This book 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。
作者簡介
暫缺《高等線性代數(shù)（第3版）》作者簡介
圖書目錄
Preface to the Third Edition，vii Preface to the Second Edition，ix Preface to the First Edition，xi Preliminaries Part 1： Preliminaries Part 2： Algebraic Structures Part I-Basic Linear Algebra 1 Vector Spaces Vector Spaces Subspaces Direct Sums Spanning Sets and Linear Independence The Dimension of a Vector Space Ordered Bases and Coordinate Matrices The Row and Column Spaces of a Matrix The C0mplexification of a Real Vector Space Exercises 2 Linear Transformations Linear Transformations The Kernel and Image of a Linear Transformation Isomorphisms The Rank Plus Nullity Theorem Linear Transformations from Fn to Fm Change of Basis Matrices The Matrix of a Linear Transformation Change of Bases for Linear Transformations Equivalence of Matrices Similarity of Matrices Similarity of Operators Invariant Subspaces and Reducing Pairs Projection Operators Topological Vector Spaces Linear Operators on Vc Exercises 3 The Isomorphism Theorems Quotient Spaces The Universal Property of Quotients and the First Isomorphism Theorem Quotient Spaces，Complements and Codimension Additional Isomorphism Theorems Linear Functionals Dual Bases Reflexivity Annihilators Operator Adjoints Exercises 4 Modules I： Basic Properties Motivation Modules Submodules Spanning Sets Linear Independence Torsion Elements Annihilators Free Modules Homomorphisms Quotient Modules The Correspondence and Isomorphism Theorems Direct Sums and Direct Summands Modules Are Not as Nice as Vector Spaces Exercises 5 Modules II： Free and Noetherian Modules The Rank of a Free Module Free Modules and Epimorphisms Noetherian Modules The Hilbert Basis Theorem Exercises 6 Modules over a Principal Ideal Domain Annihilators and Orders Cyclic Modules Free Modules over a Principal Ideal Domain Torsion-Free and Free Modules The Primary Cyclic Decomposition Theorem The Invariant Factor Decomposition Characterizing Cyclic Modules lndecomposable Modules Exercises Indecomposable Modules Exercises 159 7 The Structure of a Linear Operator The Module Associated with a Linear Operator The Primary Cyclic Decomposition of VT The Characteristic Polynomial Cyclic and Indecomposable Modules The Big Picture The Rational Canonical Form Exercises 8 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors Geometric and Algebraic Multiplicities The Jordan Canonical Form Triangularizability and Schurs Theorem Diagonalizable Operators Exercises 9 Real and Complex Inner Product Spaces Norm and Distance Isometrics Orthogonality Orthogonal and Orthonormal Sets The Projection Theorem and Best Approximations The Riesz Representation Theorem Exercises 10 Structure Theory for Normal Operators The Adjoint of a Linear Operator Orthogonal Projections Unitary Diagonalizability Normal Operators Special Types of Normal Operators Seif-Adjoint Operators Unitary Operators and Isometries The Structure of Normal Operators Functional Calculus Positive Operators The Polar Decomposition of an Operator Exercises Part Ⅱ-Topics 11 Metric Vector Spaces： The Theory of Bilinear Forms Symmetric Skew-Symmetric and Alternate Forms The Matrix ofa Bilinear Form Quadratic Forms Orthogonality Linear Functionals Orthogonal Complements and Orthogonal Direct Sums Isometrics Hyperbolic Spaces Nonsingular Completions ofa Subspace The Witt Theorems： A Preview The Classification Problem for Metric Vector Spaces Symplectic Geometry The Structure of Orthogonal Geometries： Orthogonal Bases The Classification of Orthogonal Geometries：Canonical Forms The Orthogonal Group The Witt Theorems for Orthogonal Geometries Maximal Hyperbolic Subspaces of an Orthogonal Geometry Exercises 12 Metric Spaces The Definition Open and Closed Sets Convergence in a Metric Space The Closure of a Set Dense Subsets Continuity Completeness Isometrics The Completion of a Metric Space Exercises 13 Hilbert Spaces A Brief Review Hilbert Spaces Infinite Series An Approximation Problem Hilbert Bases Fourier Expansions A Characterization of Hilbert Bases Hilbert Dimension A Characterization of Hilbert Spaces The Riesz Representation Theorem Exercises 14 Tensor Products Universality Bilinear Maps Tensor Products When Is a Tensor Product Zero? Coordinate Matrices and Rank Characterizing Vectors in a Tensor Product Defining Linear Transformations on a Tensor Product The Tensor Product of Linear Transformations Change of Base Field Multilinear Maps and Iterated Tensor Products Tensor Spaces Special Multilinear Maps Graded Algebras The Symmetric and Antisymmetric Tensor Algebras The Determinant Exercises 15 Positive Solutions to Linear Systems：Convexity and Separation Convex Closed and Compact Sets Convex Hulls Linear and Affine Hyperplanes Separation Exercises 16 Affine Geometry Affine Geometry Affine Combinations Affine Hulls The Lattice of Flats Affine Independence Affine Transformations Projective Geometry Exercises 17 Singular Values and the Moore-Penrose Inverse Singular Values The Moore-Penrose Generalized Inverse Least Squares Approximation Exercises 18 An Introduction to Algebras Motivation Associative Algebras Division Algebras Exercises 19 The Umbral Calculus Formal Power Series The Umbral Algebra Formal Power Series as Linear Operators Sheffer Sequences Examples of Sheffer Sequences Umbral Operators and Umbral Shifts Continuous Operators on the Umbral Algebra Operator Adjoints Umbral Operators and Automorphisms of the Umbral Algebra Umbral Shifts and Derivations of the Umbral Algebra The Transfer Formulas A Final Remark Exercises References Index of Symbols Index
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