高等線性代數(shù)

前言

　　Let me begin by thanking the readers of the second edition for their many helpful comments and suggestions， with special thanks to Joe Kidd and Nam Trang. For the third edition， I have corrected all known errors， polished and refined some arguments （such as the discussion of reflexivity， the rational canonical form， best approximations and the definitions of tensor products） and upgraded some proofs that were originally done only for finite-dimensional/rank cases. I have also moved some of the material on projection operators to an earlier oosition in the text.

內容概要

　　is a thorough introduction to linear algebra，for the graduate or advanced undergraduate student。 Prerequisites are limited to a knowledge of the basic properties of matrices and determinants。 However，since we cover the basics of vector spaces and linear transformations rather rapidly，a prior course in linear algebra （even at the sophomore level），along with a certain measure of "mathematical maturity，" is highly desirable。

書籍目錄

Preface to the Third Edition，viiPreface to the Second Edition，ixPreface to the First Edition，xiPreliminariesPart 1： PreliminariesPart 2： Algebraic StructuresPart I-Basic Linear Algebra1 Vector SpacesVector SpacesSubspacesDirect SumsSpanning Sets and Linear IndependenceThe Dimension of a Vector SpaceOrdered Bases and Coordinate MatricesThe Row and Column Spaces of a MatrixThe C0mplexification of a Real Vector SpaceExercises2 Linear TransformationsLinear TransformationsThe Kernel and Image of a Linear TransformationIsomorphismsThe Rank Plus Nullity TheoremLinear Transformations from Fn to FmChange of Basis MatricesThe Matrix of a Linear TransformationChange of Bases for Linear TransformationsEquivalence of MatricesSimilarity of MatricesSimilarity of OperatorsInvariant Subspaces and Reducing PairsProjection OperatorsTopological Vector SpacesLinear Operators on VcExercises3 The Isomorphism TheoremsQuotient SpacesThe Universal Property of Quotients and the First Isomorphism TheoremQuotient Spaces，Complements and CodimensionAdditional Isomorphism TheoremsLinear FunctionalsDual BasesReflexivityAnnihilatorsOperator AdjointsExercises4 Modules I： Basic PropertiesMotivationModulesSubmodulesSpanning SetsLinear IndependenceTorsion ElementsAnnihilatorsFree ModulesHomomorphismsQuotient ModulesThe Correspondence and Isomorphism TheoremsDirect Sums and Direct SummandsModules Are Not as Nice as Vector SpacesExercises5 Modules II： Free and Noetherian ModulesThe Rank of a Free ModuleFree Modules and EpimorphismsNoetherian ModulesThe Hilbert Basis TheoremExercises6 Modules over a Principal Ideal DomainAnnihilators and OrdersCyclic ModulesFree Modules over a Principal Ideal DomainTorsion-Free and Free ModulesThe Primary Cyclic Decomposition TheoremThe Invariant Factor DecompositionCharacterizing Cyclic Moduleslndecomposable ModulesExercisesIndecomposable ModulesExercises 1597 The Structure of a Linear OperatorThe Module Associated with a Linear OperatorThe Primary Cyclic Decomposition of VTThe Characteristic PolynomialCyclic and Indecomposable ModulesThe Big PictureThe Rational Canonical FormExercises8 Eigenvalues and EigenvectorsEigenvalues and EigenvectorsGeometric and Algebraic MultiplicitiesThe Jordan Canonical FormTriangularizability and Schurs TheoremDiagonalizable OperatorsExercises9 Real and Complex Inner Product SpacesNorm and DistanceIsometricsOrthogonalityOrthogonal and Orthonormal SetsThe Projection Theorem and Best ApproximationsThe Riesz Representation TheoremExercises10 Structure Theory for Normal OperatorsThe Adjoint of a Linear OperatorOrthogonal ProjectionsUnitary DiagonalizabilityNormal OperatorsSpecial Types of Normal OperatorsSeif-Adjoint OperatorsUnitary Operators and IsometriesThe Structure of Normal OperatorsFunctional CalculusPositive OperatorsThe Polar Decomposition of an OperatorExercisesPart Ⅱ-Topics11 Metric Vector Spaces： The Theory of Bilinear FormsSymmetric Skew-Symmetric and Alternate FormsThe Matrix ofa Bilinear FormQuadratic FormsOrthogonalityLinear FunctionalsOrthogonal Complements and Orthogonal Direct SumsIsometricsHyperbolic SpacesNonsingular Completions ofa SubspaceThe Witt Theorems： A PreviewThe Classification Problem for Metric Vector SpacesSymplectic GeometryThe Structure of Orthogonal Geometries： Orthogonal BasesThe Classification of Orthogonal Geometries：Canonical FormsThe Orthogonal GroupThe Witt Theorems for Orthogonal GeometriesMaximal Hyperbolic Subspaces of an Orthogonal GeometryExercises12 Metric SpacesThe DefinitionOpen and Closed SetsConvergence in a Metric SpaceThe Closure of a SetDense SubsetsContinuityCompletenessIsometricsThe Completion of a Metric SpaceExercises13 Hilbert SpacesA Brief ReviewHilbert SpacesInfinite SeriesAn Approximation ProblemHilbert BasesFourier ExpansionsA Characterization of Hilbert BasesHilbert DimensionA Characterization of Hilbert SpacesThe Riesz Representation TheoremExercises14 Tensor ProductsUniversalityBilinear MapsTensor ProductsWhen Is a Tensor Product Zero?Coordinate Matrices and RankCharacterizing Vectors in a Tensor ProductDefining Linear Transformations on a Tensor ProductThe Tensor Product of Linear TransformationsChange of Base FieldMultilinear Maps and Iterated Tensor ProductsTensor SpacesSpecial Multilinear MapsGraded AlgebrasThe Symmetric and Antisymmetric Tensor AlgebrasThe DeterminantExercises15 Positive Solutions to Linear Systems：Convexity and SeparationConvex Closed and Compact SetsConvex HullsLinear and Affine HyperplanesSeparationExercises16 Affine GeometryAffine GeometryAffine CombinationsAffine HullsThe Lattice of FlatsAffine IndependenceAffine TransformationsProjective GeometryExercises17 Singular Values and the Moore-Penrose InverseSingular ValuesThe Moore-Penrose Generalized InverseLeast Squares ApproximationExercises18 An Introduction to AlgebrasMotivationAssociative AlgebrasDivision AlgebrasExercises19 The Umbral CalculusFormal Power SeriesThe Umbral AlgebraFormal Power Series as Linear OperatorsSheffer SequencesExamples of Sheffer SequencesUmbral Operators and Umbral ShiftsContinuous Operators on the Umbral AlgebraOperator AdjointsUmbral Operators and Automorphisms of the Umbral AlgebraUmbral Shifts and Derivations of the Umbral AlgebraThe Transfer FormulasA Final RemarkExercisesReferencesIndex of SymbolsIndex

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