| 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 7 The Structure of a Linear Operator 8 Eigenvalues and Eigenvectors 9 Real and Complex Inner Product Spaces 10 Structure Theory for Normal Operators Part ll--Topics 11 Metric Vector Spaces: The Theory of Bilinear Forms 12 Metric Spaces 13 Hilbert Spaces 14 Tensor Products 15 Positive Solutions to Linear Systems:Convexity and Separation 16 Affine Geometry 17 Singular Values and the Moore-Penrose Inverse 18 An Introduction to Algebras 19 The Umbral Calculus Referenees Index of Symbols Index |
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