Linear Algebra Done Right

268 Pages·2009·3.6 MB·English

by  Sheldon Axler

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
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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