高等线性代数（第3版）

isathoroughintroductiontolinearalgebra，forthegraduateoradvancedundergraduatestudent。Prerequisitesarelimitedtoaknowledgeofthebasicpropertiesofmatricesanddeterminants。However，sincewecoverthebasicsofvectorspacesandlineartransformationsratherrapidly，apriorcourseinlinearalgebra（evenatthesophomorelevel），alongwithacertainmeasureof"mathematicalmaturity，"ishighlydesirable。

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PrefacetotheThirdEdition，vii
PrefacetotheSecondEdition，ix
PrefacetotheFirstEdition，xi
Preliminaries
Part1：Preliminaries
Part2：AlgebraicStructures

PartI-BasicLinearAlgebra
1VectorSpaces
VectorSpaces
Subspaces
DirectSums
SpanningSetsandLinearIndependence
TheDimensionofaVectorSpace
OrderedBasesandCoordinateMatrices
TheRowandColumnSpacesofaMatrix
TheC0mplexificationofaRealVectorSpace
Exercises

2LinearTransformations
LinearTransformations
TheKernelandImageofaLinearTransformation
Isomorphisms
TheRankPlusNullityTheorem
LinearTransformationsfromFntoFm
ChangeofBasisMatrices
TheMatrixofaLinearTransformation
ChangeofBasesforLinearTransformations
EquivalenceofMatrices
SimilarityofMatrices
SimilarityofOperators
InvariantSubspacesandReducingPairs
ProjectionOperators
TopologicalVectorSpaces
LinearOperatorsonVc
Exercises

3TheIsomorphismTheorems
QuotientSpaces
TheUniversalPropertyofQuotientsandtheFirstIsomorphismTheorem
QuotientSpaces，ComplementsandCodimension
AdditionalIsomorphismTheorems
LinearFunctionals
DualBases
Reflexivity
Annihilators
OperatorAdjoints
Exercises

4ModulesI：BasicProperties
Motivation
Modules
Submodules
SpanningSets
LinearIndependence
TorsionElements
Annihilators
FreeModules
Homomorphisms
QuotientModules
TheCorrespondenceandIsomorphismTheorems
DirectSumsandDirectSummands
ModulesAreNotasNiceasVectorSpaces
Exercises

5ModulesII：FreeandNoetherianModules
TheRankofaFreeModule
FreeModulesandEpimorphisms
NoetherianModules
TheHilbertBasisTheorem
Exercises

6ModulesoveraPrincipalIdealDomain
AnnihilatorsandOrders
CyclicModules
FreeModulesoveraPrincipalIdealDomain
Torsion-FreeandFreeModules
ThePrimaryCyclicDecompositionTheorem
TheInvariantFactorDecomposition
CharacterizingCyclicModules
lndecomposableModules
Exercises

IndecomposableModules
Exercises159

7TheStructureofaLinearOperator
TheModuleAssociatedwithaLinearOperator
ThePrimaryCyclicDecompositionofVT
TheCharacteristicPolynomial
CyclicandIndecomposableModules
TheBigPicture
TheRationalCanonicalForm
Exercises

8EigenvaluesandEigenvectors
EigenvaluesandEigenvectors
GeometricandAlgebraicMultiplicities
TheJordanCanonicalForm
TriangularizabilityandSchursTheorem
DiagonalizableOperators
Exercises

9RealandComplexInnerProductSpaces
NormandDistance
Isometrics
Orthogonality
OrthogonalandOrthonormalSets
TheProjectionTheoremandBestApproximations
TheRieszRepresentationTheorem
Exercises

10StructureTheoryforNormalOperators
TheAdjointofaLinearOperator
OrthogonalProjections
UnitaryDiagonalizability
NormalOperators
SpecialTypesofNormalOperators
Seif-AdjointOperators
UnitaryOperatorsandIsometries
TheStructureofNormalOperators
FunctionalCalculus
PositiveOperators
ThePolarDecompositionofanOperator
Exercises

PartⅡ-Topics
11MetricVectorSpaces：TheTheoryofBilinearForms
SymmetricSkew-SymmetricandAlternateForms
TheMatrixofaBilinearForm
QuadraticForms
Orthogonality
LinearFunctionals
OrthogonalComplementsandOrthogonalDirectSums
Isometrics
HyperbolicSpaces
NonsingularCompletionsofaSubspace
TheWittTheorems：APreview
TheClassificationProblemforMetricVectorSpaces
SymplecticGeometry
TheStructureofOrthogonalGeometries：OrthogonalBases
TheClassificationofOrthogonalGeometries：CanonicalForms
TheOrthogonalGroup
TheWittTheoremsforOrthogonalGeometries
MaximalHyperbolicSubspacesofanOrthogonalGeometry
Exercises

12MetricSpaces
TheDefinition
OpenandClosedSets
ConvergenceinaMetricSpace
TheClosureofaSet
DenseSubsets
Continuity
Completeness
Isometrics
TheCompletionofaMetricSpace
Exercises

13HilbertSpaces
ABriefReview
HilbertSpaces
InfiniteSeries
AnApproximationProblem
HilbertBases
FourierExpansions
ACharacterizationofHilbertBases
HilbertDimension
ACharacterizationofHilbertSpaces
TheRieszRepresentationTheorem
Exercises

14TensorProducts
Universality
BilinearMaps
TensorProducts
WhenIsaTensorProductZero?
CoordinateMatricesandRank
CharacterizingVectorsinaTensorProduct
DefiningLinearTransformationsonaTensorProduct
TheTensorProductofLinearTransformations
ChangeofBaseField
MultilinearMapsandIteratedTensorProducts
TensorSpaces
SpecialMultilinearMaps
GradedAlgebras
TheSymmetricandAntisymmetricTensorAlgebras
TheDeterminant
Exercises

15PositiveSolutionstoLinearSystems：ConvexityandSeparation
ConvexClosedandCompactSets
ConvexHulls
LinearandAffineHyperplanes
Separation
Exercises

16AffineGeometry
AffineGeometry
AffineCombinations
AffineHulls
TheLatticeofFlats
AffineIndependence
AffineTransformations
ProjectiveGeometry
Exercises

17SingularValuesandtheMoore-PenroseInverse
SingularValues
TheMoore-PenroseGeneralizedInverse
LeastSquaresApproximation
Exercises

18AnIntroductiontoAlgebras
Motivation
AssociativeAlgebras
DivisionAlgebras
Exercises

19TheUmbralCalculus
FormalPowerSeries
TheUmbralAlgebra
FormalPowerSeriesasLinearOperators
ShefferSequences
ExamplesofShefferSequences
UmbralOperatorsandUmbralShifts
ContinuousOperatorsontheUmbralAlgebra
OperatorAdjoints
UmbralOperatorsandAutomorphismsoftheUmbralAlgebra
UmbralShiftsandDerivationsoftheUmbralAlgebra
TheTransferFormulas
AFinalRemark
Exercises
References
IndexofSymbols
Index