A (Terse) Introduction t o Linear Algebra

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STUDENT MATHEMATICAL LIBRARY Volume 44

A (Terse) Introduction t o Linear Algebra

Yitzhak Katznelso n Yonatan R . Katznelson

#AMS AMERICAN MATHEMATICA L SOCIET Y

Providence, Rhode Islan d

Editoria l B o a r d

Gerald B . Follan d Bra d G . Osgoo d Robin Forma n (Chair ) Michae l Starbir d

2000 Mathematics Subject Classification. Primar y 15-01 .

The cove r ar t i s create d b y Noa h Katznelso n

and use d wit h permission .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages /s tml-44

Library o f Congres s Cataloging-in-Publicatio n D a t a

Katznelson, Yitzhak , 1934-A (terse ) introductio n t o linea r algebr a / Yitzha k Katznelson , Yonata n R .

Katznelson. p. cm . — (Studen t mathematica l library , ISS N 1520-912 1 ; v. 44)

Includes index . ISBN 978-0-8218-4419- 9 (alk . paper) 1. Algebras , Linear . I . Katznelson, Yonata n R. , 1961 - II . Title. III . Title:

Introduction t o linear algebra .

QA184.2.K38 200 8 512'.5—de22 2007060571

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10 9 8 7 6 5 4 3 2 1 1 3 12 11 10 09 0 8

Contents

Preface i x

1 Vecto r Spaces 1 1.1 Group s and fields 1 1.2 Vecto r spaces 4 1.3 Linea r dependence, bases, and dimension 1 4 1.4 System s of linear equations 2 2

*1.5 Norme d finite-dimensional linear spaces 3 2

2 Linea r Operators and Matrices 3 5 2.1 Linea r operators 3 5 2.2 Operato r multiplication 3 9 2.3 Matri x multiplication 4 1 2.4 Matrice s and operators 4 6 2.5 Kernel , range, nullity, and rank 5 1

*2.6 Operato r norms 5 6

3 Dualit y of Vector Spaces 5 7 3.1 Linea r functionals 5 7 3.2 Th e adjoint 6 2

4 Determinant s 6 5 4.1 Permutation s 6 5 4.2 Multilinea r maps 6 9 4.3 Alternatin g ra-forms 7 4 4.4 Determinan t of an operator 7 6 4.5 Determinan t of a matrix 7 9

V

vi Content s

5 Invarian t Subspaces 8 5 5.1 Th e characteristic polynomial 8 5 5.2 Invarian t subspaces 8 8 5.3 Th e minimal polynomial 9 3

6 Inner-Produc t Spaces 10 3 6.1 Inne r products 10 3 6.2 Dualit y and the adjoint I l l 6.3 Self-adjoin t operator s 11 3 6.4 Norma l operators 11 9 6.5 Unitar y and orthogonal operators 12 1

*6.6 Positiv e definite operators 12 7 *6.7 Pola r decomposition 12 8 *6.8 Contraction s and unitary dilations 13 2

7 Structur e Theorems 13 5 7.1 Reducin g subspaces 13 5 7.2 Semisimpl e systems 14 2 7.3 Nilpoten t operators 14 7 7.4 Th e Jordan canonical form 15 1

*7.5 Th e cyclic decomposition, general case 15 2 *7.6 Th e Jordan canonical form, general case 15 6

8 Additiona l Topics 15 9 8.1 Function s of an operator 15 9 8.2 Quadrati c forms 16 2 8.3 Perron-Frobeniu s theory 16 6 8.4 Stochasti c matrices 17 8 8.5 Representatio n of finite groups 18 0

A Appendi x 18 7 A.l Equivalenc e relations-partitions 18 7 A.2 Map s 18 8 A.3 Group s 18 9

*A.4 Grou p actions 19 4 A.5 Ring s and algebras 19 6

Contents VII

A.6 Polynomial s 20 1

Index 21 1

Symbols 21 5

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Preface

This book is a presentation of the elements of Linear Algebra that every mathematician , an d everyon e wh o use s mathematics , shoul d know. I t cover s th e core material , fro m th e basic notio n o f a finite-dimensional vector space over a general field, to the canonical form s of linear operators and their matrices, obtained by the decomposition of a general linear system into the direct sum of cyclic systems. Along the way it covers such key topics as: system s of linear equations, lin-ear operators an d matrices , determinants , duality , inne r products an d the spectral theory of operators on inner-product spaces. We conclude with a selection of additional topics, indicating some of the directions in which the core material can be applied and developed.

In it s mathematica l prerequisite s th e boo k i s elementary , i n th e sense that no previous knowledge o f linear algebr a i s assumed. I t is self-contained, an d includes a n appendix tha t provides al l the neces-sary background material : th e very basic properties o f groups , rings, and of the algebra of polynomials over a field. The book is intended, however, fo r readers with som e mathematical maturit y an d readiness to deal with abstraction and formal reasoning. I t is appropriate for an advanced undergraduate course.

As the title implies, the styl e of the book is somewha t terse . W e mean this in two senses.

First, we focus with few digressions on the principal ideas and re-sults of linear algebra qua linear algebra. The book contains fewer rou-tine numerical examples than do many other texts , and offers almos t no interspersed application s t o other fields; these shoul d b e adapte d to the readership and, if the book is used in a course, provided by the teacher.

IX

X Preface

Second, the writing itsel f tend s to be concise and to the point, to the extent that some of the proofs might be better described as detailed lists of hints. Thi s is intentional—we believe that students learn more by having to fill in some details themselves.

Besides it s style , thi s book differ s fro m man y othe r text s o n the subject in that we try to present the main ideas, whenever possible, in the context of vector spaces over a general field, F, rather than assum-ing the underlying field to be R or C. Inner-product spaces, along with the naturally associated classes of self-adjoint, normal , and unitary (or orthogonal) operators , ar e introduced late r tha n i n man y books , an d the spectral theorems for these operators, besides being fundamentall y important on their own, also serve here to pave the way for the notions of reducing and semisimplicity and, eventually, to the general structur e theorems—the Jordan form, when the underlying field is algebraically closed, and the corresponding form over general fields.

The tex t consist s o f eigh t chapter s an d a n appendix . Thes e ar e divided into sections, and further int o subsections. Definitions, propo-sitions, examples , etc. , ar e numbered accordin g t o the subsectio n i n which they appear , and no subsection has more than one object (defi -nition, theorem, etc.) o f each kind. Fo r example, Lemma 1.3. 5 i s the lemma appearin g i n subsectio n 1.3.5 , an d Theorem 1.3. 5 i s the the-orem appearin g i n the sam e subsection . Reference s t o the appendi x have the form A.x.y (for subsection y of section x, in the appendix).

Exercises appear at the end of sections, and are numbered accord -ingly, e.g., exercise ex3.1.2 is the second exercise of section 3.1.

Starred sections , subsections , an d exercises contai n materia l tha t can be skipped on first reading. Severa l of these sections , as well as parts of the additional topics (i n Chapter 8) , require some familiarit y with basic analysis, e.g., concepts like convergence and continuity.

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Index

Adjoint of a matrix, 112 of an operator, 63, 112

Algebra, 200 Alternating form, 72 Annihilator, 59 Automorphism, 7 Axiom of choice, 20

Basic partition, 175 Basis, 15

dual, 58 standard, 17

Bilinear form, 59, 63, 69 map, 69

Canonical prime-power decomposition,

140 Cauchy-Schwarz, 105 Cayley-Hamilton, 95 Character, 124 Characteristic polynomial

of a matrix, 86 of an operator, 85

Codimension, 18 Cofactor, 80 Complement, 11 Composition, 39 Congruent

matrices, 163 Conjugate

matrices, 43

operators, 40 Conjugation

group actions, 196 in S„, 67

Connecting chain, 171 Contraction, 132 Coset, 10 , 192 Cycle, 65, 66 Cyclic

decomposition, 152 system, 94 vector, 94

Decomposition cyclic, 150 , 152 general], 138 prime power, 140

Degrees of freedom, 28 Determinant

of a matrix, 79 of an operator, 76

Diagonal matrix, 44 Diagonal sum, 81,136 Diagonalizable, 49 Dimension, 17 , 18 Direct sum

formal, 1 0 of subspaces, 11

Eigenspace, 86 generalized, 142

Eigenvalue, 64, 86, 89 Eigenvector, 64, 86, 89

dominant, 168

211

212 Index

Elementary divisors, 155 Equivalence relation, 187 Euclidean space, 103

Factorization inR[jc],206 prime-power, 140, 141, 204

Field, 2 extension, 200

Fixed point, 65 Flag, 92 Fourier expansion, 125 Frobenius, 175

Gauss-Jordan elimination, 26 Gaussian elimination, 24 Group, 1 , 189

abelian, 1 cyclic, 192 dual, 125 general linear, 40, 43

Hadamard's inequality, 110 Hamel basis, 20 Hermitian

form, 103 , 163

Ideal, 197 Idempotent, 41, 11 0 Independent

subspaces, 11 , 13 vectors, 14

Inertia, law of, 16 5 Inner product, 103 Irreducible

polynomial, 203 system, 135

Isomorphism, 6

Jordan canonical form, 151 , 156

k-form, 69 Kernel, 51

Lagrange, 62 Linear

combination, 9 operator, 35 system, 40, 49

Linear equations homogeneous, 22 nonhomogeneous, 22

Markov chain, 178 reversible, 179

Matrix augmented, 25 companion, 96 derogatory, 97 diagonal, 9 Hermitian, 113 integral, 82 nonderogatory, 97 nonnegative, 166 , 170 orthogonal, 122 permutation, 44 positive, 167 self-adjoint, 11 3 skew-symmetric, 9 stochastic, 178 strongly transitive, 175 symmetric, 8 transitive, 171 triangular, 9, 81,92 unimodular, 82 unipotent, 151 unitary , 122

Minimal system, 100

Minimal polynomial, 96 for (T,v), 93

Index 213

Minmax principle, 118 Monic polynomial, 201 Multilinear

form, 69 map, 69

Nilpotent, 147 Nilspace, 142 Norm

of an operator, 56 on a vector space, 32

Normal operator, 119

Nullity, 51 Nullspace, 52

Operator derogatory, 97 induced on a quotient, 78 linear, 35 nonderogatory, 97 nonnegative definite, 12 7 nonsingular, 52 normal, 119 orthogonal, 121 positive definite, 12 7 self-adjoint, 11 3 singular, 52 unitary, 121

Order element, 192 group, 189

Orientation, 77 Orthogonal

operator, 121 projection, 10 8 vectors, 106

Orthogonal equivalence, 122 Orthonormal, 106

Period group, 174 Periodicity, 174 Permutation, 65, 189 Permutation matrix, 44 Perron, 168 Pivot column, 26 Polar decomposition, 128 Polarization, 111 Primary components, 140 Probability vector, 178 Projection, 41

along a subspace, 36 orthogonal, 108

Quadratic form, 16 2 positive definite, 16 5

Quotient space, 9

Range, 51 Rank

column, 29 of a matrix, 29 of an operator, 51 row, 25

Reduced-row-echelon form, 26 Reducing subspace, 135 Regular representation, 185 Representation, 181

equivalent, 182 faithful, 18 1 reducible, 184 regular, 185 unitary, 181

Restriction of an operator, 78 Return times, 171 Ring, 196 Row equivalence, 25

Schur decomposition, 123 Schur's lemma, 100

214

Self-adjoint algebra, 117 operator, 113

Semisimple algebra, 144 system, 142

Shift ik-shift, 14 8 standard, 148

Similar matrices, 49 operators, 49

Singular value decomposition, Singular values, 130 Solution-set, 8, 23 Span, 9, 14 Spectral mapping theorem, 89 Spectral norm, 162 , 167 Spectral Theorems, 114-12 0 Spectrum, 89, 141

joint, 117 of a matrix, 86 of an operator, 85

Square-free, 14 3 Steinitz' lemma, 17 Stochastic, 178 Subalgebra, 200 Subgroup

index, 192 normal, 193

Submatrix, 30 principal, 30

Support, 167 Sylvester matrix, 32 Symmetric form, 72 Symmetric group, 65

Tensor product, 12 Trace, 87 Transition matrix, 178

Index

Transposition, 66

Unitary group, 183 operator, 121 space, 103

Unitary dilation, 132 Unitary equivalence, 122

Vandermonde, 82 Vector space, 4

complex, 4 real, 4

Symbols

C, 3 Q,3 R, 3 T*, 124 Z 2 ,3 Z p ,3

1 7 5

ACT,44 A r ,46 A1, 36

C([0,l]),6 C~( [ - l , l ] ) , 6 CR([0,1]),6

XT>85 C(X) , 6 commfr], 4 1 Cv, 38 Cw,v, 48

d i m r , 1 8

F",4 F[x], 5 F^M, 8 Ff^,. . . ,^]^

Gh(Jf), 18 3 G L ( y ) , 4 0

height[v], 147 H0M{Y,W),2>1

3?(-T), 3 7

3f(T,W), 37

m, no JK{n;¥), 5 JK{n,m\¥), 5 minPT, 96 minPTv, 94 J({n,T), 8 2 ^^({^•}J=„5r),70 ^ r j f (r®*) , 70 ^rjs%„(r®*),72 Jt%alt{r®

k),i2

0{n), 12 2

TT ,̂ 108 &(T), 40 , 98

p(A), 29

S„, 2, 65 span [is], 9 span[7>], 88 II Wsp , 1 6 7

Typ, 78 Ty , 36

^ ( / i ) , 12 2

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Titles i n Thi s Serie s

44 Yitzha k Katznelso n an d Yonata n R . Katznelson , A (terse )

introduction t o linea r algebra , 200 8

43 Ilk a Agricol a an d Thoma s Friedrich , Elementar y geometry , 200 8

42 C . E . Silva , Invitatio n t o ergodi c theory , 200 7 41 Gar y L . Mul le n an d Car l M u m m e r t , Finit e fields an d applications ,

2007

40 Deguan g Han , Ker i Kornelson , Davi d Larson , an d Eri c Weber , Frames fo r undergraduates , 200 7

39 A le x losevich , A vie w fro m th e top : Analysis , combinatoric s an d numbe r theory, 200 7

38 B . Fristedt , N . Jain , an d N . Krylov , Filterin g an d prediction : A

primer, 200 7

37 Svet lan a Katok , p-adi c analysi s compare d wit h real , 200 7

36 Mar a D . Neuse l , Invarian t theory , 200 7

35 Jor g Bewersdorff , Galoi s theor y fo r beginners : A historica l perspective ,

2006

34 Bruc e C . Berndt , Numbe r theor y i n th e spiri t o f Ramanujan , 200 6

33 Rekh a R . Thomas , Lecture s i n geometri c combinatorics , 200 6

32 Sheldo n Katz , Enumerativ e geometr y an d strin g theory , 200 6 31 Joh n McCleary , A firs t cours e i n topology : Continuit y an d dimension ,

2006

30 Serg e Tabachnikov , Geometr y an d billiards , 200 5

29 Kristophe r Tapp , Matri x group s fo r undergraduates , 200 5

28 Emmanue l Lesigne , Head s o r tails : A n introductio n t o limi t theorem s i n

probability, 200 5

27 Reinhar d Illner , C . Sea n Bohun , Samanth a McCol lum , an d The a

van R o o d e , Mathematica l modelling : A cas e studie s approach , 200 5

26 Rober t Hardt , Editor , Si x theme s o n variation , 200 4 25 S . V . Duzhi n an d B . D . Chebotarevsky , Transformatio n group s fo r

beginners, 200 4

24 Bruc e M . Landma n an d Aaro n Robertson , Ramse y theor y o n th e integers, 200 4

23 S . K . Lando , Lecture s o n generatin g functions , 200 3

22 Andrea s Arvanitoyeorgos , A n introductio n t o Li e group s an d th e geometry o f homogeneou s spaces , 200 3

21 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

III: Integration , 200 3

20 Klau s Hulek , Elementar y algebrai c geometry , 200 3

19 A . She n an d N . K . Vereshchagin , Computabl e functions , 200 3

18 V . V . Yaschenko , Editor , Cryptography : A n introduction , 200 2

17 A . She n an d N . K . Vereshchagin , Basi c se t theory , 200 2

TITLES I N THI S SERIE S

16 Wolfgan g Kiihnel , Differentia l geometry : curve s - surface s - manifolds , second edition , 200 6

15 Ger d Fischer , Plan e algebrai c curves , 200 1

14 V . A . Vassiliev , Introductio n t o topology , 200 1

13 Frederic k J . Almgren , Jr. , Plateau' s problem : A n invitatio n t o varifol d

geometry, 200 1

12 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

II: Continuit y an d differentiation , 200 1

11 Michae l Mester ton-Gibbons , A n introductio n t o game-theoreti c modelling, 200 0

® 10 Joh n Oprea , Th e mathematic s o f soa p films : Exploration s wit h Mapl e ,

2000

9 Davi d E . Blair , Inversio n theor y an d conforma l mapping , 200 0

8 Edwar d B . Burger , Explorin g th e numbe r jungle : A journey int o

diophantine analysis , 200 0

7 Jud y L . Walker , Code s an d curves , 200 0

6 Geral d Tenenbau m an d Miche l Mende s France , Th e prim e number s

and thei r distribution , 200 0

5 Alexande r Mehlmann , Th e game' s afoot ! Gam e theor y i n myt h an d

paradox, 200 0

4 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

I: Rea l numbers , sequence s an d series , 200 0

3 Roge r Knobel , A n introductio n t o th e mathematica l theor y o f waves ,

2000

2 Gregor y F . Lawle r an d Leste r N . Coyle , Lecture s o n contemporar y

probability, 199 9

1 Charle s Radin , Mile s o f tiles , 199 9

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