Linear Algebra Done Right

352 Pages·2015·3.045 MB·English

by  Axler, Sheldon

Undergraduate Texts in Mathematics
Sheldon Axler
Linear Algebra
Done Right
Third Edition
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard:
ColinAdams,WilliamsCollege,Williamstown,MA,USA
AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney,BrandeisUniversity,Waltham,MA,USA
IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA
RogerE.Howe,YaleUniversity,NewHaven,CT,USA
DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA
JillPipher,BrownUniversity,Providence,RI,USA
FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson,UniversityofChicago,Chicago,IL,USA
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat
illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding.
For further volumes:
http://www.springer.com/series/666
Sheldon Axler
Linear Algebra Done Right
Third edition
123
SheldonAxler
Department of Mathematics
SanFranciscoStateUniversity
SanFrancisco,CA,USA
ISSN0172-6056 ISSN2197-5604(electronic)
ISBN978-3-319-11079-0 ISBN978-3-319-11080-6(eBook)
DOI10.1007/978-3-319-11080-6
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014954079
MathematicsSubjectClassification(2010):15-01,15A03,15A04,15A15,15A18,15A21
(cid:2)c SpringerInternationalPublishing2015
Thisworkissubjecttocopyright. AllrightsarereservedbythePublisher, whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor
informationstorageandretrieval,electronicadaptation, computersoftware,orbysimilarordissimilar
methodology now known or hereafter developed. Exempted from this legal reservation are brief
excerpts in connection with reviews or scholarly analysis or material supplied specifically for the
purpose of being entered and executed ona computer system, for exclusive use by the purchaser of
thework.Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsofthe
CopyrightLawofthePublisher’slocation,initscurrentversion,andpermissionforusemustalwaysbe
obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright
ClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does notimply, even inthe absence ofa specific statement, that such names are exempt
fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
TypesetbytheauthorinLaTeX
Coverfigure:ForastatementofApollonius’s Identityanditsconnectiontolinearalgebra,seethelast
exerciseinSection6.A.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Contents
PrefacefortheInstructor xi
PrefacefortheStudent xv
Acknowledgments xvii
1 VectorSpaces 1
1.A Rn andCn 2
ComplexNumbers 2
Lists 5
n
F 6
DigressiononFields 10
Exercises 1.A 11
1.B DefinitionofVectorSpace 12
Exercises 1.B 17
1.C Subspaces 18
SumsofSubspaces 20
DirectSums 21
Exercises 1.C 24
2 Finite-DimensionalVectorSpaces 27
2.A SpanandLinearIndependence 28
LinearCombinationsandSpan 28
LinearIndependence 32
Exercises 2.A 37
v
vi Contents
2.B Bases 39
Exercises 2.B 43
2.C Dimension 44
Exercises 2.C 48
3 LinearMaps 51
3.A TheVectorSpaceofLinearMaps 52
DefinitionandExamplesofLinearMaps 52
AlgebraicOperationsonL.V;W/ 55
Exercises 3.A 57
3.B NullSpacesandRanges 59
NullSpaceandInjectivity 59
RangeandSurjectivity 61
FundamentalTheoremofLinearMaps 63
Exercises 3.B 67
3.C Matrices 70
RepresentingaLinearMapbyaMatrix 70
AdditionandScalarMultiplicationofMatrices 72
MatrixMultiplication 74
Exercises 3.C 78
3.D InvertibilityandIsomorphicVectorSpaces 80
InvertibleLinearMaps 80
IsomorphicVectorSpaces 82
LinearMapsThoughtofasMatrixMultiplication 84
Operators 86
Exercises 3.D 88
3.E ProductsandQuotientsofVectorSpaces 91
ProductsofVectorSpaces 91
ProductsandDirectSums 93
QuotientsofVectorSpaces 94
Exercises 3.E 98
Contents vii
3.F Duality 101
TheDualSpaceandtheDualMap 101
TheNullSpaceandRangeoftheDualofaLinearMap 104
TheMatrixoftheDualofaLinearMap 109
TheRankofaMatrix 111
Exercises 3.F 113
4 Polynomials 117
ComplexConjugateandAbsoluteValue 118
UniquenessofCoefficientsforPolynomials 120
TheDivisionAlgorithmforPolynomials 121
ZerosofPolynomials 122
FactorizationofPolynomialsoverC 123
FactorizationofPolynomialsoverR 126
Exercises 4 129
5 Eigenvalues,Eigenvectors,andInvariantSubspaces 131
5.A InvariantSubspaces 132
EigenvaluesandEigenvectors 133
RestrictionandQuotientOperators 137
Exercises 5.A 138
5.B EigenvectorsandUpper-TriangularMatrices 143
PolynomialsAppliedtoOperators 143
ExistenceofEigenvalues 145
Upper-TriangularMatrices 146
Exercises 5.B 153
5.C EigenspacesandDiagonalMatrices 155
Exercises 5.C 160
6 InnerProductSpaces 163
6.A InnerProductsandNorms 164
InnerProducts 164
Norms 168
Exercises 6.A 175
viii Contents
6.B OrthonormalBases 180
LinearFunctionalsonInnerProductSpaces 187
Exercises 6.B 189
6.C OrthogonalComplementsandMinimizationProblems 193
OrthogonalComplements 193
MinimizationProblems 198
Exercises 6.C 201
7 OperatorsonInnerProductSpaces 203
7.A Self-AdjointandNormalOperators 204
Adjoints 204
Self-AdjointOperators 209
NormalOperators 212
Exercises 7.A 214
7.B TheSpectralTheorem 217
TheComplexSpectralTheorem 217
TheRealSpectralTheorem 219
Exercises 7.B 223
7.C PositiveOperatorsandIsometries 225
PositiveOperators 225
Isometries 228
Exercises 7.C 231
7.D PolarDecompositionandSingularValueDecomposition 233
PolarDecomposition 233
SingularValueDecomposition 236
Exercises 7.D 238
8 OperatorsonComplexVectorSpaces 241
8.A GeneralizedEigenvectorsandNilpotentOperators 242
NullSpacesofPowersofanOperator 242
GeneralizedEigenvectors 244
NilpotentOperators 248
Exercises 8.A 249
Contents ix
8.B DecompositionofanOperator 252
DescriptionofOperatorsonComplexVectorSpaces 252
MultiplicityofanEigenvalue 254
BlockDiagonalMatrices 255
SquareRoots 258
Exercises 8.B 259
8.C CharacteristicandMinimalPolynomials 261
TheCayley–HamiltonTheorem 261
TheMinimalPolynomial 262
Exercises 8.C 267
8.D JordanForm 270
Exercises 8.D 274
9 OperatorsonRealVectorSpaces 275
9.A Complexification 276
ComplexificationofaVectorSpace 276
ComplexificationofanOperator 277
TheMinimalPolynomialoftheComplexification 279
EigenvaluesoftheComplexification 280
CharacteristicPolynomialoftheComplexification 283
Exercises 9.A 285
9.B OperatorsonRealInnerProductSpaces 287
NormalOperatorsonRealInnerProductSpaces 287
IsometriesonRealInnerProductSpaces 292
Exercises 9.B 294
10 TraceandDeterminant 295
10.A Trace 296
ChangeofBasis 296
Trace: AConnectionBetweenOperatorsandMatrices 299
Exercises 10.A 304