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A (Terse) Introduction t o Linear Algebra
http://dx.doi.org/10.1090/stml/044
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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
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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
Copying an d reprinting . Individua l reader s o f this publication , an d nonprofi t libraries actin g fo r them, ar e permitted t o mak e fai r us e of the material , suc h a s to copy a chapte r fo r use in teaching o r research . Permissio n i s granted t o quot e brie f passages fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t of the sourc e i s given.
Republication, systemati c copying , or multiple reproductio n o f any materia l i n this publication i s permitted onl y unde r licens e fro m th e American Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Acquisition s Department , American Mathematica l Society , 20 1 Charles Street , Providence , Rhod e Islan d 02904 -2294, USA . Request s ca n also be made by e-mail t o reprint-permission@ams.org .
© 200 8 Yitzha k Katznelso n an d Yonatan R . Katznelso n Printed i n the United State s o f America .
@ Th e paper use d i n this boo k i s acid-free an d falls withi n th e guideline s established t o ensure permanenc e an d durability .
Visit th e AMS hom e pag e a t ht tp: / /www.ams.org /
10 9 8 7 6 5 4 3 2 1 1 3 12 11 10 09 0 8
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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
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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
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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
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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
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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
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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
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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
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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
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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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