Linear Algebra Done Right PDF
Preview Linear Algebra Done Right
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. 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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
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