1

1 0

1 −1

In the next subsection, we will expand on that example by considering more closely the property of Corollary 2.4. This includes seeing another way, the way that we will routinely use, to ﬁnd the λ’s. Exercises
2.6 Repeat Example 2.5 for the matrix from Example 2.2. 2.7 Diagonalize these upper triangular matrices. −2 1 5 4 (a) (b) 0 2 0 1 2.8 What form do the powers of a diagonal matrix have? 2.9 Give two same-sized diagonal matrices that are not similar. Must any two diﬀerent diagonal matrices come from diﬀerent similarity classes? 2.10 Give a nonsingular diagonal matrix. Can a diagonal matrix ever be singular? 2.11 Show that the inverse of a diagonal matrix is the diagonal of the the inverses, if no element on that diagonal is zero. What happens when a diagonal entry is zero? 2.12 The equation ending Example 2.5 1 1 3 2 1 1 3 0 = 0 −1 0 1 0 −1 0 1 is a bit jarring because for P we must take the ﬁrst matrix, which is shown as an inverse, and for P −1 we take the inverse of the ﬁrst matrix, so that the two −1 powers cancel and this matrix is shown without a superscript −1. (a) Check that this nicer-appearing equation holds. 3 0 0 1 = 1 0 1 −1 3 0 2 1 1 0 1 −1
−1 −1

Section II. Similarity

359

(b) Is the previous item a coincidence? Or can we always switch the P and the P −1 ? 2.13 Show that the P used to diagonalize in Example 2.5 is not unique. 2.14 Find a formula for the powers of this matrix Hint: see Exercise 8. −3 −4 1 2

2.15 Diagonalize these. 1 1 0 1 (a) (b) 0 0 1 0 2.16 We can ask how diagonalization interacts with the matrix operations. Assume that t, s : V → V are each diagonalizable. Is ct diagonalizable for all scalars c? What about t + s? t ◦ s? 2.17 Show that matrices of this form are not diagonalizable. 1 0 c 1 c=0

2.18 Show that each of these is diagonalizable. 1 2 x y (a) (b) x, y, z scalars 2 1 y z

II.3 Eigenvalues and Eigenvectors
In this subsection we will focus on the property of Corollary 2.4. 3.1 Deﬁnition A transformation t : V → V has a scalar eigenvalue λ if there is a nonzero eigenvector ζ ∈ V such that t(ζ) = λ · ζ. (“Eigen” is German for “characteristic of” or “peculiar to”; some authors call these characteristic values and vectors. No authors call them “peculiar”.) 3.2 Example The projection map     x x π y  −→ y  z 0

x, y, z ∈ C

has an eigenvalue of 1 associated with any eigenvector of the form   x y  0 where x and y are non-0 scalars. On the other hand, 2 is not an eigenvalue of π since no non-0 vector is doubled. That example shows why the ‘non-0’ appears in the deﬁnition. Disallowing 0 as an eigenvector eliminates trivial eigenvalues.

360

Chapter Five. Similarity

3.3 Example The only transformation on the trivial space {0 } is 0 → 0. This map has no eigenvalues because there are no non-0 vectors v mapped to a scalar multiple λ · v of themselves. 3.4 Example Consider the homomorphism t : P1 → P1 given by c0 + c1 x → (c0 + c1 ) + (c0 + c1 )x. The range of t is one-dimensional. Thus an application of t to a vector in the range will simply rescale that vector: c + cx → (2c) + (2c)x. That is, t has an eigenvalue of 2 associated with eigenvectors of the form c + cx where c = 0. This map also has an eigenvalue of 0 associated with eigenvectors of the form c − cx where c = 0. 3.5 Deﬁnition A square matrix T has a scalar eigenvalue λ associated with the non-0 eigenvector ζ if T ζ = λ · ζ. 3.6 Remark Although this extension from maps to matrices is obvious, there is a point that must be made. Eigenvalues of a map are also the eigenvalues of matrices representing that map, and so similar matrices have the same eigenvalues. But the eigenvectors are diﬀerent — similar matrices need not have the same eigenvectors. For instance, consider again the transformation t : P1 → P1 given by c0 + c1 x → (c0 +c1 )+(c0 +c1 )x. It has an eigenvalue of 2 associated with eigenvectors of the form c + cx where c = 0. If we represent t with respect to B = 1 + 1x, 1 − 1x 2 0 T = RepB,B (t) = 0 0 then 2 is an eigenvalue of T , associated with these eigenvectors. { c0 c1 2 0 0 0 c0 c1 = 2c0 c }={ 0 2c1 0 c0 ∈ C, c0 = 0}

On the other hand, representing t with respect to D = 2 + 1x, 1 + 0x gives S = RepD,D (t) = 3 −3 1 −1

and the eigenvectors of S associated with the eigenvalue 2 are these. { c0 c1 3 −3 1 −1 c0 c1 = 2c0 0 }={ 2c1 c1 c1 ∈ C, c1 = 0}

Thus similar matrices can have diﬀerent eigenvectors. Here is an informal description of what’s happening. The underlying transformation doubles the eigenvectors v → 2 · v. But when the matrix representing the transformation is T = RepB,B (t) then it “assumes” that column vectors are representations with respect to B. In contrast, S = RepD,D (t) “assumes” that column vectors are representations with respect to D. So the vectors that get doubled by each matrix look diﬀerent.

Section II. Similarity

361

The next example illustrates the basic tool for ﬁnding eigenvectors and eigenvalues. 3.7 Example What are the eigenvalues and eigenvectors of this matrix?   1 2 1 T =  2 0 −2 −1 2 3 To ﬁnd the scalars x such that T ζ = xζ for non-0 eigenvectors ζ, bring everything to the left-hand side      1 2 1 z1 z1  2 0 −2 z2  − x z2  = 0 −1 2 3 z3 z3 and factor (T −xI)ζ = 0. (Note that it says T −xI; the expression T −x doesn’t make sense because T is a matrix while x is a scalar.) This homogeneous linear system      z1 1−x 2 1 0  2 0−x −2  z2  = 0 −1 2 3−x z3 0 has a non-0 solution if and only if the matrix is singular. We can determine when that happens. 0 = |T − xI| = 1−x 2 −1 2 0−x 2 1 −2 3−x

= x3 − 4x2 + 4x = x(x − 2)2 The eigenvalues are λ1 = 0 and λ2 = 2. To ﬁnd the associated eigenvectors, plug in each eigenvalue. Plugging in λ1 = 0 gives          1−0 2 1 z1 0 z1 a  2 z2  = −a 0 − 0 −2  z2  = 0 =⇒ −1 2 3−0 z3 0 z3 a for a scalar parameter a = 0 (a is non-0 because eigenvectors must be non-0). In the same way, plugging in λ2 = 2 gives          1−2 2 1 z1 0 z1 b  2 z2  = 0 0 − 2 −2  z2  = 0 =⇒ −1 2 3−2 z3 0 z3 b with b = 0.

362 3.8 Example If S= π 0 1 3

Chapter Five. Similarity

(here π is not a projection map, it is the number 3.14 . . .) then π−x 0 1 3−x = (x − π)(x − 3)

so S has eigenvalues of λ1 = π and λ2 = 3. To ﬁnd associated eigenvectors, ﬁrst plug in λ1 for x: π−π 0 1 3−π z1 z2 = 0 0 =⇒ z1 z2 z1 z2 = a 0

for a scalar a = 0, and then plug in λ2 : π−3 1 0 3−3 where b = 0. 3.9 Deﬁnition The characteristic polynomial of a square matrix T is the determinant of the matrix T − xI, where x is a variable. The characteristic equation is |T − xI| = 0. The characteristic polynomial of a transformation t is the polynomial of any RepB,B (t). Exercise 30 checks that the characteristic polynomial of a transformation is well-deﬁned, that is, any choice of basis yields the same polynomial. 3.10 Lemma A linear transformation on a nontrivial vector space has at least one eigenvalue.
Proof. Any root of the characteristic polynomial is an eigenvalue. Over the

z1 z2

=

0 0

=⇒

=

−b/(π − 3) b

complex numbers, any polynomial of degree one or greater has a root. (This is the reason that in this chapter we’ve gone to scalars that are complex.) QED Notice the familiar form of the sets of eigenvectors in the above examples. 3.11 Deﬁnition The eigenspace of a transformation t associated with the eigenvalue λ is Vλ = {ζ t(ζ ) = λζ } ∪ {0 }. The eigenspace of a matrix is deﬁned analogously. 3.12 Lemma An eigenspace is a subspace.
Proof. An eigenspace must be nonempty — for one thing it contains the zero

vector — and so we need only check closure. Take vectors ζ1 , . . . , ζn from Vλ , to show that any linear combination is in Vλ t(c1 ζ1 + c2 ζ2 + · · · + cn ζn ) = c1 t(ζ1 ) + · · · + cn t(ζn ) = c1 λζ1 + · · · + cn λζn = λ(c1 ζ1 + · · · + cn ζn ) (the second equality holds even if any ζi is 0 since t(0) = λ · 0 = 0).
QED

Section II. Similarity

363

3.13 Example In Example 3.8 the eigenspace associated with the eigenvalue π and the eigenspace associated with the eigenvalue 3 are these. Vπ = { a 0 a ∈ R} V3 = { −b/π − 3 b b ∈ R}

3.14 Example In Example 3.7, these are the eigenspaces associated with the eigenvalues 0 and 2.     a b V0 = {−a a ∈ R}, V2 = {0 b ∈ R}. a b 3.15 Remark The characteristic equation is 0 = x(x−2)2 so in some sense 2 is an eigenvalue “twice”. However there are not “twice” as many eigenvectors, in that the dimension of the eigenspace is one, not two. The next example shows a case where a number, 1, is a double root of the characteristic equation and the dimension of the associated eigenspace is two. 3.16 Example With respect to the  1 0 0 represents projection.     x x π y  −→ y  0 z x, y, z ∈ C standard bases, this matrix  0 0 1 0 0 0

Its eigenspace associated with the eigenvalue 0 and its eigenspace associated with the eigenvalue 1 are easy to ﬁnd.     0 c1 V0 = { 0  c3 ∈ C} V1 = {c2  c1 , c2 ∈ C} c3 0 By the lemma, if two eigenvectors v1 and v2 are associated with the same eigenvalue then any linear combination of those two is also an eigenvector associated with that same eigenvalue. But, if two eigenvectors v1 and v2 are associated with diﬀerent eigenvalues then the sum v1 + v2 need not be related to the eigenvalue of either one. In fact, just the opposite. If the eigenvalues are diﬀerent then the eigenvectors are not linearly related. 3.17 Theorem For any set of distinct eigenvalues of a map or matrix, a set of associated eigenvectors, one per eigenvalue, is linearly independent.

364

Chapter Five. Similarity

Proof. We will use induction on the number of eigenvalues. If there is no eigen-

value or only one eigenvalue then the set of associated eigenvectors is empty or is a singleton set with a non-0 member, and in either case is linearly independent. For induction, assume that the theorem is true for any set of k distinct eigenvalues, suppose that λ1 , . . . , λk+1 are distinct eigenvalues, and let v1 , . . . , vk+1 be associated eigenvectors. If c1 v1 + · · · + ck vk + ck+1 vk+1 = 0 then after multiplying both sides of the displayed equation by λk+1 , applying the map or matrix to both sides of the displayed equation, and subtracting the ﬁrst result from the second, we have this. c1 (λk+1 − λ1 )v1 + · · · + ck (λk+1 − λk )vk + ck+1 (λk+1 − λk+1 )vk+1 = 0 The induction hypothesis now applies: c1 (λk+1 −λ1 ) = 0, . . . , ck (λk+1 −λk ) = 0. Thus, as all the eigenvalues are distinct, c1 , . . . , ck are all 0. Finally, now ck+1 must be 0 because we are left with the equation vk+1 = 0. QED 3.18 Example The eigenvalues of  2 0 −4

−2 1 8

 2 1 3

are distinct: λ1 = 1, λ2 = 2, and λ3 = 3. A set of associated eigenvectors like       2 9 2 {1 , 4 , 1} 0 4 2 is linearly independent. 3.19 Corollary An n×n matrix with n distinct eigenvalues is diagonalizable.
Proof. Form a basis of eigenvectors. Apply Corollary 2.4. QED

Exercises
3.20 For each, ﬁnd the characteristic polynomial and the eigenvalues. 10 −9 1 2 0 3 0 0 (a) (b) (c) (d) 4 −2 4 3 7 0 0 0 1 0 0 1 3.21 For each matrix, ﬁnd the characteristic equation, and the eigenvalues and associated eigenvectors. 3 0 3 2 (a) (b) 8 −1 −1 0 3.22 Find the characteristic equation, and the eigenvalues and associated eigenvectors for this matrix. Hint. The eigenvalues are complex. (e) −2 5 −1 2

Section II. Similarity

365

3.23 Find the characteristic polynomial, the eigenvalues, and the associated eigenvectors of this matrix. 1 1 1 0 0 1 0 0 1 3.24 For each matrix, ﬁnd the characteristic equation, and the eigenvalues and associated eigenvectors. 0 1 0 3 −2 0 0 1 3 0 (b) 0 (a) −2 4 −17 8 0 0 5 3.25 Let t : P2 → P2 be a0 + a1 x + a2 x2 → (5a0 + 6a1 + 2a2 ) − (a1 + 8a2 )x + (a0 − 2a2 )x2 . Find its eigenvalues and the associated eigenvectors. 3.26 Find the eigenvalues and eigenvectors of this map t : M2 → M2 . a c b d → 2c b − 2c a+c d

3.27 Find the eigenvalues and associated eigenvectors of the diﬀerentiation operator d/dx : P3 → P3 . 3.28 Prove that the eigenvalues of a triangular matrix (upper or lower triangular) are the entries on the diagonal. 3.29 Find the formula for the characteristic polynomial of a 2×2 matrix. 3.30 Prove that the characteristic polynomial of a transformation is well-deﬁned. 3.31 (a) Can any non-0 vector in any nontrivial vector space be a eigenvector? That is, given a v = 0 from a nontrivial V , is there a transformation t : V → V and a scalar λ ∈ R such that t(v) = λv? (b) Given a scalar λ, can any non-0 vector in any nontrivial vector space be an eigenvector associated with the eigenvalue λ? 3.32 Suppose that t : V → V and T = RepB,B (t). Prove that the eigenvectors of T associated with λ are the non-0 vectors in the kernel of the map represented (with respect to the same bases) by T − λI. 3.33 Prove that if a, . . . , d are all integers and a + b = c + d then a c b d

has integral eigenvalues, namely a + b and a − c. 3.34 Prove that if T is nonsingular and has eigenvalues λ1 , . . . , λn then T −1 has eigenvalues 1/λ1 , . . . , 1/λn . Is the converse true? 3.35 Suppose that T is n×n and c, d are scalars. (a) Prove that if T has the eigenvalue λ with an associated eigenvector v then v is an eigenvector of cT + dI associated with eigenvalue cλ + d. (b) Prove that if T is diagonalizable then so is cT + dI. 3.36 Show that λ is an eigenvalue of T if and only if the map represented by T − λI is not an isomorphism. 3.37 [Strang 80] (a) Show that if λ is an eigenvalue of A then λk is an eigenvalue of Ak . (b) What is wrong with this proof generalizing that? “If λ is an eigenvalue of A and µ is an eigenvalue for B, then λµ is an eigenvalue for AB, for, if Ax = λx and Bx = µx then ABx = Aµx = µAxµλx”?

366

Chapter Five. Similarity

3.38 Do matrix-equivalent matrices have the same eigenvalues? 3.39 Show that a square matrix with real entries and an odd number of rows has at least one real eigenvalue. 3.40 Diagonalize. −1 2 2 2 2 2 −3 −6 −6 3.41 Suppose that P is a nonsingular n×n matrix. Show that the similarity transformation map tP : Mn×n → Mn×n sending T → P T P −1 is an isomorphism. ? 3.42 Show that if A is an n square matrix and each row (column) sums to c then c is a characteristic root of A. [Math. Mag., Nov. 1967]

Section III. Nilpotence

367

III

Nilpotence

The goal of this chapter is to show that every square matrix is similar to one that is a sum of two kinds of simple matrices. The prior section focused on the ﬁrst kind, diagonal matrices. We now consider the other kind.

III.1 Self-Composition
This subsection is optional, although it is necessary for later material in this section and in the next one. A linear transformations t : V → V , because it has the same domain and codomain, can be iterated.∗ That is, compositions of t with itself such as t2 = t◦t and t3 = t ◦ t ◦ t are deﬁned.

v t(v ) t2 (v )

Note that this power notation for the linear transformation functions dovetails with the notation that we’ve used earlier for their square matrix representations because if RepB,B (t) = T then RepB,B (tj ) = T j . 1.1 Example For the derivative map d/dx : P3 → P3 given by a + bx + cx2 + dx3 −→ b + 2cx + 3dx2 the second power is the second derivative a + bx + cx2 + dx3 −→ 2c + 6dx the third power is the third derivative a + bx + cx2 + dx3 −→ 6d and any higher power is the zero map. 1.2 Example This transformation of the space of 2×2 matrices a c

d/dx

d2 /dx2

d3 /dx3

b d

−→

t

b d

a 0

368 has this second power a c and this third power. a c b d −→
t3

Chapter Five. Similarity

b d

−→

t2

a b 0 0 b a 0 0

After that, t4 = t2 and t5 = t3 , etc. These examples suggest that on iteration more and more zeros appear until there is a settling down. The next result makes this precise. 1.3 Lemma For any transformation t : V → V , the rangespaces of the powers form a descending chain V ⊇ R(t) ⊇ R(t2 ) ⊇ · · · and the nullspaces form an ascending chain. {0 } ⊆ N (t) ⊆ N (t2 ) ⊆ · · · Further, there is a k such that for powers less than k the subsets are proper (if j < k then R(tj ) ⊃ R(tj+1 ) and N (tj ) ⊂ N (tj+1 )), while for powers greater than k the sets are equal (if j ≥ k then R(tj ) = R(tj+1 ) and N (tj ) = N (tj+1 )).
Proof. We will do the rangespace half and leave the rest for Exercise 13. Recall,

however, that for any map the dimension of its rangespace plus the dimension of its nullspace equals the dimension of its domain. So if the rangespaces shrink then the nullspaces must grow. That the rangespaces form chains is clear because if w ∈ R(tj+1 ), so that w = tj+1 (v), then w = tj ( t(v) ) and so w ∈ R(tj ). To verify the “further” property, ﬁrst observe that if any pair of rangespaces in the chain are equal R(tk ) = R(tk+1 ) then all subsequent ones are also equal R(tk+1 ) = R(tk+2 ), etc. This is because if t : R(tk+1 ) → R(tk+2 ) is the same map, with the same domain, as t : R(tk ) → R(tk+1 ) and it therefore has the same range: R(tk+1 ) = R(tk+2 ) (and induction shows that it holds for all higher powers). So if the chain of rangespaces ever stops being strictly decreasing then it is stable from that point onward. But the chain must stop decreasing. Each rangespace is a subspace of the one before it. For it to be a proper subspace it must be of strictly lower dimension (see Exercise 11). These spaces are ﬁnite-dimensional and so the chain can fall for only ﬁnitely-many steps, that is, the power k is at most the dimension of V. QED 1.4 Example The derivative map a + bx + cx2 + dx3 −→ b + 2cx + 3dx2 of Example 1.1 has this chain of rangespaces P3 ⊃ P2 ⊃ P1 ⊃ P0 ⊃ {0 } = {0 } = · · ·
d/dx

Section III. Nilpotence and this chain of nullspaces. {0 } ⊂ P0 ⊂ P1 ⊂ P2 ⊂ P3 = P3 = · · ·

369

1.5 Example The transformation π : C3 → C3 projecting onto the ﬁrst two coordinates     c1 c1 π c2  −→ c2  c3 0 has C3 ⊃ R(π) = R(π 2 ) = · · · and {0 } ⊂ N (π) = N (π 2 ) = · · · . 1.6 Example Let t : P2 → P2 be the map c0 + c1 x + c2 x2 → 2c0 + c2 x. As the lemma describes, on iteration the rangespace shrinks R(t0 ) = P2 R(t) = {a + bx a, b ∈ C} R(t2 ) = {a a ∈ C}

and then stabilizes R(t2 ) = R(t3 ) = · · · , while the nullspace grows N (t0 ) = {0} N (t) = {cx c ∈ C} N (t2 ) = {cx + d c, d ∈ C}

and then stabilizes N (t2 ) = N (t3 ) = · · · . This graph illustrates Lemma 1.3. The horizontal axis gives the power j of a transformation. The vertical axis gives the dimension of the rangespace of tj as the distance above zero — and thus also shows the dimension of the nullspace as the distance below the gray horizontal line, because the two add to the dimension n of the domain. n

rank(tj ) n

0 1 2 j Power j of the transformation

As sketched, on iteration the rank falls and with it the nullity grows until the two reach a steady state. This state must be reached by the n-th iterate. The steady state’s distance above zero is the dimension of the generalized rangespace and its distance below n is the dimension of the generalized nullspace. 1.7 Deﬁnition Let t be a transformation on an n-dimensional space. The generalized rangespace (or the closure of the rangespace) is R∞ (t) = R(tn ) The generalized nullspace (or the closure of the nullspace) is N∞ (t) = N (tn ).

370 Exercises

Chapter Five. Similarity

1.8 Give the chains of rangespaces and nullspaces for the zero and identity transformations. 1.9 For each map, give the chain of rangespaces and the chain of nullspaces, and the generalized rangespace and the generalized nullspace. (a) t0 : P2 → P2 , a + bx + cx2 → b + cx2 (b) t1 : R2 → R2 , a 0 → b a (c) t2 : P2 → P2 , a + bx + cx2 → b + cx + ax2 (d) t3 : R3 → R3 , a a b → a c b 1.10 Prove that function composition is associative (t ◦ t) ◦ t = t ◦ (t ◦ t) and so we can write t3 without specifying a grouping. 1.11 Check that a subspace must be of dimension less than or equal to the dimension of its superspace. Check that if the subspace is proper (the subspace does not equal the superspace) then the dimension is strictly less. (This is used in the proof of Lemma 1.3.) 1.12 Prove that the generalized rangespace R∞ (t) is the entire space, and the generalized nullspace N∞ (t) is trivial, if the transformation t is nonsingular. Is this ‘only if’ also? 1.13 Verify the nullspace half of Lemma 1.3. 1.14 Give an example of a transformation on a three dimensional space whose range has dimension two. What is its nullspace? Iterate your example until the rangespace and nullspace stabilize. 1.15 Show that the rangespace and nullspace of a linear transformation need not be disjoint. Are they ever disjoint?

III.2 Strings
This subsection is optional, and requires material from the optional Direct Sum subsection. The prior subsection shows that as j increases, the dimensions of the R(tj )’s fall while the dimensions of the N (tj )’s rise, in such a way that this rank and nullity split the dimension of V . Can we say more; do the two split a basis — is V = R(tj ) ⊕ N (tj )? The answer is yes for the smallest power j = 0 since V = R(t0 ) ⊕ N (t0 ) = V ⊕ {0}. The answer is also yes at the other extreme. 2.1 Lemma Where t : V → V is a linear transformation, the space is the direct sum V = R∞ (t) ⊕ N∞ (t). That is, both dim(V ) = dim(R∞ (t)) + dim(N∞ (t)) and R∞ (t) ∩ N∞ (t) = {0 }.

Section III. Nilpotence

371

Proof. We will verify the second sentence, which is equivalent to the ﬁrst. The

ﬁrst clause, that the dimension n of the domain of tn equals the rank of tn plus the nullity of tn , holds for any transformation and so we need only verify the second clause. Assume that v ∈ R∞ (t) ∩ N∞ (t) = R(tn ) ∩ N (tn ), to prove that v is 0. Because v is in the nullspace, tn (v) = 0. On the other hand, because R(tn ) = R(tn+1 ), the map t : R∞ (t) → R∞ (t) is a dimension-preserving homomorphism and therefore is one-to-one. A composition of one-to-one maps is one-to-one, and so tn : R∞ (t) → R∞ (t) is one-to-one. But now — because only 0 is sent by a one-to-one linear map to 0 — the fact that tn (v) = 0 implies that v = 0. QED 2.2 Note Technically we should distinguish the map t : V → V from the map t : R∞ (t) → R∞ (t) because the domains or codomains might diﬀer. The second one is said to be the restriction ∗ of t to R(tk ). We shall use later a point from that proof about the restriction map, namely that it is nonsingular. In contrast to the j = 0 and j = n cases, for intermediate powers the space V might not be the direct sum of R(tj ) and N (tj ). The next example shows that the two can have a nontrivial intersection. 2.3 Example Consider the transformation of C2 deﬁned by this action on the elements of the standard basis. 1 0 The vector e2 = 0 1 −→
n

0 1

0 1

−→

n

0 0

N = RepE2 ,E2 (n) =

0 1

0 0

is in both the rangespace and nullspace. Another way to depict this map’s action is with a string. e1 → e2 → 0 2.4 Example A map n : C4 → C4 whose action on E4 is given by the string ˆ e1 → e2 → e3 → e4 → 0 has R(ˆ ) ∩ N (ˆ ) equal to the span [{e4 }], has R(ˆ 2 ) ∩ N (ˆ 2 ) = [{e3 , e4 }], and n n n n has R(ˆ 3 ) ∩ N (ˆ 3 ) = [{e4 }]. The matrix representation is all zeros except for n n some subdiagonal ones.   0 0 0 0 1 0 0 0 ˆ  N = RepE4 ,E4 (ˆ ) =  n 0 1 0 0 0 0 1 0

372

Chapter Five. Similarity

2.5 Example Transformations can act via more than one string. A transformation t acting on a basis B = β1 , . . . , β5 by β1 → β2 → β3 → 0 β4 → β5 → 0 is represented by a matrix that is all zeros except for blocks of subdiagonal ones   0 0 0 0 0 1 0 0 0 0   RepB,B (t) = 0 1 0 0 0   0 0 0 0 0 0 0 0 1 0 (the lines just visually organize the blocks). In those three examples all vectors are eventually transformed to zero. 2.6 Deﬁnition A nilpotent transformation is one with a power that is the zero map. A nilpotent matrix is one with a power that is the zero matrix. In either case, the least such power is the index of nilpotency. 2.7 Example In Example 2.3 the index of nilpotency is two. In Example 2.4 it is four. In Example 2.5 it is three. 2.8 Example The diﬀerentiation map d/dx : P2 → P2 is nilpotent of index three since the third derivative of any quadratic polynomial is zero. This map’s action is described by the string x2 → 2x → 2 → 0 and taking the basis B = x2 , 2x, 2 gives this representation.   0 0 0 RepB,B (d/dx) = 1 0 0 0 1 0 Not all nilpotent matrices are all zeros except for blocks of subdiagonal ones. ˆ 2.9 Example With the matrix N from Example 2.4, and this four-vector basis         1 0 1 0 0 2 1 0 D =  , , ,  1 1 1 0 0 0 0 1 a change of basis operation produces this representation with respect to D, D.  1 0  1 0 0 2 1 0 1 1 1 0  0 0 0 1  0 0 1 0 0 0 1 0 0 0 0 1  0 1 0 0  0 1 0 0 0 2 1 0 1 1 1 0 −1  0 −1 −3 0  = −2 0 1 2 0 −2 −1 1 1 5 3 −2  0 0  0 0

Section III. Nilpotence The new matrix is nilpotent; it’s fourth power is the zero matrix since ˆ ˆ ˆ ˆ ˆ ˆ (P N P −1 )4 = P N P −1 · P N P −1 · P N P −1 · P N P −1 = P N 4 P −1 ˆ and N 4 is the zero matrix.

373

The goal of this subsection is Theorem 2.13, which shows that the prior example is prototypical in that every nilpotent matrix is similar to one that is all zeros except for blocks of subdiagonal ones. 2.10 Deﬁnition Let t be a nilpotent transformation on V . A t-string generated by v ∈ V is a sequence v, t(v), . . . , tk−1 (v) . This sequence has length k. A t-string basis is a basis that is a concatenation of t-strings. 2.11 Example In Example 2.5, the t-strings β1 , β2 , β3 and β4 , β5 , of length three and two, can be concatenated to make a basis for the domain of t. 2.12 Lemma If a space has a t-string basis then the longest string in it has length equal to the index of nilpotency of t.
Proof. Suppose not. Those strings cannot be longer; if the index is k then tk sends any vector — including those starting the string — to 0. So suppose instead that there is a transformation t of index k on some space, such that the space has a t-string basis where all of the strings are shorter than length k. Because t has index k, there is a vector v such that tk−1 (v) = 0. Represent v as a linear combination of basis elements and apply tk−1 . We are supposing that tk−1 sends each basis element to 0 but that it does not send v to 0. That is impossible. QED

We shall show that every nilpotent map has an associated string basis. Then our goal theorem, that every nilpotent matrix is similar to one that is all zeros except for blocks of subdiagonal ones, is immediate, as in Example 2.5. Looking for a counterexample, a nilpotent map without an associated string basis that is disjoint, will suggest the idea for the proof. Consider the map t : C5 → C5 with this action.   e1 0 0 0 0 0 → 0 0 0 0 0 e3 → 0   RepE5 ,E5 (t) = 1 1 0 0 0 e2 →   0 0 0 0 0 e4 → e5 → 0 0 0 0 1 0 Even after ommitting the zero vector, these three strings aren’t disjoint, but that doesn’t end hope of ﬁnding a t-string basis. It only means that E5 will not do for the string basis. To ﬁnd a basis that will do, we ﬁrst ﬁnd the number and lengths of its strings. Since t’s index of nilpotency is two, Lemma 2.12 says that at least one

374

Chapter Five. Similarity

string in the basis has length two. Thus the map must act on a string basis in one of these two ways. β1 → β2 → 0 β3 → β4 → 0 β5 → 0 β1 β3 β4 β5 → → → → β2 → 0 0 0 0

Now, the key point. A transformation with the left-hand action has a nullspace of dimension three since that’s how many basis vectors are sent to zero. A transformation with the right-hand action has a nullspace of dimension four. Using the matrix representation above, calculation of t’s nullspace  x −x   N (t) = { z  x, z, r ∈ C}    0  r  shows that it is three-dimensional, meaning that we want the left-hand action. To produce a string basis, ﬁrst pick β2 and β4 from R(t) ∩ N (t)   0 0   β2 = 1   0 0   0 0   β4 = 0   0 1

(other choices are possible, just be sure that {β2 , β4 } is linearly independent). For β5 pick a vector from N (t) that is not in the span of {β2 , β4 }.  1 −1   β5 =  0    0 0 Finally, take β1 and β3 such that t(β1 ) = β2 and t(β3 ) = β4 .   0 1   β1 = 0   0 0   0 0   β3 = 0   1 0 

Section III. Nilpotence Now, with respect to B = β1 , . . . , β5 , the matrix of t is as desired.   0 0 0 0 0 1 0 0 0 0   RepB,B (t) = 0 0 0 0 0   0 0 1 0 0 0 0 0 0 0

375

2.13 Theorem Any nilpotent transformation t is associated with a t-string basis. While the basis is not unique, the number and the length of the strings is determined by t. This illustrates the proof. Basis vectors are categorized into kind 1, kind 2, and kind 3. They are also shown as squares or circles, according to whether they are in the nullspace or not. k k k 3 → 1 → ··· ··· → 1 → 1 → 0 k k k 3 → 1 → ··· ··· → 1 → 1 → 0 . . . k→ 1 → · · · → 1 → 1 → 0 k k 3 2 → 0 . . . 2 → 0
Proof. Fix a vector space V ; we will argue by induction on the index of nilpo-

tency of t : V → V . If that index is 1 then t is the zero map and any basis is a string basis β1 → 0, . . . , βn → 0. For the inductive step, assume that the theorem holds for any transformation with an index of nilpotency between 1 and k − 1 and consider the index k case. First observe that the restriction to the rangespace t : R(t) → R(t) is also nilpotent, of index k − 1. Apply the inductive hypothesis to get a string basis for R(t), where the number and length of the strings is determined by t. B = β1 , t(β1 ), . . . , th1 (β1 ) β2 , . . . , th2 (β2 ) ··· βi , . . . , thi (βi )

(In the illustration these are the basis vectors of kind 1, so there are i strings shown with this kind of basis vector.) Second, note that taking the ﬁnal nonzero vector in each string gives a basis C = th1 (β1 ), . . . , thi (βi ) for R(t) ∩ N (t). (These are illustrated with 1’s in squares.) For, a member of R(t) is mapped to zero if and only if it is a linear combination of those basis vectors that are mapped to zero. Extend C to a basis for all of N (t). ˆ C = C ξ1 , . . . , ξp ˆ (The ξ’s are the vectors of kind 2 so that C is the set of squares.) While many choices are possible for the ξ’s, their number p is determined by the map t as it is the dimension of N (t) minus the dimension of R(t) ∩ N (t).

376

Chapter Five. Similarity

ˆ Finally, B C is a basis for R(t)+N (t) because any sum of something in the rangespace with something in the nullspace can be represented using elements ˆ of B for the rangespace part and elements of C for the part from the nullspace. Note that dim R(t) + N (t) = dim(R(t)) + dim(N (t)) − dim(R(t) ∩ N (t)) = rank(t) + nullity(t) − i = dim(V ) − i ˆ and so B C can be extended to a basis for all of V by the addition of i more vectors. Speciﬁcally, remember that each of β1 , . . . , βi is in R(t), and extend ˆ B C with vectors v1 , . . . , vi such that t(v1 ) = β1 , . . . , t(vi ) = βi . (In the illustration, these are the 3’s.) The check that linear independence is preserved by this extension is Exercise 29. QED 2.14 Corollary Every nilpotent matrix is similar to a matrix that is all zeros except for blocks of subdiagonal ones. That is, every nilpotent map is represented with respect to some basis by such a matrix. This form is unique in the sense that if a nilpotent matrix is similar to two such matrices then those two simply have their blocks ordered diﬀerently. Thus this is a canonical form for the similarity classes of nilpotent matrices provided that we order the blocks, say, from longest to shortest. 2.15 Example The matrix M= 1 1 −1 −1

has an index of nilpotency of two, as this calculation shows. p 1 2 Mp 1 −1 M= 1 −1 0 0 M2 = 0 0 N (M p ) x x ∈ C} { x C2

The calculation also describes how a map m represented by M must act on any string basis. With one map application the nullspace has dimension one and so one vector of the basis is sent to zero. On a second application, the nullspace has dimension two and so the other basis vector is sent to zero. Thus, the action of the map is β1 → β2 → 0 and the canonical form of the matrix is this. 0 1 0 0

We can exhibit such a m-string basis and the change of basis matrices witnessing the matrix similarity. For the basis, take M to represent m with respect

Section III. Nilpotence

377

to the standard bases, pick a β2 ∈ N (m) and also pick a β1 so that m(β1 ) = β2 . β2 = 1 1 β1 = 1 0

(If we take M to be a representative with respect to some nonstandard bases then this picking step is just more messy.) Recall the similarity diagram.
2 C2 −− w.r.t. E2 − − → Cw.r.t. E2 M     id P id P 2 Cw.r.t. B m

− − → C2 −− w.r.t.

m

B

The canonical form equals RepB,B (m) = P M P −1 , where P −1 = RepB,E2 (id) = 1 0 1 1 P = (P −1 )−1 = 1 0 −1 1

and the veriﬁcation of the matrix calculation is routine. 1 0 −1 1 1 1 −1 −1 1 0 1 1 = 0 1 0 0

2.16 Example The matrix 

0 1  −1  0 1

0 0 0 0 1 1 1 0 0 −1

0 0 −1 0 1

 0 0  1  0 −1

is nilpotent. These calculations show the nullspaces growing. p Np  0 0 0 0 0 1 0 0 0 0   −1 1 1 −1 1    0 1 0 0 0 1 0 −1 1  −1  0 0 0 0 0  0 0 0 0 0    1 0 0 0 0    1 0 0 0 0 0 0 0 0 0 –zero matrix–  1 0  0    {u − v  u, v ∈ C}    u  v   0 y    { z  y, z, u, v ∈ C}   u v C5  N (N p ) 

2

3

That table shows that any string basis must satisfy: the nullspace after one map application has dimension two so two basis vectors are sent directly to zero,

378

Chapter Five. Similarity

the nullspace after the second application has dimension four so two additional basis vectors are sent to zero by the second iteration, and the nullspace after three applications is of dimension ﬁve so the ﬁnal basis vector is sent to zero in three hops. β1 → β2 → β3 → 0 β4 → β5 → 0 To produce such a basis, ﬁrst pick two independent vectors from N (n)     0 0 0 0     β3 = 1 β5 = 0     1 1 1 0 then add β2 , β4 ∈ N (n2 ) such that n(β2 ) = β3 and n(β4 ) = β5     0 0 1 1     β2 = 0 β4 = 0     1 0 0 0 and ﬁnish by adding β1 ∈ N (n3 ) = C5 ) such that n(β1 ) = β2 .   1 0   β1 = 1   0 0 Exercises
2.17 What is the index of nilpotency of the left-shift operator, here acting on the space of triples of reals? (x, y, z) → (0, x, y) 2.18 For each string basis state the index of nilpotency and give the dimension of the rangespace and nullspace of each iteration of the nilpotent map. (a) β1 → β2 → 0 β3 → β4 → 0 (b) β1 → β2 → β3 → 0 β4 → 0 β5 → 0 β6 → 0 (c) β1 → β2 → β3 → 0 Also give the canonical form of the matrix. 2.19 Decide which of these matrices are nilpotent.

Section III. Nilpotence
−3 −3 −3 2 2 2 1 1 1 1 3 5 1 0 2 4 −1 7

379

(a)

−2 −1

4 2

(b)

3 1

1 3

(c)

(d)

45 −22 −19 33 −16 −14 69 −34 −29 2.20 Find the canonical form of this  0 0  0 0 0 (e)

matrix. 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0

1 1  0 0 0

2.21 Consider the matrix from Example 2.16. (a) Use the action of the map on the string basis to give the canonical form. (b) Find the change of basis matrices that bring the matrix to canonical form. (c) Use the answer in the prior item to check the answer in the ﬁrst item. 2.22 Each of these matrices is nilpotent. 0 0 0 −1 1 −1 1/2 −1/2 1 0 1 (a) (b) 0 −1 1 (c) 1/2 −1/2 0 −1 1 1 −1 1 Put each in canonical form. 2.23 Describe the eﬀect of left or right multiplication by a matrix that is in the canonical form for nilpotent matrices. 2.24 Is nilpotence invariant under similarity? That is, must a matrix similar to a nilpotent matrix also be nilpotent? If so, with the same index? 2.25 Show that the only eigenvalue of a nilpotent matrix is zero. 2.26 Is there a nilpotent transformation of index three on a two-dimensional space? 2.27 In the proof of Theorem 2.13, why isn’t the proof’s base case that the index of nilpotency is zero? 2.28 Let t : V → V be a linear transformation and suppose v ∈ V is such that tk (v) = 0 but tk−1 (v) = 0. Consider the t-string v, t(v), . . . , tk−1 (v) . (a) Prove that t is a transformation on the span of the set of vectors in the string, that is, prove that t restricted to the span has a range that is a subset of the span. We say that the span is a t-invariant subspace. (b) Prove that the restriction is nilpotent. (c) Prove that the t-string is linearly independent and so is a basis for its span. (d) Represent the restriction map with respect to the t-string basis. 2.29 Finish the proof of Theorem 2.13. 2.30 Show that the terms ‘nilpotent transformation’ and ‘nilpotent matrix’, as given in Deﬁnition 2.6, ﬁt with each other: a map is nilpotent if and only if it is represented by a nilpotent matrix. (Is it that a transformation is nilpotent if an only if there is a basis such that the map’s representation with respect to that basis is a nilpotent matrix, or that any representation is a nilpotent matrix?) 2.31 Let T be nilpotent of index four. How big can the rangespace of T 3 be? 2.32 Recall that similar matrices have the same eigenvalues. Show that the converse does not hold. 2.33 Prove a nilpotent matrix is similar to one that is all zeros except for blocks of super-diagonal ones.

380

Chapter Five. Similarity

2.34 Prove that if a transformation has the same rangespace as nullspace. then the dimension of its domain is even. 2.35 Prove that if two nilpotent matrices commute then their product and sum are also nilpotent. 2.36 Consider the transformation of Mn×n given by tS (T ) = ST − T S where S is an n×n matrix. Prove that if S is nilpotent then so is tS . 2.37 Show that if N is nilpotent then I − N is invertible. Is that ‘only if’ also?

Section IV. Jordan Form

381

IV

Jordan Form

This section uses material from three optional subsections: Direct Sum, Determinants Exist, and Other Formulas for the Determinant. The chapter on linear maps shows that every h : V → W can be represented by a partial-identity matrix with respect to some bases B ⊂ V and D ⊂ W . This chapter revisits this issue in the special case that the map is a linear transformation t : V → V . Of course, the general result still applies but with the codomain and domain equal we naturally ask about having the two bases also be equal. That is, we want a canonical form to represent transformations as RepB,B (t). After a brief review section, we began by noting that a block partial identity form matrix is not always obtainable in this B, B case. We therefore considered the natural generalization, diagonal matrices, and showed that if its eigenvalues are distinct then a map or matrix can be diagonalized. But we also gave an example of a matrix that cannot be diagonalized and in the section prior to this one we developed that example. We showed that a linear map is nilpotent — if we take higher and higher powers of the map or matrix then we eventually get the zero map or matrix — if and only if there is a basis on which it acts via disjoint strings. That led to a canonical form for nilpotent matrices. Now, this section concludes the chapter. We will show that the two cases we’ve studied are exhaustive in that for any linear transformation there is a basis such that the matrix representation RepB,B (t) is the sum of a diagonal matrix and a nilpotent matrix in its canonical form.

IV.1 Polynomials of Maps and Matrices
Recall that the set of square matrices is a vector space under entry-by-entry addition and scalar multiplication and that this space Mn×n has dimension n2 . 2 Thus, for any n×n matrix T the n2 +1-member set {I, T, T 2 , . . . , T n } is linearly 2 dependent and so there are scalars c0 , . . . , cn2 such that cn2 T n + · · · + c1 T + c0 I is the zero matrix. 1.1 Remark This observation is small but important. It says that every transformation exhibits a generalized nilpotency: the powers of a square matrix cannot climb forever without a “repeat”. 1.2 Example Rotation of plane vectors π/6 radians counterclockwise is represented with respect to the standard basis by √ 3/2 √ −1/2 T = 1/2 3/2 and verifying that 0T 4 + 0T 3 + 1T 2 − 2T − 1I equals the zero matrix is easy.

382

Chapter Five. Similarity

1.3 Deﬁnition For any polynomial f (x) = cn xn + · · · + c1 x + c0 , where t is a linear transformation then f (t) is the transformation cn tn + · · · + c1 t + c0 (id) on the same space and where T is a square matrix then f (T ) is the matrix cn T n + · · · + c1 T + c0 I. 1.4 Remark If, for instance, f (x) = x − 3, then most authors write in the identity matrix: f (T ) = T − 3I. But most authors don’t write in the identity map: f (t) = t − 3. In this book we shall also observe this convention. Of course, if T = RepB,B (t) then f (T ) = RepB,B (f (t)), which follows from the relationships T j = RepB,B (tj ), and cT = RepB,B (ct), and T1 + T2 = RepB,B (t1 + t2 ). As Example 1.2 shows, there may be polynomials of degree smaller than n2 that zero the map or matrix. 1.5 Deﬁnition The minimal polynomial m(x) of a transformation t or a square matrix T is the polynomial of least degree and with leading coeﬃcient 1 such that m(t) is the zero map or m(T ) is the zero matrix. A minimal polynomial always exists by the observation opening this subsection. A minimal polynomial is unique by the ‘with leading coeﬃcient 1’ clause. This is because if there are two polynomials m(x) and m(x) that are both of the ˆ minimal degree to make the map or matrix zero (and thus are of equal degree), and both have leading 1’s, then their diﬀerence m(x) − m(x) has a smaller deˆ gree than either and still sends the map or matrix to zero. Thus m(x) − m(x) is ˆ the zero polynomial and the two are equal. (The leading coeﬃcient requirement also prevents a minimal polynomial from being the zero polynomial.) 1.6 Example We can see that m(x) = x2 − 2x − 1 is minimal for the matrix of Example 1.2 by computing the powers of T up to the power n2 = 4. √ √ 1/2 − 3/2 0 −1 −1/2 − 3/2 2 3 4 T = √ T = T = √ 1 0 3/2 1/2 3/2 −1/2 Next, put c4 T 4 + c3 T 3 + c2 T 2 + c1 T + c0 I equal to the zero matrix √ −(1/2)c4 + √ (1/2)c2 + ( 3/2)c1 + c0 = 0 √ −(√3/2)c4 − c3 − (√3/2)c2 − (1/2)c1 =0 ( 3/2)c4 + c3 + ( 3/2)c2 + √ (1/2)c1 =0 −(1/2)c4 + (1/2)c2 + ( 3/2)c1 + c0 = 0 and use Gauss’ method. c4 − √ c2 − c3 + 3c2 + √ 3c1 − √2c0 = 0 2c1 + 3c0 = 0

Setting c4 , c3 , and c2 to zero forces c1 and c0 to also come out as zero. To get a leading one, the most we can do is to set c4 and c3 to zero. Thus the minimal polynomial is quadratic.

Section IV. Jordan Form

383

Using the method of that example to ﬁnd the minimal polynomial of a 3×3 matrix would mean doing Gaussian reduction on a system with nine equations in ten unknowns. We shall develop an alternative. To begin, note that we can break a polynomial of a map or a matrix into its components. 1.7 Lemma Suppose that the polynomial f (x) = cn xn + · · · + c1 x + c0 factors as k(x − λ1 )q1 · · · (x − λ )q . If t is a linear transformation then these two are equal maps. cn tn + · · · + c1 t + c0 = k · (t − λ1 )q1 ◦ · · · ◦ (t − λ )q Consequently, if T is a square matrix then f (T ) and k ·(T −λ1 I)q1 · · · (T −λ I)q are equal matrices.
Proof. This argument is by induction on the degree of the polynomial. The

cases where the polynomial is of degree 0 and 1 are clear. The full induction argument is Exercise 1.7 but the degree two case gives its sense. A quadratic polynomial factors into two linear terms f (x) = k(x − λ1 ) · (x − λ2 ) = k(x2 + (λ1 + λ2 )x + λ1 λ2 ) (the roots λ1 and λ2 might be equal). We can check that substituting t for x in the factored and unfactored versions gives the same map. k · (t − λ1 ) ◦ (t − λ2 ) (v) = k · (t − λ1 ) (t(v) − λ2 v) = k · t(t(v)) − t(λ2 v) − λ1 t(v) − λ1 λ2 v = k · t ◦ t (v) − (λ1 + λ2 )t(v) + λ1 λ2 v = k · (t2 − (λ1 + λ2 )t + λ1 λ2 ) (v) The third equality holds because the scalar λ2 comes out of the second term, as t is linear. QED In particular, if a minimial polynomial m(x) for a transformation t factors as m(x) = (x − λ1 )q1 · · · (x − λ )q then m(t) = (t − λ1 )q1 ◦ · · · ◦ (t − λ )q is the zero map. Since m(t) sends every vector to zero, at least one of the maps t − λi sends some nonzero vectors to zero. So, too, in the matrix case — if m is minimal for T then m(T ) = (T − λ1 I)q1 · · · (T − λ I)q is the zero matrix and at least one of the matrices T −λi I sends some nonzero vectors to zero. Rewording both cases: at least some of the λi are eigenvalues. (See Exercise 29.) Recall how we have earlier found eigenvalues. We have looked for λ such that T v = λv by considering the equation 0 = T v−xv = (T −xI)v and computing the determinant of the matrix T − xI. That determinant is a polynomial in x, the characteristic polynomial, whose roots are the eigenvalues. The major result of this subsection, the next result, is that there is a connection between this characteristic polynomial and the minimal polynomial. This results expands on the prior paragraph’s insight that some roots of the minimal polynomial are eigenvalues by asserting that every root of the minimal polynomial is an eigenvalue and further that every eigenvalue is a root of the minimal polynomial (this is because it says ‘1 ≤ qi ’ and not just ‘0 ≤ qi ’).

384

Chapter Five. Similarity

1.8 Theorem (Cayley-Hamilton) If the characteristic polynomial of a transformation or square matrix factors into k · (x − λ1 )p1 (x − λ2 )p2 · · · (x − λ )p then its minimal polynomial factors into (x − λ1 )q1 (x − λ2 )q2 · · · (x − λ )q where 1 ≤ qi ≤ pi for each i between 1 and . The proof takes up the next three lemmas. Although they are stated only in matrix terms, they apply equally well to maps. We give the matrix version only because it is convenient for the ﬁrst proof. The ﬁrst result is the key — some authors call it the Cayley-Hamilton Theorem and call Theorem 1.8 above a corollary. For the proof, observe that a matrix of polynomials can be thought of as a polynomial with matrix coeﬃcients. 2x2 + 3x − 1 x2 + 2 2 2 3x + 4x + 1 4x + x + 1 = 2 3 1 2 3 x + 4 4 0 −1 x+ 1 1 2 1

1.9 Lemma If T is a square matrix with characteristic polynomial c(x) then c(T ) is the zero matrix.
Proof. Let C be T − xI, the matrix whose determinant is the characteristic

polynomial c(x) = cn xn + · · · + c1 x + c0 .  t1,1 − x t1,2  t2,1 t2,2 − x  C= .  . . ... .. . tn,n − x Recall that the product of the adjoint of a matrix with the matrix itself is the determinant of that matrix times the identity. c(x) · I = adj(C)C = adj(C)(T − xI) = adj(C)T − adj(C) · x (∗)     

The entries of adj(C) are polynomials, each of degree at most n − 1 since the minors of a matrix drop a row and column. Rewrite it, as suggested above, as adj(C) = Cn−1 xn−1 + · · · + C1 x + C0 where each Ci is a matrix of scalars. The left and right ends of equation (∗) above give this. cn Ixn + cn−1 Ixn−1 + · · · + c1 Ix + c0 I = (Cn−1 T )xn−1 + · · · + (C1 T )x + C0 T − Cn−1 xn − Cn−2 xn−1 − · · · − C0 x

Section IV. Jordan Form Equate the coeﬃcients of xn , the coeﬃcients of xn−1 , etc. cn I = −Cn−1 cn−1 I = −Cn−2 + Cn−1 T . . . c1 I = −C0 + C1 T c0 I = C0 T

385

Multiply (from the right) both sides of the ﬁrst equation by T n , both sides of the second equation by T n−1 , etc. Add. The result on the left is cn T n + cn−1 T n−1 + · · · + c0 I, and the result on the right is the zero matrix. QED We sometimes refer to that lemma by saying that a matrix or map satisﬁes its characteristic polynomial. 1.10 Lemma Where f (x) is a polynomial, if f (T ) is the zero matrix then f (x) is divisible by the minimal polynomial of T . That is, any polynomial satisﬁed by T is divisable by T ’s minimal polynomial.
Proof. Let m(x) be minimal for T . The Division Theorem for Polynomials gives f (x) = q(x)m(x) + r(x) where the degree of r is strictly less than the degree of m. Plugging T in shows that r(T ) is the zero matrix, because T satisﬁes both f and m. That contradicts the minimality of m unless r is the zero polynomial. QED

Combining the prior two lemmas gives that the minimal polynomial divides the characteristic polynomial. Thus, any root of the minimal polynomial is also a root of the characteristic polynomial. That is, so far we have that if m(x) = (x − λ1 )q1 . . . (x − λi )qi then c(x) must has the form (x − λ1 )p1 . . . (x − λi )pi (x − λi+1 )pi+1 . . . (x − λ )p where each qj is less than or equal to pj . The proof of the Cayley-Hamilton Theorem is ﬁnished by showing that in fact the characteristic polynomial has no extra roots λi+1 , etc. 1.11 Lemma Each linear factor of the characteristic polynomial of a square matrix is also a linear factor of the minimal polynomial.
Proof. Let T be a square matrix with minimal polynomial m(x) and assume

that x − λ is a factor of the characteristic polynomial of T , that is, assume that λ is an eigenvalue of T . We must show that x − λ is a factor of m, that is, that m(λ) = 0. In general, where λ is associated with the eigenvector v, for any polynomial function f (x), application of the matrix f (T ) to v equals the result of multiplying v by the scalar f (λ). (For instance, if T has eigenvalue λ associated with the eigenvector v and f (x) = x2 + 2x + 3 then (T 2 + 2T + 3) (v) = T 2 (v) + 2T (v) + 3v = λ2 · v + 2λ · v + 3 · v = (λ2 + 2λ + 3) · v.) Now, as m(T ) is the zero matrix, 0 = m(T )(v) = m(λ) · v and therefore m(λ) = 0. QED

386

Chapter Five. Similarity

1.12 Example We can use the Cayley-Hamilton Theorem to help ﬁnd the minimal polynomial of this matrix.   2 0 0 1 1 2 0 2   T = 0 0 2 −1 0 0 0 1 First, its characteristic polynomial c(x) = (x − 1)(x − 2)3 can be found with the usual determinant. Now, the Cayley-Hamilton Theorem says that T ’s minimal polynomial is either (x − 1)(x − 2) or (x − 1)(x − 2)2 or (x − 1)(x − 2)3 . We can decide among the choices just by computing:      1 0 0 1 0 0 0 1 0 0 0 0  1 1 0 2   1 0 0 2   1 0 0 1     (T − 1I)(T − 2I) =  0 0 1 −1 0 0 0 −1 = 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 and  0 1 (T − 1I)(T − 2I)2 =  0 0 0 0 0 0 0 0 0 0  0 0 1 1  0 0 0 0 0 0 0 0 0 0 0 0   0 1 2  0 = −1 0 0 −1 0 0 0 0 0 0 0 0  0 0  0 0

and so m(x) = (x − 1)(x − 2)2 . Exercises
1.13 What are the possible minimal polynomials if a matrix has the given characteristic polynomial? (a) 8 · (x − 3)4 (b) (1/3) · (x + 1)3 (x − 4) (c) −1 · (x − 2)2 (x − 5)2 (d) 5 · (x + 3)2 (x − 1)(x − 2)2 What is the degree of each possibility? 1.14 Find the minimal polynomial of each matrix. 3 0 0 3 0 0 3 0 0 2 0 1 (a) 1 3 0 (b) 1 3 0 (c) 1 3 0 (d) 0 6 2 0 0 4 0 0 3 0 1 3 0 0 2   −1 4 0 0 0 0 2 2 1 3 0 0 0    −4 −1 0 0 (e) 0 6 2 (f )  0  3 −9 −4 2 −1 0 0 2 1 5 4 1 4 1.15 Find the minimal polynomial of this matrix. 0 0 1 1 0 0 0 1 0

1.16 What is the minimal polynomial of the diﬀerentiation operator d/dx on Pn ?

Section IV. Jordan Form
1.17 Find the minimal polynomial of matrices of this form

387

λ 1  0

0 λ 1

0 0 λ

...

0 0

   

.. 0 ...

. λ 1

     0

0

λ

where the scalar λ is ﬁxed (i.e., is not a variable). 1.18 What is the minimal polynomial of the transformation of Pn that sends p(x) to p(x + 1)? 1.19 What is the minimal polynomial of the map π : C3 → C3 projecting onto the ﬁrst two coordinates? 1.20 Find a 3×3 matrix whose minimal polynomial is x2 . 1.21 What is wrong with this claimed proof of Lemma 1.9: “if c(x) = |T − xI| then c(T ) = |T − T I| = 0”? [Cullen] 1.22 Verify Lemma 1.9 for 2×2 matrices by direct calculation. 1.23 Prove that the minimal polynomial of an n × n matrix has degree at most n (not n2 as might be guessed from this subsection’s opening). Verify that this maximum, n, can happen. 1.24 The only eigenvalue of a nilpotent map is zero. Show that the converse statement holds. 1.25 What is the minimal polynomial of a zero map or matrix? Of an identity map or matrix? 1.26 Interpret the minimal polynomial of Example 1.2 geometrically. 1.27 What is the minimal polynomial of a diagonal matrix? 1.28 A projection is any transformation t such that t2 = t. (For instance, the transformation of the plane R2 projecting each vector onto its ﬁrst coordinate will, if done twice, result in the same value as if it is done just once.) What is the minimal polynomial of a projection? 1.29 The ﬁrst two items of this question are review. (a) Prove that the composition of one-to-one maps is one-to-one. (b) Prove that if a linear map is not one-to-one then at least one nonzero vector from the domain is sent to the zero vector in the codomain. (c) Verify the statement, excerpted here, that preceeds Theorem 1.8. . . . if a minimial polynomial m(x) for a transformation t factors as m(x) = (x − λ1 )q1 · · · (x − λ )q then m(t) = (t − λ1 )q1 ◦ · · · ◦ (t − λ )q is the zero map. Since m(t) sends every vector to zero, at least one of the maps t − λi sends some nonzero vectors to zero. . . . Rewording . . . : at least some of the λi are eigenvalues. 1.30 True or false: for a transformation on an n dimensional space, if the minimal polynomial has degree n then the map is diagonalizable. 1.31 Let f (x) be a polynomial. Prove that if A and B are similar matrices then f (A) is similar to f (B). (a) Now show that similar matrices have the same characteristic polynomial. (b) Show that similar matrices have the same minimal polynomial.

388
(c) Decide if these are similar. 1 2 3 3 4 1 −1 1

Chapter Five. Similarity

1.32 (a) Show that a matrix is invertible if and only if the constant term in its minimal polynomial is not 0. (b) Show that if a square matrix T is not invertible then there is a nonzero matrix S such that ST and T S both equal the zero matrix. 1.33 (a) Finish the proof of Lemma 1.7. (b) Give an example to show that the result does not hold if t is not linear. 1.34 Any transformation or square matrix has a minimal polynomial. Does the converse hold?

IV.2 Jordan Canonical Form
This subsection moves from the canonical form for nilpotent matrices to the one for all matrices. We have shown that if a map is nilpotent then all of its eigenvalues are zero. We can now prove the converse. 2.1 Lemma A linear transformation whose only eigenvalue is zero is nilpotent.
Proof. If a transformation t on an n-dimensional space has only the single

eigenvalue of zero then its characteristic polynomial is xn . The Cayley-Hamilton Theorem says that a map satisﬁes its characteristic polynimial so tn is the zero map. Thus t is nilpotent. QED We have a canonical form for nilpotent matrices, that is, for each matrix whose single eigenvalue is zero: each such matrix is similar to one that is all zeroes except for blocks of subdiagonal ones. (To make this representation unique we can ﬁx some arrangement of the blocks, say, from longest to shortest.) We next extend this to all single-eigenvalue matrices. Observe that if t’s only eigenvalue is λ then t − λ’s only eigenvalue is 0 because t(v) = λv if and only if (t − λ) (v) = 0 · v. The natural way to extend the results for nilpotent matrices is to represent t − λ in the canonical form N , and try to use that to get a simple representation T for t. The next result says that this try works. 2.2 Lemma If the matrices T − λI and N are similar then T and N + λI are also similar, via the same change of basis matrices. P T P −1 − P P −1 (λI) since the diagonal matrix λI commutes with anything, and so N = P T P −1 − λI. Therefore N + λI = P T P −1 , as required. QED
Proof. With N = P (T − λI)P −1 = P T P −1 − P (λI)P −1 we have N =

Section IV. Jordan Form 2.3 Example The characteristic polynomial of T = 2 1 −1 4

389

is (x − 3)2 and so T has only the single eigenvalue 3. Thus for T − 3I = −1 1 −1 1

the only eigenvalue is 0, and T − 3I is nilpotent. The null spaces are routine to ﬁnd; to ease this computation we take T to represent the transformation t : C2 → C2 with respect to the standard basis (we shall maintain this convention for the rest of the chapter). N (t − 3) = { −y y y ∈ C} N ((t − 3)2 ) = C2

The dimensions of these null spaces show that the action of an associated map t − 3 on a string basis is β1 → β2 → 0. Thus, the canonical form for t − 3 with one choice for a string basis is RepB,B (t − 3) = N = 0 1 0 0 B= 1 −2 , 1 2

and by Lemma 2.2, T is similar to this matrix. Rept (B, B) = N + 3I = 3 1 0 3

We can produce the similarity computation. Recall from the Nilpotence section how to ﬁnd the change of basis matrices P and P −1 to express N as P (T − 3I)P −1 . The similarity diagram
2 −− C2 w.r.t. E2 − − → Cw.r.t. E2 T −3I     id P id P 2 Cw.r.t. B t−3

− − → C2 −− w.r.t.
N

t−3

B

describes that to move from the lower left to the upper left we multiply by P −1 = RepE2 ,B (id)
−1

= RepB,E2 (id) =

1 1

−2 2

and to move from the upper right to the lower right we multiply by this matrix. P = 1 1 −2 2
−1

=

1/2 −1/4

1/2 1/4

390 So the similarity is expressed by 3 1 0 3 = 1/2 −1/4 1/2 1/4 2 1 −1 4

Chapter Five. Similarity

1 1

−2 2

which is easily checked. 2.4 Example This matrix has characteristic polynomial (x − 4)4   4 1 0 −1 0 3 0 1   T = 0 0 4 0  1 0 0 5 and so has the single eigenvalue 4. The nullities of t − 4 are: the null space of t − 4 has dimension two, the null space of (t − 4)2 has dimension three, and the null space of (t − 4)3 has dimension four. Thus, t − 4 has the action on a string basis of β1 → β2 → β3 → 0 and β4 → 0. This gives the canonical form N for t − 4, which in turn gives the form for t.   4 0 0 0 1 4 0 0  N + 4I =  0 1 4 0 0 0 0 4 An array that is all zeroes, except for some number λ down the diagonal and blocks of subdiagonal ones, is a Jordan block. We have shown that Jordan block matrices are canonical representatives of the similarity classes of singleeigenvalue matrices. 2.5 Example The 3 × 3 matrices whose only eigenvalue is 1/2 separate into three similarity classes. The three classes have these canonical representatives.       1/2 0 0 1/2 0 0 1/2 0 0  0  1 1/2 0   1 1/2 0  1/2 0  0 0 1/2 0 0 1/2 0 1 1/2 In particular, this matrix  1/2  0 0  0 0 1/2 0  1 1/2

belongs to the similarity class represented by the middle one, because we have adopted the convention of ordering the blocks of subdiagonal ones from the longest block to the shortest. We will now ﬁnish the program of this chapter by extending this work to cover maps and matrices with multiple eigenvalues. The best possibility for general maps and matrices would be if we could break them into a part involving

Section IV. Jordan Form

391

their ﬁrst eigenvalue λ1 (which we represent using its Jordan block), a part with λ2 , etc. This ideal is in fact what happens. For any transformation t : V → V , we shall break the space V into the direct sum of a part on which t−λ1 is nilpotent, plus a part on which t − λ2 is nilpotent, etc. More precisely, we shall take three steps to get to this section’s major theorem and the third step shows that V = N∞ (t − λ1 ) ⊕ · · · ⊕ N∞ (t − λ ) where λ1 , . . . , λ are t’s eigenvalues. Suppose that t : V → V is a linear transformation. Note that the restriction∗ of t to a subspace M need not be a linear transformation on M because there may be an m ∈ M with t(m) ∈ M . To ensure that the restriction of a transformation to a ‘part’ of a space is a transformation on the partwe need the next condition. 2.6 Deﬁnition Let t : V → V be a transformation. A subspace M is t invariant if whenever m ∈ M then t(m) ∈ M (shorter: t(M ) ⊆ M ). Two examples are that the generalized null space N∞ (t) and the generalized range space R∞ (t) of any transformation t are invariant. For the generalized null space, if v ∈ N∞ (t) then tn (v) = 0 where n is the dimension of the underlying space and so t(v) ∈ N∞ (t) because tn ( t(v) ) is zero also. For the generalized range space, if v ∈ R∞ (t) then v = tn (w) for some w and then t(v) = tn+1 (w) = tn ( t(w) ) shows that t(v) is also a member of R∞ (t). Thus the spaces N∞ (t − λi ) and R∞ (t − λi ) are t − λi invariant. Observe also that t − λi is nilpotent on N∞ (t − λi ) because, simply, if v has the property that some power of t − λi maps it to zero — that is, if it is in the generalized null space — then some power of t − λi maps it to zero. The generalized null space N∞ (t − λi ) is a ‘part’ of the space on which the action of t − λi is easy to understand. The next result is the ﬁrst of our three steps. It establishes that t − λj leaves t − λi ’s part unchanged. 2.7 Lemma A subspace is t invariant if and only if it is t − λ invariant for any scalar λ. In particular, where λi is an eigenvalue of a linear transformation t, then for any other eigenvalue λj , the spaces N∞ (t − λi ) and R∞ (t − λi ) are t − λj invariant.
Proof. For the ﬁrst sentence we check the two implications of the ‘if and only

if’ separately. One of them is easy: if the subspace is t − λ invariant for any λ then taking λ = 0 shows that it is t invariant. For the other implication suppose that the subspace is t invariant, so that if m ∈ M then t(m) ∈ M , and let λ be any scalar. The subspace M is closed under linear combinations and so if t(m) ∈ M then t(m) − λm ∈ M . Thus if m ∈ M then (t − λ) (m) ∈ M , as required. The second sentence follows straight from the ﬁrst. Because the two spaces are t − λi invariant, they are therefore t invariant. From this, applying the ﬁrst sentence again, we conclude that they are also t − λj invariant. QED

392

Chapter Five. Similarity

The second step of the three that we will take to prove this section’s major result makes use of an additional property of N∞ (t − λi ) and R∞ (t − λi ), that they are complementary. Recall that if a space is the direct sum of two others V = N ⊕ R then any vector v in the space breaks into two parts v = n + r where n ∈ N and r ∈ R, and recall also that if BN and BR are bases for N and R then the concatenation BN BR is linearly independent (and so the two parts of v do not “overlap”). The next result says that for any subspaces N and R that are complementary as well as t invariant, the action of t on v breaks into the “non-overlapping” actions of t on n and on r. 2.8 Lemma Let t : V → V be a transformation and let N and R be t invariant complementary subspaces of V . Then t can be represented by a matrix with blocks of square submatrices T1 and T2 T1 Z1 Z2 T2 }dim(N )-many rows }dim(R)-many rows

where Z1 and Z2 are blocks of zeroes.
Proof. Since the two subspaces are complementary, the concatenation of a basis

for N and a basis for R makes a basis B = ν1 , . . . , νp , µ1 , . . . , µq for V . We shall show that the matrix   . . . . . .   ··· RepB (t(µq )) RepB,B (t) = RepB (t(ν1 ))   . . . . . .

has the desired form. Any vector v ∈ V is in N if and only if its ﬁnal q components are when it is represented with respect to B. As N is t invariant, each vectors RepB (t(ν1 )), . . . , RepB (t(νp )) has that form. Hence the lower RepB,B (t) is all zeroes. The argument for the upper right is similar.

zeroes of the left of
QED

To see that t has been decomposed into its action on the parts, observe that the restrictions of t to the subspaces N and R are represented, with respect to the obvious bases, by the matrices T1 and T2 . So, with subspaces that are invariant and complementary, we can split the problem of examining a linear transformation into two lower-dimensional subproblems. The next result illustrates this decomposition into blocks. 2.9 Lemma If T is a matrices with square submatrices T1 and T2 T = T1 Z1 Z2 T2

where the Z’s are blocks of zeroes, then |T | = |T1 | · |T2 |.

Section IV. Jordan Form

393

Proof. Suppose that T is n×n, that T1 is p×p, and that T2 is q ×q. In the permutation formula for the determinant

|T | =
permutations φ

t1,φ(1) t2,φ(2) · · · tn,φ(n) sgn(φ)

each term comes from a rearrangement of the column numbers 1, . . . , n into a new order φ(1), . . . , φ(n). The upper right block Z2 is all zeroes, so if a φ has at least one of p + 1, . . . , n among its ﬁrst p column numbers φ(1), . . . , φ(p) then the term arising from φ is zero, e.g., if φ(1) = n then t1,φ(1) t2,φ(2) . . . tn,φ(n) = 0 · t2,φ(2) . . . tn,φ(n) = 0. So the above formula reduces to a sum over all permutations with two halves: any signiﬁcant φ is the composition of a φ1 that rearranges only 1, . . . , p and a φ2 that rearranges only p + 1, . . . , p + q. Now, the distributive law (and the fact that the signum of a composition is the product of the signums) gives that this |T1 | · |T2 | = t1,φ1 (1) · · · tp,φ1 (p) sgn(φ1 )
perms φ1 of 1,...,p

·

tp+1,φ2 (p+1) · · · tp+q,φ2 (p+q) sgn(φ2 )
perms φ2 of p+1,...,p+q

equals |T | =

signiﬁcant φ t1,φ(1) t2,φ(2)

· · · tn,φ(n) sgn(φ).

QED

2.10 Example 2 1 0 0 0 2 0 0 0 0 3 0 0 2 0 = 1 0 3 0 3 · 2 0 0 = 36 3

From Lemma 2.9 we conclude that if two subspaces are complementary and t invariant then t is nonsingular if and only if its restrictions to both subspaces are nonsingular. Now for the promised third, ﬁnal, step to the main result. 2.11 Lemma If a linear transformation t : V → V has the characteristic polynomial (x − λ1 )p1 . . . (x − λ )p then (1) V = N∞ (t − λ1 ) ⊕ · · · ⊕ N∞ (t − λ ) and (2) dim(N∞ (t − λi )) = pi .
Proof. Because dim(V ) is the degree p1 + · · · + p of the characteristic poly-

nomial, to establish statement (1) we need only show that statement (2) holds and that N∞ (t − λi ) ∩ N∞ (t − λj ) is trivial whenever i = j. For the latter, by Lemma 2.7, both N∞ (t−λi ) and N∞ (t−λj ) are t invariant. Notice that an intersection of t invariant subspaces is t invariant and so the restriction of t to N∞ (t − λi ) ∩ N∞ (t − λj ) is a linear transformation. But both t − λi and t − λj are nilpotent on this subspace and so if t has any eigenvalues

394

Chapter Five. Similarity

on the intersection then its “only” eigenvalue is both λi and λj . That cannot be, so this restriction has no eigenvalues: N∞ (t − λi ) ∩ N∞ (t − λj ) is trivial (Lemma 3.10 shows that the only transformation without any eigenvalues is on the trivial space). To prove statement (2), ﬁx the index i. Decompose V as N∞ (t − λi ) ⊕ R∞ (t − λi ) and apply Lemma 2.8. T = T1 Z1 Z2 T2 }dim( N∞ (t − λi ) )-many rows }dim( R∞ (t − λi ) )-many rows

By Lemma 2.9, |T − xI| = |T1 − xI| · |T2 − xI|. By the uniqueness clause of the Fundamental Theorem of Arithmetic, the determinants of the blocks have the same factors as the characteristic polynomial |T1 − xI| = (x − λ1 )q1 . . . (x − λ )q and |T2 − xI| = (x − λ1 )r1 . . . (x − λ )r , and the sum of the powers of these factors is the power of the factor in the characteristic polynomial: q1 + r1 = p1 , . . . , q + r = p . Statement (2) will be proved if we will show that qi = pi and that qj = 0 for all j = i, because then the degree of the polynomial |T1 − xI| — which equals the dimension of the generalized null space — is as required. For that, ﬁrst, as the restriction of t − λi to N∞ (t − λi ) is nilpotent on that space, the only eigenvalue of t on it is λi . Thus the characteristic equation of t on N∞ (t − λi ) is |T1 − xI| = (x − λi )qi . And thus qj = 0 for all j = i. Now consider the restriction of t to R∞ (t − λi ). By Note II.2.2, the map t − λi is nonsingular on R∞ (t − λi ) and so λi is not an eigenvalue of t on that subspace. Therefore, x − λi is not a factor of |T2 − xI|, and so qi = pi . QED Our major result just translates those steps into matrix terms. 2.12 Theorem Any square matrix is similar to one in Jordan form   Jλ1 –zeroes–   Jλ2     ..   .     Jλ −1 –zeroes– Jλ where each Jλ is the Jordan block associated with the eigenvalue λ of the original matrix (that is, is all zeroes except for λ’s down the diagonal and some subdiagonal ones).
Proof. Given an n×n matrix T , consider the linear map t : Cn → Cn that it

represents with respect to the standard bases. Use the prior lemma to write Cn = N∞ (t − λ1 ) ⊕ · · · ⊕ N∞ (t − λ ) where λ1 , . . . , λ are the eigenvalues of t. Because each N∞ (t − λi ) is t invariant, Lemma 2.8 and the prior lemma show that t is represented by a matrix that is all zeroes except for square blocks along the diagonal. To make those blocks into Jordan blocks, pick each Bλi to be a string basis for the action of t − λi on N∞ (t − λi ). QED

Section IV. Jordan Form

395

Jordan form is a canonical form for similarity classes of square matrices, provided that we make it unique by arranging the Jordan blocks from least eigenvalue to greatest and then arranging the subdiagonal 1 blocks inside each Jordan block from longest to shortest. 2.13 Example This matrix has the characteristic polynomial (x − 2)2 (x − 6).   2 0 1 T = 0 6 2 0 0 2 We will handle the eigenvalues 2 and 6 separately. Computation of the powers, and the null spaces and nullities, of T − 2I is routine. (Recall from Example 2.3 the convention of taking T to represent a transformation, here t : C3 → C3 , with respect to the standard basis.) power p 1 (T − 2I)p   0 0 1   0 4 2 0 0 0   0 0 0   0 16 8 0 0 0   0 0 0   0 64 32 0 0 0 N ((t − 2)p )   x   { 0  x ∈ C} 0  x z –same– —  2   {−z/2 x, z ∈ C} nullity 1

2

3

So the generalized null space N∞ (t − 2) has dimension two. We’ve noted that the restriction of t − 2 is nilpotent on this subspace. From the way that the nullities grow we know that the action of t − 2 on a string basis β1 → β2 → 0. Thus the restriction can be represented in the canonical form     1 −2 0 0 N2 = = RepB,B (t − 2) B2 =  1  ,  0  1 0 −2 0 where many choices of basis are possible. Consequently, the action of the restriction of t to N∞ (t − 2) is represented by this matrix. J2 = N2 + 2I = RepB2 ,B2 (t) = 2 1 0 2

The second eigenvalue’s computations are easier. Because the power of x − 6 in the characteristic polynomial is one, the restriction of t − 6 to N∞ (t − 6) must be nilpotent of index one. Its action on a string basis must be β3 → 0 and since it is the zero map, its canonical form N6 is the 1×1 zero matrix. Consequently,

396

Chapter Five. Similarity

the canonical form J6 for the action of t on N∞ (t−6) is the 1×1 matrix with the single entry 6. For the basis we can use any nonzero vector from the generalized null space.   0 B6 = 1 0 Taken together, these two give that the Jordan form of T is   2 0 0 RepB,B (t) = 1 2 0 0 0 6 where B is the concatenation of B2 and B6 . 2.14 Example Contrast the prior example with   2 2 1 T = 0 6 2 0 0 2 which has the same characteristic polynomial (x − 2)2 (x − 6). While the characteristic polynomial is the same, power p 1 (T − 2I)p   0 2 1   0 4 2 0  2 0  0 0 0 8 16 0 0  4  8 0 N ((t − 2)p )  x   {−z/2 x, z ∈ C}  z –same– — nullity 2

here the action of t − 2 is stable after only one application — the restriction of of t − 2 to N∞ (t − 2) is nilpotent of index only one. (So the contrast with the prior example is that while the characteristic polynomial tells us to look at the action of the t − 2 on its generalized null space, the characteristic polynomial does not describe completely its action and we must do some computations to ﬁnd, in this example, that the minimal polynomial is (x − 2)(x − 6).) The restriction of t − 2 to the generalized null space acts on a string basis as β1 → 0 and β2 → 0, and we get this Jordan block associated with the eigenvalue 2. J2 = 2 0 0 2

For the other eigenvalue, the arguments for the second eigenvalue of the prior example apply again. The restriction of t − 6 to N∞ (t − 6) is nilpotent of index one (it can’t be of index less than one, and since x − 6 is a factor of

Section IV. Jordan Form

397

the characteristic polynomial to the power one it can’t be of index more than one either). Thus t − 6’s canonical form N6 is the 1×1 zero matrix, and the associated Jordan block J6 is the 1×1 matrix with entry 6. Therefore, T is diagonalizable.  2 RepB,B (t) = 0 0 0 2 0  0 0 6       1 0 3 B6 = 0 ,  1  , 4 0 −2 0

B = B2

(Checking that the third vector in B is in the nullspace of t − 6 is routine.) 2.15 Example A bit of computing with  −1 0  T = 0  3 1 4 3 −4 −9 5  0 0 0 0 0 0  −1 0 0   −4 2 −1 4 1 4

shows that its characteristic polynomial is (x − 3)3 (x + 1)2 . This table power p N ((t − 3)p ) nullity    4 0 0 0 −(u + v)/2    0 0 0 0  −(u + v)/2    −4 −4 0 0  { (u + v)/2  u, v ∈ C} 2       −9 −4 −1 −1  u  1 5 4 1 1 v     −z 16 −16 0 0 0     0 0 0 0 −z   0      0 3 { z  z, u, v ∈ C} 16 16 0 0         16 0 0 −16 32 u v 0 −16 −16 0 0   −64 64 0 0 0    0 0 0 0 0    0 –same– — −64 −64 0 0      64 −128 −64 0 0 0 64 64 0 0 (T − 3I)p  −4  0  0   3

1

2

3

shows that the restriction of t − 3 to N∞ (t − 3) acts on a string basis via the two strings β1 → β2 → 0 and β3 → 0. A similar calculation for the other eigenvalue

398 power p  0  0  0   3 1  0  0  0   8 8 (T + 1I)p 4 4 −4 −9 5 16 16 −16 24 0 0 0 −4 4 0 0 0 16 0 0 0 3 1 0 0 0 8 8 0   0  0   −1 5  0  0  0   −8 24

Chapter Five. Similarity N ((t + 1)p )  −(u + v)    0    { −v  u, v ∈ C}     u   v  nullity

1

2

2

–same–

−40 −16

shows that the restriction of t + 1 to its generalized null space acts on a string basis via the two separate strings β4 → 0 and β5 → 0. Therefore T is similar to this Jordan form matrix.   −1 0 0 0 0  0 −1 0 0 0   0 0 3 0 0   0 0 1 3 0 0 0 0 0 3 We close with the statement that the subjects considered earlier in this Chpater are indeed, in this sense, exhaustive. 2.16 Corollary Every square matrix is similar to the sum of a diagonal matrix and a nilpotent matrix. Exercises
2.17 Do the check for Example 2.3. 2.18 Each matrix is in Jordan form. State its characteristic polynomial and its minimal polynomial. 2 0 0 3 0 0 3 0 −1 0 0 (a) (b) (c) 1 2 (d) 1 3 0 1 3 0 −1 0 0 −1/2 0 1 3     3 0 0 0 4 0 0 0 5 0 0 0  1 3 0 0  1 4 0 (e)   (f ) 0 0 −4 0  (g) 0 2 0 0 0 3 0 0 0 3 0 0 1 3 0 0 1 −4     5 0 0 0 5 0 0 0 0 2 0 0 0 2 0 0  (h)  (i)  0 0 2 0 0 1 2 0 0 0 0 3 0 0 0 3 2.19 Find the Jordan form from the given data. (a) The matrix T is 5×5 with the single eigenvalue 3. The nullities of the powers are: T − 3I has nullity two, (T − 3I)2 has nullity three, (T − 3I)3 has nullity four, and (T − 3I)4 has nullity ﬁve.

Section IV. Jordan Form

399

(b) The matrix S is 5×5 with two eigenvalues. For the eigenvalue 2 the nullities are: S − 2I has nullity two, and (S − 2I)2 has nullity four. For the eigenvalue −1 the nullities are: S + 1I has nullity one. 2.20 Find the change of basis matrices for each example. (a) Example 2.13 (b) Example 2.14 (c) Example 2.15 2.21 Find the Jordan form and a Jordan basis for each matrix. −10 4 (a) −25 10 5 −4 9 −7 4 0 0 (c) 2 1 3 5 0 4 5 4 3 0 −3 (d) −1 1 −2 1 9 7 3 (e) −9 −7 −4 4 4 4 2 2 −1 1 (f ) −1 −1 −1 −2 2   7 1 2 2  1 4 −1 −1 (g)  −2 1 5 −1 1 1 2 8 2.22 Find all possible Jordan forms of a transformation with characteristic polynomial (x − 1)2 (x + 2)2 . 2.23 Find all possible Jordan forms of a transformation with characteristic polynomial (x − 1)3 (x + 2). 2.24 Find all possible Jordan forms of a transformation with characteristic polynomial (x − 2)3 (x + 1) and minimal polynomial (x − 2)2 (x + 1). 2.25 Find all possible Jordan forms of a transformation with characteristic polynomial (x − 2)4 (x + 1) and minimal polynomial (x − 2)2 (x + 1). 2.26 Diagonalize these. 1 1 0 1 (a) (b) 0 0 1 0 2.27 Find the Jordan matrix representing the diﬀerentiation operator on P3 . 2.28 Decide if these two are similar. 1 −1 −1 0 4 −3 1 −1 (b) 2.29 Find the Jordan form of this matrix. 0 −1 1 0 Also give a Jordan basis. 2.30 How many similarity classes are there for 3×3 matrices whose only eigenvalues are −3 and 4?

400

Chapter Five. Similarity

2.31 Prove that a matrix is diagonalizable if and only if its minimal polynomial has only linear factors. 2.32 Give an example of a linear transformation on a vector space that has no non-trivial invariant subspaces. 2.33 Show that a subspace is t − λ1 invariant if and only if it is t − λ2 invariant. 2.34 Prove or disprove: two n×n matrices are similar if and only if they have the same characteristic and minimal polynomials. 2.35 The trace of a square matrix is the sum of its diagonal entries. (a) Find the formula for the characteristic polynomial of a 2×2 matrix. (b) Show that trace is invariant under similarity, and so we can sensibly speak of the ‘trace of a map’. (Hint: see the prior item.) (c) Is trace invariant under matrix equivalence? (d) Show that the trace of a map is the sum of its eigenvalues (counting multiplicities). (e) Show that the trace of a nilpotent map is zero. Does the converse hold? 2.36 To use Deﬁnition 2.6 to check whether a subspace is t invariant, we seemingly have to check all of the inﬁnitely many vectors in a (nontrivial) subspace to see if they satisfy the condition. Prove that a subspace is t invariant if and only if its subbasis has the property that for all of its elements, t(β) is in the subspace. 2.37 Is t invariance preserved under intersection? Under union? Complementation? Sums of subspaces? 2.38 Give a way to order the Jordan blocks if some of the eigenvalues are complex numbers. That is, suggest a reasonable ordering for the complex numbers. 2.39 Let Pj (R) be the vector space over the reals of degree j polynomials. Show that if j ≤ k then Pj (R) is an invariant subspace of Pk (R) under the diﬀerentiation operator. In P7 (R), does any of P0 (R), . . . , P6 (R) have an invariant complement? 2.40 In Pn (R), the vector space (over the reals) of degree n polynomials, E = {p(x) ∈ Pn (R) p(−x) = p(x) for all x} and O = {p(x) ∈ Pn (R) p(−x) = −p(x) for all x} are the even and the odd polynomials; p(x) = x2 is even while p(x) = x3 is odd. Show that they are subspaces. Are they complementary? Are they invariant under the diﬀerentiation transformation? 2.41 Lemma 2.8 says that if M and N are invariant complements then t has a representation in the given block form (with respect to the same ending as starting basis, of course). Does the implication reverse? 2.42 A matrix S is the square root of another T if S 2 = T . Show that any nonsingular matrix has a square root.

Topic: Method of Powers

401

Topic: Method of Powers
In practice, calculating eigenvalues and eigenvectors is a diﬃcult problem. Finding, and solving, the characteristic polynomial of the large matrices often encountered in applications is too slow and too hard. Other techniques, indirect ones that avoid the characteristic polynomial, are used. Here we shall see such a method that is suitable for large matrices that are ‘sparse’ (the great majority of the entries are zero). Suppose that the n×n matrix T has the n distinct eigenvalues λ1 , λ2 , . . . , λn . Then Rn has a basis that is composed of the associated eigenvectors ζ1 , . . . , ζn . For any v ∈ Rn , where v = c1 ζ1 + · · · + cn ζn , iterating T on v gives these. T v = c1 λ1 ζ1 + c2 λ2 ζ2 + · · · + cn λn ζn T 2 v = c1 λ2 ζ1 + c2 λ2 ζ2 + · · · + cn λ2 ζn 1 2 n T 3 v = c1 λ3 ζ1 + c2 λ3 ζ2 + · · · + cn λ3 ζn 1 2 n . . . T k v = c1 λk ζ1 + c2 λk ζ2 + · · · + cn λk ζn 1 2 n If one of the eigenvaluse, say, λ1 , has a larger absolute value than any of the other eigenvalues then its term will dominate the above expression. Put another way, dividing through by λk gives this, 1 λk λk T kv 2 = c1 ζ1 + c2 k ζ2 + · · · + cn n ζn k λ1 λ1 λk 1 and, because λ1 is assumed to have the largest absolute value, as k gets larger the fractions go to zero. Thus, the entire expression goes to c1 ζ1 . That is (as long as c1 is not zero), as k increases, the vectors T k v will tend toward the direction of the eigenvectors associated with the dominant eigenvalue, and, consequently, the ratios of the lengths T k v / T k−1 v will tend toward that dominant eigenvalue. For example (sample computer code for this follows the exercises), because the matrix 3 0 T = 8 −1 is triangular, its eigenvalues are just the entries on the diagonal, 3 and −1. Arbitrarily taking v to have the components 1 and 1 gives v 1 1 Tv 3 7 T 2v 9 17 ··· ··· T 9v 19 683 39 367 T 10 v 59 049 118 097

and the ratio between the lengths of the last two is 2.999 9. Two implementation issues must be addressed. The ﬁrst issue is that, instead of ﬁnding the powers of T and applying them to v, we will compute v1 as T v and

402

Chapter Five. Similarity

then compute v2 as T v1 , etc. (i.e., we never separately calculate T 2 , T 3 , etc.). These matrix-vector products can be done quickly even if T is large, provided that it is sparse. The second issue is that, to avoid generating numbers that are so large that they overﬂow our computer’s capability, we can normalize the vi ’s at each step. For instance, we can divide each vi by its length (other possibilities are to divide it by its largest component, or simply by its ﬁrst component). We thus implement this method by generating w0 = v0 / v0 v1 = T w0 w1 = v1 / v1 v2 = T w2 . . . wk−1 = vk−1 / vk−1 vk = T wk until we are satisﬁed. Then the vector vk is an approximation of an eigenvector, and the approximation of the dominant eigenvalue is the ratio vk / wk−1 = vk . One way we could be ‘satisﬁed’ is to iterate until our approximation of the eigenvalue settles down. We could decide, for instance, to stop the iteration process not after some ﬁxed number of steps, but instead when vk diﬀers from vk−1 by less than one percent, or when they agree up to the second signiﬁcant digit. The rate of convergence is determined by the rate at which the powers of λ2 /λ1 go to zero, where λ2 is the eigenvalue of second largest norm. If that ratio is much less than one then convergence is fast, but if it is only slightly less than one then convergence can be quite slow. Consequently, the method of powers is not the most commonly used way of ﬁnding eigenvalues (although it is the simplest one, which is why it is here as the illustration of the possibility of computing eigenvalues without solving the characteristic polynomial). Instead, there are a variety of methods that generally work by ﬁrst replacing the given matrix T with another that is similar to it and so has the same eigenvalues, but is in some reduced form such as tridiagonal form: the only nonzero entries are on the diagonal, or just above or below it. Then special techniques can be used to ﬁnd the eigenvalues. Once the eigenvalues are known, the eigenvectors of T can be easily computed. These other methods are outside of our scope. A good reference is [Goult, et al.] Exercises
1 Use ten iterations to estimate the largest eigenvalue of these matrices, starting from the vector with components 1 and 2. Compare the answer with the one obtained by solving the characteristic equation.

Topic: Method of Powers
1 0 5 4 3 −1 2 0

403

(a)

(b)

2 Redo the prior exercise by iterating until vk − vk−1 has absolute value less than 0.01 At each step, normalize by dividing each vector by its length. How many iterations are required? Are the answers signiﬁcantly diﬀerent? 3 Use ten iterations to estimate the largest eigenvalue of these matrices, starting from the vector with components 1, 2, and 3. Compare the answer with the one obtained by solving the characteristic equation. −1 2 2 4 0 1 2 2 2 (b) (a) −2 1 0 −3 −6 −6 −2 0 1 4 Redo the prior exercise by iterating until vk − vk−1 has absolute value less than 0.01. At each step, normalize by dividing each vector by its length. How many iterations does it take? Are the answers signiﬁcantly diﬀerent? 5 What happens if c1 = 0? That is, what happens if the initial vector does not to have any component in the direction of the relevant eigenvector? 6 How can the method of powers be adopted to ﬁnd the smallest eigenvalue?

Computer Code This is the code for the computer algebra system Octave that was used to do the calculation above. (It has been lightly edited to remove blank lines, etc.)
>T=[3, 0; 8, -1] T= 3 0 8 -1 >v0=[1; 2] v0= 1 1 >v1=T*v0 v1= 3 7 >v2=T*v1 v2= 9 17 >T9=T**9 T9= 19683 0 39368 -1 >T10=T**10 T10= 59049 0 118096 1 >v9=T9*v0 v9=

404
19683 39367 >v10=T10*v0 v10= 59049 118096 >norm(v10)/norm(v9) ans=2.9999

Chapter Five. Similarity

Remark: we are ignoring the power of Octave here; there are built-in functions to automatically apply quite sophisticated methods to ﬁnd eigenvalues and eigenvectors. Instead, we are using just the system as a calculator.

Topic: Stable Populations

405

Topic: Stable Populations
Imagine a reserve park with animals from a species that we are trying to protect. The park doesn’t have a fence and so animals cross the boundary, both from the inside out and in the other direction. Every year, 10% of the animals from inside of the park leave, and 1% of the animals from the outside ﬁnd their way in. We can ask if we can ﬁnd a stable level of population for this park: is there a population that, once established, will stay constant over time, with the number of animals leaving equal to the number of animals entering? To answer that question, we must ﬁrst establish the equations. Let the year n population in the park be pn and in the rest of the world be rn . pn+1 = .90pn + .01rn rn+1 = .10pn + .99rn We can set this system up as a matrix equation (see the Markov Chain topic). pn+1 rn+1 = .90 .01 .10 .99 pn rn

Now, “stable level” means that pn+1 = pn and rn+1 = rn , so that the matrix equation vn+1 = T vn becomes v = T v. We are therefore looking for eigenvectors for T that are associated with the eigenvalue 1. The equation (I − T )v = 0 is .10 .01 .10 .01 p r = 0 0

which gives the eigenspace: vectors with the restriction that p = .1r. Coupled with additional information, that the total world population of this species is is p + r = 110 000, we ﬁnd that the stable state is p = 10, 000 and r = 100, 000. If we start with a park population of ten thousand animals, so that the rest of the world has one hundred thousand, then every year ten percent (a thousand animals) of those inside will leave the park, and every year one percent (a thousand) of those from the rest of the world will enter the park. It is stable, self-sustaining. Now imagine that we are trying to gradually build up the total world population of this species. We can try, for instance, to have the world population grow at a rate of 1% per year. In this case, we can take a “stable” state for the park’s population to be that it also grows at 1% per year. The equation vn+1 = 1.01 · vn = T vn leads to ((1.01 · I) − T )v = 0, which gives this system. .11 .01 .10 .02 p r = 0 0

The matrix is nonsingular, and so the only solution is p = 0 and r = 0. Thus, there is no (usable) initial population that we can establish at the park and expect that it will grow at the same rate as the rest of the world.

406

Chapter Five. Similarity

Knowing that an annual world population growth rate of 1% forces an unstable park population, we can ask which growth rates there are that would allow an initial population for the park that will be self-sustaining. We consider λv = T v and solve for λ. 0= λ − .9 .10 .01 = (λ − .9)(λ − .99) − (.10)(.01) = λ2 − 1.89λ + .89 λ − .99

A shortcut to factoring that quadratic is our knowledge that λ = 1 is an eigenvalue of T , so the other eigenvalue is .89. Thus there are two ways to have a stable park population (a population that grows at the same rate as the population of the rest of the world, despite the leaky park boundaries): have a world population that is does not grow or shrink, and have a world population that shrinks by 11% every year. So this is one meaning of eigenvalues and eigenvectors — they give a stable state for a system. If the eigenvalue is 1 then the system is static. If the eigenvalue isn’t 1 then the system is either growing or shrinking, but in a dynamically-stable way. Exercises
1 What initial population for the park discussed above should be set up in the case where world populations are allowed to decline by 11% every year? 2 What will happen to the population of the park in the event of a growth in world population of 1% per year? Will it lag the world growth, or lead it? Assume that the inital park population is ten thousand, and the world population is one hunderd thousand, and calculate over a ten year span. 3 The park discussed above is partially fenced so that now, every year, only 5% of the animals from inside of the park leave (still, about 1% of the animals from the outside ﬁnd their way in). Under what conditions can the park maintain a stable population now? 4 Suppose that a species of bird only lives in Canada, the United States, or in Mexico. Every year, 4% of the Canadian birds travel to the US, and 1% of them travel to Mexico. Every year, 6% of the US birds travel to Canada, and 4% go to Mexico. From Mexico, every year 10% travel to the US, and 0% go to Canada. (a) Give the transition matrix. (b) Is there a way for the three countries to have constant populations? (c) Find all stable situations.

Topic: Linear Recurrences

407

Topic: Linear Recurrences
In 1202 Leonardo of Pisa, also known as Fibonacci, posed this problem. A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? This moves past an elementary exponential growth model for population increase to include the fact that there is an initial period where newborns are not fertile. However, it retains other simplyﬁng assumptions, such as that there is no gestation period and no mortality. The number of newborn pairs that will appear in the upcoming month is simply the number of pairs that were alive last month, since those will all be fertile, having been alive for two months. The number of pairs alive next month is the sum of the number alive last month and the number of newborns. f (n + 1) = f (n) + f (n − 1) where f (0) = 1, f (1) = 1

The is an example of a recurrence relation (it is called that because the values of f are calculated by looking at other, prior, values of f ). From it, we can easily answer Fibonacci’s twelve-month question. month pairs 0 1 1 1 2 2 3 3 4 5 5 8 6 13 7 21 8 34 9 55 10 89 11 144 12 233

The sequence of numbers deﬁned by the above equation (of which the ﬁrst few are listed) is the Fibonacci sequence. The material of this chapter can be used to give a formula with which we can can calculate f (n + 1) without having to ﬁrst ﬁnd f (n), f (n − 1), etc. For that, observe that the recurrence is a linear relationship and so we can give a suitable matrix formulation of it. 1 1 1 0 f (n) f (n − 1) = f (n + 1) f (n) where f (1) f (0) = 1 1

Then, where we write T for the matrix and vn for the vector with components f (n+1) and f (n), we have that vn = T n v0 . The advantage of this matrix formulation is that by diagonalizing T we get a fast way to compute its powers: where T = P DP −1 we have T n = P Dn P −1 , and the n-th power of the diagonal matrix D is the diagonal matrix whose entries that are the n-th powers of the entries of D. The characteristic equation of√ is λ2 − λ − 1. The quadratic formula gives T √ its roots as (1 + 5)/2 and (1 − 5)/2. Diagonalizing gives this. 1 1 1 0 =
√ 1+ 5 2 √ 1− 5 2 √ 1+ 5 2

1

1

0

1− 5 2

0 √

1 √ 5 −1 √ 5

√ − 1− 55 2 √ 1+ 5 √ 2 5

408

Chapter Five. Similarity

Introducing the vectors and taking the n-th power, we have f (n + 1) f (n) = = 1 1 1 1 0
n

f (1) f (0)
√ 1− 5 2 √ n 1+ 5 2

√ 1+ 5 2

1

0

1− 5 2

0 √

n

1 √ 5 −1 √ 5

√ − 1− 55 2 √ 1+ 5 √ 2 5

f (1) f (0)

We can compute f (n) from the second component of that equation. 1 f (n) = √ 5 √ 1+ 5 2
n

√ 1− 5 2

n

√ Notice that f is dominated by its ﬁrst term because (1 − 5)/2 is less than one, so its powers go to zero. Although we have extended the elementary model of population growth by adding a delay period before the onset of fertility, we nonetheless still get an (asmyptotically) exponential function. In general, a linear recurrence relation has the form f (n + 1) = an f (n) + an−1 f (n − 1) + · · · + an−k f (n − k) (it is also called a diﬀerence equation). This recurrence relation is homogeneous because there is no constant term; i.e, it can be put into the form 0 = −f (n+1)+ an f (n)+an−1 f (n−1)+· · ·+an−k f (n−k). This is said to be a relation of order k. The relation, along with the initial conditions f (0), . . . , f (k) completely determine a sequence. For instance, the Fibonacci relation is of order 2 and it, along with the two initial conditions f (0) = 1 and f (1) = 1, determines the Fibonacci sequence simply because we can compute any f (n) by ﬁrst computing f (2), f (3), etc. In this Topic, we shall see how linear algebra can be used to solve linear recurrence relations. First, we deﬁne the vector space in which we are working. Let V be the set of functions f from the natural numbers N = {0, 1, 2, . . .} to the real numbers. (Below we shall have functions with domain {1, 2, . . .}, that is, without 0, but it is not an important distinction.) Putting the initial conditions aside for a moment, for any recurrence, we can consider the subset S of V of solutions. For example, without initial conditions, in addition to the function f given above, the Fibonacci relation is also solved by the function g whose ﬁrst few values are g(0) = 1, g(1) = 1, g(2) = 3, g(3) = 4, and g(4) = 7. The subset S is a subspace of V . It is nonempty because the zero function is a solution. It is closed under addition since if f1 and f2 are solutions, then an+1 (f1 + f2 )(n + 1) + · · · + an−k (f1 + f2 )(n − k) = (an+1 f1 (n + 1) + · · · + an−k f1 (n − k)) + (an+1 f2 (n + 1) + · · · + an−k f2 (n − k)) = 0.

Topic: Linear Recurrences And, it is closed under scalar multiplication since an+1 (rf1 )(n + 1) + · · · + an−k (rf1 )(n − k)

409

= r(an+1 f1 (n + 1) + · · · + an−k f1 (n − k)) =r·0 = 0. We can give the dimension of S. Consider this map from the set of functions S to the set of vectors Rk .   f (0) f (1)   f → .   .  . f (k) Exercise 3 shows that this map is linear. Because, as noted above, any solution of the recurrence is uniquely determined by the k initial conditions, this map is one-to-one and onto. Thus it is an isomorphism, and thus S has dimension k, the order of the recurrence. So (again, without any initial conditions), we can describe the set of solutions of any linear homogeneous recurrence relation of degree k by taking linear combinations of only k linearly independent functions. It remains to produce those functions. For that, we express the recurrence f (n + 1) = an f (n) + · · · + an−k f (n − k) with a matrix equation.  an 1  0  0  . . . 0 an−1 0 1 0 . . . 0 an−2 0 0 1 0 . . . an−k+1 ... 0  an−k     0  f (n) f (n + 1)    f (n − 1)  f (n)     =   . . . .     . . .  .  f (n − k) f (n − k + 1) . 0

. ...

..

1

In trying to ﬁnd the characteristic function of the matrix, we can see the pattern in the 2×2 case an − λ 1 and 3×3 case.  an − λ  1 0 an−1 −λ 1  an−2 0  = −λ3 + an λ2 + an−1 λ + an−2 −λ an−1 −λ = λ2 − an λ − an−1

410

Chapter Five. Similarity

Exercise 4 shows that the characteristic equation is this. an − λ 1 0 0 . . . 0 an−1 −λ 1 0 . . . 0 an−2 0 −λ 1 0 ... ... an−k+1 0 an−k 0

. ...
k

..

1

. . . −λ

= ±(−λ + an λk−1 + an−1 λk−2 + · · · + an−k+1 λ + an−k ) We call that the polynomial ‘associated’ with the recurrence relation. (We will be ﬁnding the roots of this polynomial and so we can drop the ± as irrelevant.) If −λk + an λk−1 + an−1 λk−2 + · · · + an−k+1 λ + an−k has no repeated roots then the matrix is diagonalizable and we can, in theory, get a formula for f (n) as in the Fibonacci case. But, because we know that the subspace of solutions has dimension k, we do not need to do the diagonalization calculation, provided that we can exhibit k linearly independent functions satisfying the relation. n Where r1 , r2 , . . . , rk are the distinct roots, consider the functions fr1 (n) = r1 n through frk (n) = rk of powers of those roots. Exercise 5 shows that each is a solution of the recurrence and that the k of them form a linearly independent set. So, given the homogeneous linear recurrence f (n + 1) = an f (n) + · · · + an−k f (n − k) (that is, 0 = −f (n + 1) + an f (n) + · · · + an−k f (n − k)) we consider the associated equation 0 = −λk + an λk−1 + · · · + an−k+1 λ + an−k . We ﬁnd its roots r1 , . . . , rk , and if those roots are distinct then any solution of the relation n n n has the form f (n) = c1 r1 + c2 r2 + · · · + ck rk for c1 , . . . , cn ∈ R. (The case of repeated roots is also easily done, but we won’t cover it here — see any text on Discrete Mathematics.) Now, given some initial conditions, so that we are interested in a particular solution, we can solve for c1 , . . . , cn . For instance, the polynomial associated √ with the Fibonacci relation is −λ2 + λ + 1, whose roots are (1 ± 5)/2 and so √ any solution of the Fibonacci equation has the form f (n) = c1 ((1 + 5)/2)n + √ c2 ((1 − 5)/2)n . Including the initial conditions for the cases n = 0 and n = 1 gives c1 + c2 = 1 √ √ (1 + 5/2)c1 + (1 − 5/2)c2 = 1 √ √ which yields c1 = 1/ 5 and c2 = −1/ 5, as was calculated above. We close by considering the nonhomogeneous case, where the relation has the form f (n + 1) = an f (n) + an−1 f (n − 1) + · · · + an−k f (n − k) + b for some nonzero b. As in the ﬁrst chapter of this book, only a small adjustment is needed to make the transition from the homogeneous case. This classic example illustrates. In 1883, Edouard Lucas posed the following problem. In the great temple at Benares, beneath the dome which marks the center of the world, rests a brass plate in which are ﬁxed three diamond needles, each a cubit high and as thick as the body of a

Topic: Linear Recurrences bee. On one of these needles, at the creation, God placed sixty four disks of pure gold, the largest disk resting on the brass plate, and the others getting smaller and smaller up to the top one. This is the Tower of Bramah. Day and night unceasingly the priests transfer the disks from one diamond needle to another according to the ﬁxed and immutable laws of Bramah, which require that the priest on duty must not move more than one disk at a time and that he must place this disk on a needle so that there is no smaller disk below it. When the sixty-four disks shall have been thus transferred from the needle on which at the creation God placed them to one of the other needles, tower, temple, and Brahmins alike will crumble into dusk, and with a thunderclap the world will vanish. (Translation of [De Parville] from [Ball & Coxeter].)

411

How many disk moves will it take? Instead of tackling the sixty four disk problem right away, we will consider the problem for smaller numbers of disks, starting with three. To begin, all three disks are on the same needle.

After moving the small disk to the far needle, the mid-sized disk to the middle needle, and then moving the small disk to the middle needle we have this.

Now we can move the big disk over. Then, to ﬁnish, we repeat the process of moving the smaller disks, this time so that they end up on the third needle, on top of the big disk. So the thing to see is that to move the very largest disk, the bottom disk, at a minimum we must: ﬁrst move the smaller disks to the middle needle, then move the big one, and then move all the smaller ones from the middle needle to the ending needle. Those three steps give us this recurence. T (n + 1) = T (n) + 1 + T (n) = 2T (n) + 1 We can easily get the ﬁrst few values of T . where T (1) = 1

412 n T (n) 1 1 2 3 3 7 4 15 5 31 6 63 7 127

Chapter Five. Similarity 8 255 9 511 10 1023

We recognize those as being simply one less than a power of two. To derive this equation instead of just guessing at it, we write the original relation as −1 = −T (n + 1) + 2T (n), consider the homogeneous relation 0 = −T (n) + 2T (n − 1), get its associated polynomial −λ + 2, which obviously has the single, unique, root of r1 = 2, and conclude that functions satisfying the homogeneous relation take the form T (n) = c1 2n . That’s the homogeneous solution. Now we need a particular solution. Because the nonhomogeneous relation −1 = −T (n + 1) + 2T (n) is so simple, in a few minutes (or by remembering the table) we can spot the particular solution T (n) = −1 (there are other particular solutions, but this one is easily spotted). So we have that — without yet considering the initial condition — any solution of T (n + 1) = 2T (n) + 1 is the sum of the homogeneous solution and this particular solution: T (n) = c1 2n − 1. The initial condition T (1) = 1 now gives that c1 = 1, and we’ve gotten the formula that generates the table: the n-disk Tower of Hanoi problem requires a minimum of 2n − 1 moves. Finding a particular solution in more complicated cases is, naturally, more complicated. A delightful and rewarding, but challenging, source on recurrence relations is [Graham, Knuth, Patashnik]., For more on the Tower of Hanoi, [Ball & Coxeter] or [Gardner 1957] are good starting points. So is [Hofstadter]. Some computer code for trying some recurrence relations follows the exercises. Exercises
1 Solve each homogeneous linear recurrence relations. (a) f (n + 1) = 5f (n) − 6f (n − 1) (b) f (n + 1) = 4f (n − 1) (c) f (n + 1) = 6f (n) + 7f (n − 1) + 6f (n − 2) 2 Give a formula for the relations of the prior exercise, with these initial conditions. (a) f (0) = 1, f (1) = 1 (b) f (0) = 0, f (1) = 1 (c) f (0) = 1, f (1) = 1, f (2) = 3. 3 Check that the isomorphism given betwween S and Rk is a linear map. It is argued above that this map is one-to-one. What is its inverse? 4 Show that the characteristic equation of the matrix is as stated, that is, is the polynomial associated with the relation. (Hint: expanding down the ﬁnal column, and using induction will work.) 5 Given a homogeneous linear recurrence relation f (n + 1) = an f (n) + · · · + an−k f (n − k), let r1 , . . . , rk be the roots of the associated polynomial. n (a) Prove that each function fri (n) = rk satisﬁes the recurrence (without initial conditions). (b) Prove that no ri is 0. (c) Prove that the set {fr1 , . . . , frk } is linearly independent.

Topic: Linear Recurrences

413

6 (This refers to the value T (64) = 18, 446, 744, 073, 709, 551, 615 given in the computer code below.) Transferring one disk per second, how many years would it take the priests at the Tower of Hanoi to ﬁnish the job?

Computer Code This code allows the generation of the ﬁrst few values of a function deﬁned by a recurrence and initial conditions. It is in the Scheme dialect of LISP (speciﬁcally, it was written for A. Jaﬀer’s free scheme interpreter SCM, although it should run in any Scheme implementation). First, the Tower of Hanoi code is a straightforward implementation of the recurrence.
(define (tower-of-hanoi-moves n) (if (= n 1) 1 (+ (* (tower-of-hanoi-moves (- n 1)) 2) 1) ) )

(Note for readers unused to recursive code: to compute T (64), the computer is told to compute 2 ∗ T (63) − 1, which requires, of course, computing T (63). The computer puts the ‘times 2’ and the ‘plus 1’ aside for a moment to do that. It computes T (63) by using this same piece of code (that’s what ‘recursive’ means), and to do that is told to compute 2 ∗ T (62) − 1. This keeps up (the next step is to try to do T (62) while the other arithmetic is held in waiting), until, after 63 steps, the computer tries to compute T (1). It then returns T (1) = 1, which now means that the computation of T (2) can proceed, etc., up until the original computation of T (64) ﬁnishes.) The next routine calculates a table of the ﬁrst few values. (Some language notes: ’() is the empty list, that is, the empty sequence, and cons pushes something onto the start of a list. Note that, in the last line, the procedure proc is called on argument n.)
(define (first-few-outputs proc n) (first-few-outputs-helper proc n ’()) ) ; (define (first-few-outputs-aux proc n lst) (if (< n 1) lst (first-few-outputs-aux proc (- n 1) (cons (proc n) lst)) ) )

The session at the SCM prompt went like this.
>(first-few-outputs tower-of-hanoi-moves 64) Evaluation took 120 mSec (1 3 7 15 31 63 127 255 511 1023 2047 4095 8191 16383 32767 65535 131071 262143 524287 1048575 2097151 4194303 8388607 16777215 33554431 67108863 134217727 268435455 536870911 1073741823 2147483647 4294967295 8589934591 17179869183 34359738367 68719476735 137438953471 274877906943 549755813887

414

Chapter Five. Similarity

1099511627775 2199023255551 4398046511103 8796093022207 17592186044415 35184372088831 70368744177663 140737488355327 281474976710655 562949953421311 1125899906842623 2251799813685247 4503599627370495 9007199254740991 18014398509481983 36028797018963967 72057594037927935 144115188075855871 288230376151711743 576460752303423487 1152921504606846975 2305843009213693951 4611686018427387903 9223372036854775807 18446744073709551615)

This is a list of T (1) through T (64). (The 120 mSec came on a 50 mHz ’486 running in an XTerm of XWindow under Linux. The session was edited to put line breaks between numbers.)

Appendix
Mathematics is made of arguments (reasoned discourse that is, not crockerythrowing). This section is a reference to the most used techniques. A reader having trouble with, say, proof by contradiction, can turn here for an outline of that method. But this section gives only a sketch. For more, these are classics: Methods of Logic by Quine, Induction and Analogy in Mathematics by P´lya, and Naive o Set Theory by Halmos.

IV.3 Propositions
The point at issue in an argument is the proposition. Mathematicians usually write the point in full before the proof and label it either Theorem for major points, Corollary for points that follow immediately from a prior one, or Lemma for results chieﬂy used to prove other results. The statements expressing propositions can be complex, with many subparts. The truth or falsity of the entire proposition depends both on the truth value of the parts, and on the words used to assemble the statement from its parts. Not. For example, where P is a proposition, ‘it is not the case that P ’ is true provided that P is false. Thus, ‘n is not prime’ is true only when n is the product of smaller integers. We can picture the ‘not’ operation with a Venn diagram.

P

Where the box encloses all natural numbers, and inside the circle are the primes, the shaded area holds numbers satisfying ‘not P ’. To prove that a ‘not P ’ statement holds, show that P is false. A-1

A-2 And. Consider the statement form ‘P and Q’. For the statement to be true both halves must hold: ‘7 is prime and so is 3’ is true, while ‘7 is prime and 3 is not’ is false. Here is the Venn diagram for ‘P and Q’.

P

Q

To prove ‘P and Q’, prove that each half holds. Or. A ‘P or Q’ is true when either half holds: ‘7 is prime or 4 is prime’ is true, while ‘7 is not prime or 4 is prime’ is false. We take ‘or’ inclusively so that if both halves are true ‘7 is prime or 4 is not’ then the statement as a whole is true. (In everyday speech, sometimes ‘or’ is meant in an exclusive way — “Eat your vegetables or no dessert” does not intend both halves to hold — but we will not use ‘or’ in that way.) The Venn diagram for ‘or’ includes all of both circles.

P

Q

To prove ‘P or Q’, show that in all cases at least one half holds (perhaps sometimes one half and sometimes the other, but always at least one). If-then. An ‘if P then Q’ statement (sometimes written ‘P materially implies Q’ or just ‘P implies Q’ or ‘P =⇒ Q’) is true unless P is true while Q is false. Thus ‘if 7 is prime then 4 is not’ is true while ‘if 7 is prime then 4 is also prime’ is false. (Contrary to its use in casual speech, in mathematics ‘if P then Q’ does not connote that P precedes Q or causes Q.) More subtly, in mathematics ‘if P then Q’ is true when P is false: ‘if 4 is prime then 7 is prime’ and ‘if 4 is prime then 7 is not’ are both true statements, sometimes said to be vacuously true. We adopt this convention because we want statements like ‘if a number is a perfect square then it is not prime’ to be true, for instance when the number is 5 or when the number is 6. The diagram

P

Q

A-3 shows that Q holds whenever P does (another phrasing is ‘P is suﬃcient to give Q’). Notice again that if P does not hold, Q may or may not be in force. There are two main ways to establish an implication. The ﬁrst way is direct: assume that P is true and, using that assumption, prove Q. For instance, to show ‘if a number is divisible by 5 then twice that number is divisible by 10’, assume that the number is 5n and deduce that 2(5n) = 10n. The second way is indirect: prove the contrapositive statement: ‘if Q is false then P is false’ (rephrased, ‘Q can only be false when P is also false’). As an example, to show ‘if a number is prime then it is not a perfect square’, argue that if it were a square p = n2 then it could be factored p = n · n where n < p and so wouldn’t be prime (of course p = 0 or p = 1 don’t give n < p but they are nonprime by deﬁnition). Note two things about this statement form. First, an ‘if P then Q’ result can sometimes be improved by weakening P or strengthening Q. Thus, ‘if a number is divisible by p2 then its square is also divisible by p2 ’ could be upgraded either by relaxing its hypothesis: ‘if a number is divisible by p then its square is divisible by p2 ’, or by tightening its conclusion: ‘if a number is divisible by p2 then its square is divisible by p4 ’. Second, after showing ‘if P then Q’, a good next step is to look into whether there are cases where Q holds but P does not. The idea is to better understand the relationship between P and Q, with an eye toward strengthening the proposition. Equivalence. An if-then statement cannot be improved when not only does P imply Q, but also Q implies P . Some ways to say this are: ‘P if and only if Q’, ‘P iﬀ Q’, ‘P and Q are logically equivalent’, ‘P is necessary and suﬃcient to give Q’, ‘P ⇐⇒ Q’. For example, ‘a number is divisible by a prime if and only if that number squared is divisible by the prime squared’. The picture here shows that P and Q hold in exactly the same cases.

P

Q

Although in simple arguments a chain like “P if and only if R, which holds if and only if S . . . ” may be practical, typically we show equivalence by showing the ‘if P then Q’ and ‘if Q then P ’ halves separately.

IV.4 Quantiﬁers
Compare these two statements about natural numbers: ‘there is an x such that x is divisible by x2 ’ is true, while ‘for all numbers x, that x is divisible by x2 ’ is false. We call the ‘there is’ and ‘for all’ preﬁxes quantiﬁers.

A-4 For all. The ‘for all’ preﬁx is the universal quantiﬁer, symbolized ∀. Venn diagrams aren’t very helpful with quantiﬁers, but in a sense the box we draw to border the diagram shows the universal quantiﬁer since it dilineates the universe of possible members.

To prove that a statement holds in all cases, we must show that it holds in each case. Thus, to prove ‘every number divisible by p has its square divisible by p2 ’, take a single number of the form pn and square it (pn)2 = p2 n2 . This is a “typical element” or “generic element” proof. This kind of argument requires that we are careful to not assume properties for that element other than those in the hypothesis — for instance, this type of wrong argument is a common mistake: “if n is divisible by a prime, say 2, so that n = 2k then n2 = (2k)2 = 4k 2 and the square of the number is divisible by the square of the prime”. That is an argument about the case p = 2, but it isn’t a proof for general p. There exists. We will also use the existential quantiﬁer, symbolized ∃ and read ‘there exists’. As noted above, Venn diagrams are not much help with quantiﬁers, but a picture of ‘there is a number such that P ’ would show both that there can be more than one and that not all numbers need satisfy P .

P

An existence proposition can be proved by producing something satisfying 5 the property: once, to settle the question of primality of 22 + 1, Euler produced its divisor 641. But there are proofs showing that something exists without saying how to ﬁnd it; Euclid’s argument given in the next subsection shows there are inﬁnitely many primes without naming them. In general, while demonstrating existence is better than nothing, giving an example is better, and an exhaustive list of all instances is great. Still, mathematicians take what they can get. Finally, along with “Are there any?” we often ask “How many?” That is why the issue of uniqueness often arises in conjunction with questions of existence. Many times the two arguments are simpler if separated, so note that just as proving something exists does not show it is unique, neither does proving something is unique show that it exists. (Obviously ‘the natural number with

A-5 more factors than any other’ would be unique, but in fact no such number exists.)

IV.5 Techniques of Proof

Induction. Many proofs are iterative, “Here’s why the statement is true for for the case of the number 1, it then follows for 2, and from there to 3, and so on . . . ”. These are called proofs by induction. Such a proof has two steps. In the base step the proposition is established for some ﬁrst number, often 0 or 1. Then in the inductive step we assume that the proposition holds for numbers up to some k and deduce that it then holds for the next number k + 1. Here is an example.
We will prove that 1 + 2 + 3 + · · · + n = n(n + 1)/2. For the base step we must show that the formula holds when n = 1. That’s easy, the sum of the ﬁrst 1 number does indeed equal 1(1 + 1)/2. For the inductive step, assume that the formula holds for the numbers 1, 2, . . . , k. That is, assume all of these instances of the formula. 1 = 1(1 + 1)/2 and and 1 + 2 = 2(2 + 1)/2 . . . and 1 + · · · + k = k(k + 1)/2 1 + 2 + 3 = 3(3 + 1)/2

From this assumption we will deduce that the formula therefore also holds in the k + 1 next case. The deduction is straightforward algebra. 1 + 2 + · · · + k + (k + 1) = (k + 1)(k + 2) k(k + 1) + (k + 1) = 2 2

We’ve shown in the base case that the above proposition holds for 1. We’ve shown in the inductive step that if it holds for the case of 1 then it also holds for 2; therefore it does hold for 2. We’ve also shown in the inductive step that if the statement holds for the cases of 1 and 2 then it also holds for the next case 3, etc. Thus it holds for any natural number greater than or equal to 1. Here is another example.
We will prove that every integer greater than 1 is a product of primes. The base step is easy: 2 is the product of a single prime. For the inductive step assume that each of 2, 3, . . . , k is a product of primes, aiming to show k + 1 is also a product of primes. There are two

A-6
possibilities: (i) if k + 1 is not divisible by a number smaller than itself then it is a prime and so is the product of primes, and (ii) if k + 1 is divisible then its factors can be written as a product of primes (by the inductive hypothesis) and so k +1 can be rewritten as a product of primes. That ends the proof. (Remark. The Prime Factorization Theorem of Number Theory says that not only does a factorization exist, but that it is unique. We’ve shown the easy half.)

There are two things to note about the ‘next number’ in an induction argument. For one thing, while induction works on the integers, it’s no good on the reals. There is no ‘next’ real. The other thing is that we sometimes use induction to go down, say, from 10 to 9 to 8, etc., down to 0. So ‘next number’ could mean ‘next lowest number’. Of course, at the end we have not shown the fact for all natural numbers, only for those less than or equal to 10. Contradiction. Another technique of proof is to show something is true by showing it can’t be false.
The classic example is Euclid’s, that there are inﬁnitely many primes. Suppose there are only ﬁnitely many primes p1 , . . . , pk . Consider p1 · p2 . . . pk + 1. None of the primes on this supposedly exhaustive list divides that number evenly, each leaves a remainder of 1. But every number is a product of primes so this can’t be. Thus there cannot be only ﬁnitely many primes.

Every proof by contradiction has the same form: assume that the proposition is false and derive some contradiction to known facts.
√ Another example is this proof that 2 is not a rational number. √ Suppose that 2 = m/n. 2n2 = m2 Factor out the 2’s: n = 2kn · n and m = 2km · m and rewrite. ˆ ˆ 2 · (2kn · n)2 = (2km · m)2 ˆ ˆ The Prime Factorization Theorem says that there must be the same number of factors of 2 on both sides, but there are an odd number 1 + 2kn on the left and an even number 2km on the right. That’s a contradiction, so a rational with a square of 2 cannot be.

Both of these examples aimed to prove something doesn’t exist. A negative proposition often suggests a proof by contradiction.

A-7

IV.6 Sets, Functions, and Relations
Sets. Mathematicians work with collections called sets. A set can be given as a listing between curly braces as in {1, 4, 9, 16}, or, if that’s unwieldy, by using set-builder notation as in {x x5 − 3x3 + 2 = 0} (read “the set of all x such that . . . ”). We name sets with capital roman letters as with the primes P = {2, 3, 5, 7, 11, . . . }, except for a few special sets such as the real numbers R, and the complex numbers C. To denote that something is an element (or member ) of a set we use ‘ ∈ ’, so that 7 ∈ {3, 5, 7} while 8 ∈ {3, 5, 7}. What distinguishes a set from any other type of collection is the Principle of Extensionality, that two sets with the same elements are equal. Because of this principle, in a set repeats collapse {7, 7} = {7} and order doesn’t matter {2, π} = {π, 2}. We use ‘⊂’ for the subset relationship: {2, π} ⊂ {2, π, 7} and ‘⊆’ for subset or equality (if A is a subset of B but A = B then A is a proper subset of B). These symbols may be ﬂipped, for instance {2, π, 5} ⊃ {2, 5}. Because of Extensionality, to prove that two sets are equal A = B, just show that they have the same members. Usually we show mutual inclusion, that both A ⊆ B and A ⊇ B. Set operations. Venn diagrams are handy here. For instance, x ∈ P can be pictured

P x

and ‘P ⊆ Q’ looks like this.

P

Q

Note that this is a repeat of the diagram for ‘if . . . then . . . ’ propositions. That’s because ‘P ⊆ Q’ means ‘if x ∈ P then x ∈ Q’. In general, for every propositional logic operator there is an associated set operator. For instance, the complement of P is P comp = {x not(x ∈ P )}

P

A-8 the union is P ∪ Q = {x (x ∈ P ) or (x ∈ Q)}

P

Q

and the intersection is P ∩ Q = {x (x ∈ P ) and (x ∈ Q)}.

P

Q

When two sets share no members their intersection is the empty set {}, symbolized ∅. Any set has the empty set for a subset, by the ‘vacuously true’ property of the deﬁnition of implication. Sequences. We shall also use collections where order does matter and where repeats do not collapse. These are sequences, denoted with angle brackets: 2, 3, 7 = 2, 7, 3 . A sequence of length 2 is sometimes called an ordered pair and written with parentheses: (π, 3). We also sometimes say ‘ordered triple’, ‘ordered 4-tuple’, etc. The set of ordered n-tuples of elements of a set A is denoted An . Thus the set of pairs of reals is R2 . Functions. We ﬁrst see functions in elementary Algebra, where they are presented as formulas (e.g., f (x) = 16x2 − 100), but progressing to more advanced Mathematics reveals more general functions — trigonometric ones, exponential and logarithmic ones, and even constructs like absolute value that involve piecing together parts — and we see that functions aren’t formulas, instead the key idea is that a function associates with its input x a single output f (x). Consequently, a function or map is deﬁned to be a set of ordered pairs (x, f (x) ) such that x suﬃces to determine f (x), that is: if x1 = x2 then f (x1 ) = f (x2 ) (this requirement is referred to by saying a function is well-deﬁned ).∗ Each input x is one of the function’s arguments and each output f (x) is a value. The set of all arguments is f ’s domain and the set of output values is its range. Usually we don’t need know what is and is not in the range and we instead work with a superset of the range, the codomain. The notation for a function f with domain X and codomain Y is f : X → Y .
∗ More

on this is in the section on isomorphisms

A-9

We sometimes instead use the notation x −→ 16x2 − 100, read ‘x maps under f to 16x2 − 100’, or ‘16x2 − 100 is the image of x’. Some maps, like x → sin(1/x), can be thought of as combinations of simple maps, here, g(y) = sin(y) applied to the image of f (x) = 1/x. The composition of g : Y → Z with f : X → Y , is the map sending x ∈ X to g( f (x) ) ∈ Z. It is denoted g ◦ f : X → Z. This deﬁnition only makes sense if the range of f is a subset of the domain of g. Observe that the identity map id : Y → Y deﬁned by id(y) = y has the property that for any f : X → Y , the composition id ◦ f is equal to f . So an identity map plays the same role with respect to function composition that the number 0 plays in real number addition, or that the number 1 plays in multiplication. In line with that analogy, deﬁne a left inverse of a map f : X → Y to be a function g : range(f ) → X such that g ◦ f is the identity map on X. Of course, a right inverse of f is a h : Y → X such that f ◦ h is the identity. A map that is both a left and right inverse of f is called simply an inverse. An inverse, if one exists, is unique because if both g1 and g2 are inverses of f then g1 (x) = g1 ◦ (f ◦ g2 )(x) = (g1 ◦ f ) ◦ g2 (x) = g2 (x) (the middle equality comes from the associativity of function composition), so we often call it “the” inverse, written f −1 . For instance, the inverse of the function f : R → R given by f (x) = 2x − 3 is the function f −1 : R → R given by f −1 (x) = (x + 3)/2. The superscript ‘f −1 ’ notation for function inverse can be confusing — it doesn’t mean 1/f (x). It is used because it ﬁts into a larger scheme. Functions that have the same codomain as domain can be iterated, so that where f : X → X, we can consider the composition of f with itself: f ◦ f , and f ◦ f ◦ f , etc. Naturally enough, we write f ◦ f as f 2 and f ◦ f ◦ f as f 3 , etc. Note that the familiar exponent rules for real numbers obviously hold: f i ◦ f j = f i+j and (f i )j = f i·j . The relationship with the prior paragraph is that, where f is invertible, writing f −1 for the inverse and f −2 for the inverse of f 2 , etc., gives that these familiar exponent rules continue to hold, once f 0 is deﬁned to be the identity map. If the codomain Y equals the range of f then we say that the function is onto. A function has a right inverse if and only if it is onto (this is not hard to check). If no two arguments share an image, if x1 = x2 implies that f (x1 ) = f (x2 ), then the function is one-to-one. A function has a left inverse if and only if it is one-to-one (this is also not hard to check). By the prior paragraph, a map has an inverse if and only if it is both onto and one-to-one; such a function is a correspondence. It associates one and only one element of the domain with each element of the range (for example, ﬁnite

f

A-10 sets must have the same number of elements to be matched up in this way). Because a composition of one-to-one maps is one-to-one, and a composition of onto maps is onto, a composition of correspondences is a correspondence. We sometimes want to shrink the domain of a function. For instance, we may take the function f : R → R given by f (x) = x2 and, in order to have an ˆ inverse, limit input arguments to nonnegative reals f : R+ → R. Technically, ˆ f is a diﬀerent function than f ; we call it the restriction of f to the smaller domain. A ﬁnal point on functions: neither x nor f (x) need be a number. As an example, we can think of f (x, y) = x + y as a function that takes the ordered pair (x, y) as its argument. Relations. Some familiar operations are obviously functions: addition maps (5, 3) to 8. But what of ‘<’ or ‘=’ ? We here take the approach of rephrasing ‘3 < 5’ to ‘(3, 5) is in the relation <’. That is, deﬁne a binary relation on a set A to be a set of ordered pairs of elements of A. For example, the < relation is the set {(a, b) a < b}; some elements of that set are (3, 5), (3, 7), and (1, 100). Another binary relation on the natural numbers is equality; this relation is formally written as the set {. . . , (−1, −1), (0, 0), (1, 1), . . .}. Still another example is ‘closer than 10’, the set {(x, y) |x − y| < 10}. Some members of that relation are (1, 10), (10, 1), and (42, 44). Neither (11, 1) nor (1, 11) is a member. Those examples illustrate the generality of the deﬁnition. All kinds of relationships (e.g., ‘both numbers even’ or ‘ﬁrst number is the second with the digits reversed’) are covered under the deﬁnition. Equivalence Relations. We shall need to say, formally, that two objects are alike in some way. While these alike things aren’t identical, they are related (e.g., two integers that ‘give the same remainder when divided by 2’). A binary relation {(a, b), . . .} is an equivalence relation when it satisﬁes (1) reﬂexivity: any object is related to itself; (2) symmetry: if a is related to b then b is related to a; (3) transitivity: if a is related to b and b is related to c then a is related to c. (To see that these conditions formalize being the same, read them again, replacing ‘is related to’ with ‘is like’.) Some examples (on the integers): ‘=’ is an equivalence relation, ‘<’ does not satisfy symmetry, ‘same sign’ is a equivalence, while ‘nearer than 10’ fails transitivity. Partitions. In ‘same sign’ {(1, 3), (−5, −7), (−1, −1), . . .} there are two kinds of pairs, the ﬁrst with both numbers positive and the second with both negative. So integers fall into exactly one of two classes, positive or negative. A partition of a set S is a collection of subsets {S1 , S2 , . . .} such that every element of S is in one and only one Si : S1 ∪ S2 ∪ . . . = S, and if i is not equal to j then Si ∩ Sj = ∅. Picture S being decomposed into distinct parts.

A-11

S1 S0 S3

S2

...

Thus, the ﬁrst paragraph says ‘same sign’ partitions the integers into the positives and the negatives. Similarly, the equivalence relation ‘=’ partitions the integers into one-element sets. Another example is the fractions. Of course, 2/3 and 4/6 are equivalent fractions. That is, for the set S = {n/d n, d ∈ Z and d = 0}, we deﬁne two elements n1 /d1 and n2 /d2 to be equivalent if n1 d2 = n2 d1 . We can check that this is an equivalence relation, that is, that it satisﬁes the above three conditions. With that, S is divided up into parts.
.1/1 .2/2 .0/1 .0/3 .4/3 .8/6 .2/4 .−2/−4

...

Before we show that equivalence relations always give rise to partitions, we ﬁrst illustrate the argument. Consider the relationship between two integers of ‘same parity’, the set {(−1, 3), (2, 4), (0, 0), . . .} (i.e., ‘give the same remainder when divided by 2’). We want to say that the natural numbers split into two pieces, the evens and the odds, and inside a piece each member has the same parity as each other. So for each x we deﬁne the set of numbers associated with it: Sx = {y (x, y) ∈ ‘same parity’}. Some examples are S1 = {. . . , −3, −1, 1, 3, . . .}, and S4 = {. . . , −2, 0, 2, 4, . . .}, and S−1 = {. . . , −3, −1, 1, 3, . . .}. These are the parts, e.g., S1 is the odds. Theorem. An equivalence relation induces a partition on the underlying set.
Proof. Call the set S and the relation R. In line with the illustration in the

paragraph above, for each x ∈ S deﬁne Sx = {y (x, y) ∈ R}. Observe that, as x is a member if Sx , the union of all these sets is S. So we will be done if we show that distinct parts are disjoint: if Sx = Sy then Sx ∩ Sy = ∅. We will verify this through the contrapositive, that is, we wlll assume that Sx ∩ Sy = ∅ in order to deduce that Sx = Sy . Let p be an element of the intersection. Then by deﬁnition of Sx and Sy , the two (x, p) and (y, p) are members of R, and by symmetry of this relation (p, x) and (p, y) are also members of R. To show that Sx = Sy we will show each is a subset of the other. Assume that q ∈ Sx so that (q, x) ∈ R. Use transitivity along with (x, p) ∈ R to conclude that (q, p) is also an element of R. But (p, y) ∈ R so another use of transitivity gives that (q, y) ∈ R. Thus q ∈ Sy . Therefore q ∈ Sx implies q ∈ Sy , and so Sx ⊆ Sy . The same argument in the other direction gives the other inclusion, and so the two sets are equal, completing the contrapositive argument. QED

A-12 We call each part of a partition an equivalence class (or informally, ‘part’). We somtimes pick a single element of each equivalence class to be the class representative.

...

Usually when we pick representatives we have some natural scheme in mind. In that case we call them the canonical representatives. An example is the simplest form of a fraction. We’ve deﬁned 3/5 and 9/15 to be equivalent fractions. In everyday work we often use the ‘simplest form’ or ‘reduced form’ fraction as the class representatives.
1/1 1/2

0/1 4/3

...

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Index
accuracy of Gauss’ method, 68–71 rounding error, 69 addition vector, 80 additive inverse, 80 adjoint matrix, 328 angle, 42 antipodal, 340 antisymmetric matrix, 138 argument, A-8 arrow diagram, 217, 233, 238, 242, 353 augmented matrix, 14 automorphism, 163 dilation, 163 reﬂection, 163 rotation, 163 back-substitution, 5 base step of induction, A-5 basis, 112–123 change of, 238 deﬁnition, 112 orthogonal, 256 orthogonalization, 257 orthonormal, 258 standard, 113, 352 standard over the complex numbers, 352 string, 373 best ﬁt line, 269 block matrix, 311 box, 321 orientation, 321 sense, 321 volume, 321 C language, 68 canonical form for matrix equivalence, 245 for nilpotent matrices, 376 for row equivalence, 58 for similarity, 394 canonical representative, A-12 Cauchy-Schwartz Inequality, 41 Cayley-Hamilton theorem, 384 central projection, 337 change of basis, 238–249 characteristic vectors, values, 359 characteristic equation, 362 characteristic polynomial, 362 characterized, 172 characterizes, 246 Chemistry problem, 1, 9 chemistry problem, 22 circuits parallel, 73 series, 73 series-parallel, 74 class equivalence, A-12 closure, 95 of nullspace, 369 of rangespace, 369 codomain, A-8 cofactor, 327 column, 13 rank, 125 vector, 15 column rank full, 130 column space, 125 complement, A-7 complementary subspaces, 135 orthogonal, 263 complex numbers vector space over, 90 component, 15

composition, A-9 self, 367 computer algebra systems, 62–63 concatenation, 133 conditioning number, 71 congruent ﬁgures, 287 congruent plane ﬁgures, 287 contradiction, A-6 contrapositive, A-3 convex set, 183 coordinates homogeneous, 340 with respect to a basis, 115 corollary, A-1 correspondence, 161, A-9 coset, 193 Cramer’s rule, 331–333 cross product, 298 crystals, 142–145 diamond, 143 graphite, 143 salt, 142 unit cell, 143 da Vinci, Leonardo, 337 determinant, 294, 299–318 cofactor, 327 Cramer’s rule, 332 deﬁnition, 299 exists, 309, 315 Laplace expansion, 327 minor, 327 permutation expansion, 308, 312, 334 diagonal matrix, 210, 225 diagonalizable, 356–359 diﬀerence equation, 408 homogeneous, 408 dilation, 163, 276 representing, 203 dimension, 120 physical, 152 direct map, 290 direct sum, 131 ), 139 deﬁnition, 135 external, 168 internal, 168 of two subspaces, 135 direction vector, 35 distance-preserving, 287

division theorem, 350 domain, A-8 dot product, 40 double precision, 69 dual space, 194 echelon form, 5 free variable, 12 leading variable, 5 reduced, 47 eigenspace, 362 eigenvalue, eigenvector of a matrix, 360 of a transformation, 359 element, A-7 elementary matrix, 227, 275 elementary reduction operations, 4 pivoting, 4 rescaling, 4 swapping, 4 elementary row operations, 4 empty set, A-8 entry, 13 equivalence class, A-12 canonical representative, A-12 relation, A-10 representative, A-12 equivalence relation, A-10, A-11 isomorphism, 169 matrix equivalence, 244 matrix similarity, 353 row equivalence, 50 equivalent statements, A-3 Erlanger Program, 287 Euclid, 287 even functions, 99, 137 even polynomials, 400 external direct sum, 168 Fibonacci sequence, 407 ﬁeld, 140–141 deﬁnition, 140 ﬁnite-dimensional vector space, 119 ﬂat, 36 form, 56 free variable, 12 full column rank, 130 full row rank, 130

function, A-8 inverse image, 185 argument, A-8 codomain, A-8 composition, 216, A-9 correspondence, A-9 domain, A-8 even, 99 identity, A-9 inverse, 232, A-9 left inverse, 231 multilinear, 304 odd, 99 one-to-one, A-9 onto, A-9 range, A-8 restriction, A-10 right inverse, 231 structure preserving, 161, 165 seehomomorphism, 176 two-sided inverse, 232 value, A-8 well-deﬁned, A-8 zero, 177 Fundamental Theorem of Linear Algebra, 268 Gauss’ method, 2 accuracy, 68–71 back-substitution, 5 elementary operations, 4 Gauss-Jordan, 47 Gauss-Jordan, 47 generalized nullspace, 369 generalized rangespace, 369 Geometry of Linear Maps, 274–280 Gram-Schmidt process, 255–260 historyless Markov chain, 281 homogeneous coordinate vector, 340 homogeneous coordinates, 292 homogeneous equation, 21 homomorphism, 176 composition, 216 matrix representing, 195–205 nonsingular, 190, 208 nullity, 188 nullspace, 188 rangespace, 184

rank, 207 zero, 177 ideal line, 343 ideal point, 343 identity function, A-9 matrix, 225 if-then statement, A-2 ill-conditioned, 69 image under a function, A-9 improper subspace, 92 incidence matrix, 229 index of nilpotency, 372 induction, 23, A-5 inductive step of induction, A-5 inherited operations, 82 inner product, 40 Input-Output Analysis, 64–67 internal direct sum, 135, 168 intersection, A-8 invariant subspace, 379 invariant subspace deﬁnition, 391 inverse, 232, A-9 additive, 80 exists, 232 left, 232, A-9 matrix, 329 right, 232, A-9 two-sided, A-9 inverse function, 232 inverse image, 185 inversion, 313 isometry, 287 isomorphism, 159–175 characterized by dimension, 172 deﬁnition, 161 of a space with itself, 163 Jordan block, 390 Jordan form, 381–400 represents similarity classes, 394 kernel, 188 Kirchhoﬀ’s Laws, 73

Klein, F., 287 Laplace expansion, 326–330 computes determinant, 327 leading variable, 5 least squares, 269–273 lemma, A-1 length, 39 Leontief, W., 64 line best ﬁt, 269 in projective plane, 341 line at inﬁnity, 343 line of best ﬁt, 269–273 linear transpose operation, 130 linear combination, 52 Linear Combination Lemma, 53 linear equation, 2 coeﬃcients, 2 constant, 2 homogeneous, 21 inconsistent systems, 269 satisﬁed by a vector, 15 solution of, 2 Gauss’ method, 3 Gauss-Jordan, 47 solutions of Cramer’s rule, 332 system of, 2 linear map dilation, 276 reﬂection, 290 rotation, 274, 290 seehomomorphism, 176 skew, 277 trace, 400 linear recurrence, 408 linear recurrences, 407–414 linear relationship, 102 linear surface, 36 linear transformation seetransformation, 179 linearly dependent, 102 linearly independent, 102 LINPACK, 62 map, A-8 distance-preserving, 287 extended linearly, 173

self composition, 367 Maple, 62 Markov chain, 281 historyless, 281 Markov chains, 281–286 Markov matrix, 285 material implication, A-2 Mathematica, 62 mathematical induction, 23, A-5 MATLAB, 62 matrix, 13 adjoint, 328 antisymmetric, 138 augmented, 14 block, 245, 311 change of basis, 238 characteristic polynomial, 362 cofactor, 327 column, 13 column space, 125 conditioning number, 71 determinant, 294, 299 diagonal, 210, 225 diagonalizable, 356 diagonalized, 244 elementary reduction, 227, 275 entry, 13 equivalent, 244 identity, 221, 225 incidence, 229 inverse, 329 main diagonal, 225 Markov, 230, 285 matrix-vector product, 198 minimal polynomial, 221, 382 minor, 327 multiplication, 216 nilpotent, 372 nonsingular, 27, 208 orthogonal, 289 orthonormal, 287–292 permutation, 226 rank, 207 representation, 197 row, 13 row equivalence, 50 row rank, 124 row space, 124 scalar multiple, 213 similar, 324

similarity, 353 singular, 27 skew-symmetric, 311 submatrix, 303 sum, 213 symmetric, 118, 138, 214, 221, 229, 268 trace, 214, 230, 400 transition, 281 transpose, 19, 126, 214 triangular, 204, 230, 330 unit, 223 Vandermonde, 311 matrix equivalence, 242–249 canonical form, 245 deﬁnition, 244 matrix:form, 56 mean arithmetic, 44 geometric, 44 member, A-7 method of powers, 401–404 minimal polynomial, 221, 382 minor, 327 morphism, 161 multilinear, 304 multiplication matrix-matrix, 216 matrix-vector, 198 MuPAD, 62 mutual inclusion, A-7 natural representative, A-12 networks, 72–77 Kirchhoﬀ’s Laws, 73 nilpotent, 370–380 canonical form for, 376 deﬁnition, 372 matrix, 372 transformation, 372 nilpotentcy index, 372 nonsingular, 208, 232 homomorphism, 190 matrix, 27 normalize, 258 nullity, 188 nullspace, 188 closure of, 369 generalized, 369

Octave, 62 odd functions, 99, 137 one-to-one function, A-9 onto function, A-9 opposite map, 290 order of a recurrence, 408 ordered pair, A-8 orientation, 321, 324 orthogonal, 42 basis, 256 complement, 263 mutually, 255 projection, 263 orthogonal matrix, 289 orthogonalization, 257 orthonormal basis, 258 orthonormal matrix, 287–292 pair ordered, A-8 parallelepiped, 321 parallelogram rule, 34 parameter, 13 partial pivoting, 70 partition, A-10–A-12 matrix equivalence classes, 244, 246 row equivalence classes, 51 partitions into isomorphism classes, 170 permutation, 307 inversions, 313 matrix, 226 signum, 314 permutation expansion, 308, 312, 334 perp, 263 perpendicular, 42 perspective triangles, 343 Physics problem, 1 pivoting full, 70 partial scaled, 70 pivoting on rows, 4 plane ﬁgure, 287 congruence, 287 point at inﬁnity, 343 in projective plane, 340

polynomial even, 400 minimal, 382 of map, matrix, 381 polynomials division theorem, 350 populations, stable, 405–406 potential, 72 powers, method of, 401–404 preserves structure, 176 probability vector, 281 projection, 176, 185, 250, 268, 387 along a subspace, 260 central, 337 vanishing point, 337 into a line, 251 into a subspace, 260 orthogonal, 251, 263 Projective Geometry, 337–347 projective geometry Duality Principle, 342 projective plane ideal line, 343 ideal point, 343 lines, 341 proof techniques induction, 23 proper subset, A-7 proper subspace, 92 proposition, A-1 propositions equivalent, A-3 quantiﬁer, A-3 existential, A-4 universal, A-4 quantiﬁers, A-3 range, A-8 rangespace, 184 closure of, 369 generalized, 369 rank, 127, 207 column, 125 of a homomorphism, 184, 189 recurrence, 327, 408 homogeneous, 408 initial conditions, 408 reduced echelon form, 47

reﬂection, 290 glide, 290 reﬂection (or ﬂip) about a line, 163 reﬂexivity, A-10 relation, A-10 equivalence, A-10 reﬂexive, A-10 symmetric, A-10 transitive, A-10 relationship linear, 102 representation of a matrix, 197 of a vector, 115 representative, A-12 canonical, A-12 for row equivalence classes, 58 of matrix equivalence classes, 245 of similarity classes, 395 rescaling rows, 4 resistance, 72 resistance:equivalent, 76 resistor, 72 restriction, A-10 rigid motion, 287 rotation, 274, 290 rotation (or turning), 163 represented, 200 row, 13 rank, 124 vector, 15 row equivalence, 50 row rank full, 130 row space, 124 scalar, 80 scalar multiple matrix, 213 vector, 15, 34, 80 scalar product, 40 scaled partial pivoting, 70 Schwartz Inequality, 41 SciLab, 62 self composition of maps, 367 sense, 321 sequence, A-8 concatenation, 133 set, A-7

complement, A-7 element, A-7 empty, A-8 intersection, A-8 member, A-7 union, A-8 sets, A-7 dependent, independent, 102 empty, 104 mutual inclusion, A-7 proper subset, A-7 span of, 95 subset, A-7 sgn seesignum, 314 signum, 314 similar, 298, 324 canonical form, 394 similar matrices, 353 similar triangles, 290 similarity, 353–366 similarity transformation, 366 single precision, 68 singular matrix, 27 size, 319, 321 skew, 277 skew-symmetric, 311 span, 95 of a singleton, 99 spin, 148 square root, 400 stable populations, 405–406 standard basis, 113 state, 281 absorbtive, 281 Statics problem, 5 string, 373 basis, 373 of basis vectors, 371 structure preservation, 176 submatrix, 303 subspace, 91–100 closed, 93 complementary, 135 deﬁnition, 91 direct sum, 135 improper, 92 independence, 135

invariant, 391 proper, 92 sum, 131 sum of matrices, 213 of subspaces, 131 vector, 15, 34, 80 summation notation for permutation expansion, 308 swapping rows, 4 symmetric matrix, 118, 138, 214, 221 symmetry, A-10 system of linear equations, 2 Gauss’ method, 2 solving, 2 theorem, A-1 trace, 214, 230, 400 transformation characteristic polynomial, 362 composed with itself, 367 diagonalizable, 356 eigenspace, 362 eigenvalue, eigenvector, 359 Jordan form for, 394 minimal polynomial, 382 nilpotent, 372 canonical representative, 376 projection, 387 size change, 321 transition matrix, 281 transitivity, A-10 translation, 287 transpose, 19, 126 determinant, 309, 317 interaction with sum and scalar multiplication, 214 Triangle Inequality, 40 triangles similar, 290 triangular matrix, 230 Triangularization, 204 trivial space, 83, 113 turning map, 163 union, A-8 unit matrix, 223 vacuously true, A-2 value, A-8

Vandermonde matrix, 311 vanishing point, 337 vector, 15, 33 angle, 42 canonical position, 34 column, 15 component, 15 cross product, 298 direction, 35 dot product, 40 free, 33 homogeneous coordinate, 340 length, 39 orthogonal, 42 probability, 281 representation of, 115, 238 row, 15 satisﬁes an equation, 15 scalar multiple, 15, 34, 80 sum, 15, 34, 80 unit, 44 zero, 22, 80 vector space, 80–100 basis, 112 closure, 80 complex scalars, 90 deﬁnition, 80 dimension, 120 dual, 194 ﬁnite dimensional, 119 homomorphism, 176 isomorphism, 161 map, 176 over complex numbers, 349 subspace, 91 trivial, 83, 113 Venn diagram, A-1 voltage drop, 73 volume, 321 voting paradox, 146 majority cycle, 146 rational preference order, 146 voting paradoxes, 146–151 spin, 148 well-deﬁned, A-8 Wheatstone bridge, 74 zero divisor, 221

zero zero zero zero

divison, 237 divisor, 221 homomorphism, 177 vector, 22, 80