Table Of Content

Linear Algebra Done Right, Second Edition Sheldon Axler Springer Contents PrefacetotheInstructor ix PrefacetotheStudent xiii Acknowledgments xv Chapter1 VectorSpaces 1 ComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 2 DefinitionofVectorSpace. . . . . . . . . . . . . . . . . . . . . . 4 PropertiesofVectorSpaces . . . . . . . . . . . . . . . . . . . . . 11 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 SumsandDirectSums . . . . . . . . . . . . . . . . . . . . . . . . 14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Chapter2 Finite-DimensionalVectorSpaces 21 SpanandLinearIndependence . . . . . . . . . . . . . . . . . . . 22 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter3 LinearMaps 37 DefinitionsandExamples . . . . . . . . . . . . . . . . . . . . . . 38 NullSpacesandRanges . . . . . . . . . . . . . . . . . . . . . . . 41 TheMatrixofaLinearMap . . . . . . . . . . . . . . . . . . . . . 48 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 v vi Contents Chapter4 Polynomials 63 Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ComplexCoefficients . . . . . . . . . . . . . . . . . . . . . . . . 67 RealCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Chapter5 EigenvaluesandEigenvectors 75 InvariantSubspaces . . . . . . . . . . . . . . . . . . . . . . . . . 76 PolynomialsAppliedtoOperators . . . . . . . . . . . . . . . . . 80 Upper-TriangularMatrices . . . . . . . . . . . . . . . . . . . . . 81 DiagonalMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 InvariantSubspacesonRealVectorSpaces . . . . . . . . . . . 91 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Chapter6 Inner-ProductSpaces 97 InnerProducts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 OrthonormalBases . . . . . . . . . . . . . . . . . . . . . . . . . . 106 OrthogonalProjectionsandMinimizationProblems . . . . . . 111 LinearFunctionalsandAdjoints . . . . . . . . . . . . . . . . . . 117 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Chapter7 OperatorsonInner-ProductSpaces 127 Self-AdjointandNormalOperators . . . . . . . . . . . . . . . . 128 TheSpectralTheorem . . . . . . . . . . . . . . . . . . . . . . . . 132 NormalOperatorsonRealInner-ProductSpaces . . . . . . . . 138 PositiveOperators . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 PolarandSingular-ValueDecompositions . . . . . . . . . . . . 152 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Chapter8 OperatorsonComplexVectorSpaces 163 GeneralizedEigenvectors . . . . . . . . . . . . . . . . . . . . . . 164 TheCharacteristicPolynomial . . . . . . . . . . . . . . . . . . . 168 DecompositionofanOperator . . . . . . . . . . . . . . . . . . . 173 Contents vii SquareRoots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 TheMinimalPolynomial . . . . . . . . . . . . . . . . . . . . . . . 179 JordanForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Chapter9 OperatorsonRealVectorSpaces 193 EigenvaluesofSquareMatrices . . . . . . . . . . . . . . . . . . . 194 BlockUpper-TriangularMatrices . . . . . . . . . . . . . . . . . . 195 TheCharacteristicPolynomial . . . . . . . . . . . . . . . . . . . 198 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Chapter10 TraceandDeterminant 213 ChangeofBasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 DeterminantofanOperator . . . . . . . . . . . . . . . . . . . . 222 DeterminantofaMatrix . . . . . . . . . . . . . . . . . . . . . . . 225 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 SymbolIndex 247 Index 249 Preface to the Instructor You are probably about to teach a course that will give students their second exposure to linear algebra. During their first brush with thesubject,yourstudentsprobablyworkedwithEuclideanspacesand matrices. Incontrast,thiscoursewillemphasizeabstractvectorspaces andlinearmaps. The audacious title of this book deserves an explanation. Almost alllinearalgebrabooksusedeterminantstoprovethateverylinearop- eratoronafinite-dimensionalcomplexvectorspacehasaneigenvalue. Determinantsaredifficult,nonintuitive,andoftendefinedwithoutmo- tivation. Toprovethetheoremaboutexistenceofeigenvaluesoncom- plexvectorspaces,mostbooksmustdefinedeterminants,provethata linearmapisnotinvertibleifandonlyifitsdeterminantequals0,and then define the characteristic polynomial. This tortuous (torturous?) pathgivesstudentslittlefeelingforwhyeigenvaluesmustexist. In contrast, the simple determinant-free proofs presented here of- fer more insight. Once determinants have been banished to the end of the book, a new route opens to the main goal of linear algebra— understandingthestructureoflinearoperators. This book starts at the beginning of the subject, with no prerequi- sitesotherthantheusualdemandforsuitablemathematicalmaturity. Even if your students have already seen some of the material in the first few chapters, they may be unaccustomed to working exercises of the type presented here, most of which require an understanding of proofs. • VectorspacesaredefinedinChapter1,andtheirbasicproperties aredeveloped. • Linear independence, span, basis, and dimension are defined in Chapter 2, which presents the basic theory of finite-dimensional vectorspaces. ix x PrefacetotheInstructor • Linear maps are introduced in Chapter 3. The key result here is that for a linear map T, the dimension of the null space of T plusthedimensionoftherangeofT equalsthedimensionofthe domainofT. • The part of the theory of polynomials that will be needed to un- derstand linear operators is presented in Chapter 4. If you take class time going through the proofs in this chapter (which con- tains no linear algebra), then you probably will not have time to cover some important aspects of linear algebra. Your students will already be familiar with the theorems about polynomials in this chapter, so you can ask them to read the statements of the results but not the proofs. The curious students will read some oftheproofsanyway,whichiswhytheyareincludedinthetext. • The idea of studying a linear operator by restricting it to small subspacesleadsinChapter5toeigenvectors. Thehighlightofthe chapterisasimpleproofthatoncomplexvectorspaces,eigenval- uesalwaysexist. Thisresultisthenusedtoshowthateachlinear operator on a complex vector space has an upper-triangular ma- trix with respect to some basis. Similar techniques are used to show that every linear operator on a real vector space has an in- variantsubspaceofdimension1or2. Thisresultisusedtoprove thateverylinearoperatoronanodd-dimensionalrealvectorspace hasaneigenvalue. Allthisisdonewithoutdefiningdeterminants orcharacteristicpolynomials! • Inner-product spaces are defined in Chapter 6, and their basic propertiesaredevelopedalongwithstandardtoolssuchasortho- normal bases, the Gram-Schmidt procedure, and adjoints. This chapter also shows how orthogonal projections can be used to solvecertainminimizationproblems. • Thespectraltheorem,whichcharacterizesthelinearoperatorsfor which there exists an orthonormal basis consisting of eigenvec- tors, is the highlight of Chapter 7. The work in earlier chapters pays off here with especially simple proofs. This chapter also dealswithpositiveoperators,linearisometries,thepolardecom- position,andthesingular-valuedecomposition.