| Preface to the Second Edition |
| Preface to the First Edition |
| Preliminaries |
| Algebraic Structures / Part 1： |
| Basic Linear Algebra / Part I： |
| Vector Spaces / 1： |
| 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 Complexification of a Real Vector Space |
| Exercises |
| Linear Transformations / 2： |
| Isomorphisms |
| The Kernel and Image of a Linear Transformation |
| Linear Transformations from F[superscript n] to F[superscript m] |
| The Rank Plus Nullity Theorem |
| 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 Paris |
| Topological Vector Spaces |
| Linear Operators on V[superscript C] |
| The Isomorphism Theorems / 3： |
| 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 |
| Modules I: Basic Properties / 4： |
| Modules |
| Motivation |
| Submodules |
| Spanning Sets |
| Linear Independence |
| Torsion Elements |
| Free Modules |
| Homomorphisms |
| Quotient Modules |
| The Correspondence and Isomorphism Theorems |
| Direct Sums and Direct Summands |
| Modules Are Not As Nice As Vector Spaces |
| Modules II: Free and Noetherian Modules / 5： |
| The Rank of a Free Module |
| Free Modules and Epimorphisms |
| Noetherian Modules |
| The Hilbert Basis Theorem |
| Modules over a Principal Ideal Domain / 6： |
| Annihilators and Orders |
| Cyclic Modules |
| Free Modules over a Principal Ideal Domain |
| Torsion-Free and Free Modules |
| Prelude to Decomposition: Cyclic Modules |
| The First Decomposition |
| A Look Ahead |
| The Primary Decomposition |
| The Cyclic Decomposition of a Primary Module |
| The Primary Cyclic Decomposition Theorem |
| The Invariant Factor Decomposition |
| The Structure of a Linear Operator / 7： |
| A Brief Review |
| The Module Associated with a Linear Operator |
| Orders and the Minimal Polynomial |
| Cyclic Submodules and Cyclic Subspaces |
| Summary |
| The Decomposition of V[subscript tau] |
| The Rational Canonical Form |
| Eigenvalues and Eigenvectors / 8： |
| The Characteristic Polynomial of an Operator |
| Geometric and Algebraic Multiplicities |
| The Jordan Canonical Form |
| Triangularizability and Schur's Lemma |
| Diagonalizable Operators |
| Projections |
| The Algebra of Projections |
| Resolutions of the Identity |
| Spectral Resolutions |
| Projections and Invariance |
| Real and Complex Inner Product Spaces / 9： |
| Norm and Distance |
| Isometries |
| Orthogonality |
| Orthogonal and Orthonormal Sets |
| The Projection Theorem and Best Approximations |
| Orthogonal Direct Sums |
| The Riesz Representation Theorem |
| Structure Theory for Normal Operators / 10： |
| The Adjoint of a Linear Operator |
| Unitary Diagonalizability |
| Normal Operators |
| Special Types of Normal Operators |
| Self-Adjoint Operators |
| Unitary Operators and Isometries |
| The Structure of Normal Operators |
| Matrix Versions |
| Orthogonal Projections |
| Orthogonal Resolutions of the Identity |
| The Spectral Theorem |
| Spectral Resolutions and Functional Calculus |
| Positive Operators |
| The Polar Decomposition of an Operator |
| Topics / Part II： |
| Metric Vector Spaces: The Theory of Bilinear Forms / 11： |
| Symmetric, Skew-Symmetric and Alternate Forms |
| The Matrix of a Bilinear Form |
| Quadratic Forms |
| Orthogonal Complements and Orthogonal Direct Sums |
| Hyperbolic Spaces |
| Nonsingular Completions of a 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's Theorems for Orthogonal Geometries |
| Maximal Hyperbolic Subspaces of an Orthogonal Geometry |
| Metric Spaces / 12： |
| The Definition |
| Open and Closed Sets |
| Convergence in a Metric Space |
| The Closure of a Set |
| Dense Subsets |
| Continuity |
| Completeness |
| The Completion of a Metric Space |
| Hilbert Spaces / 13： |
| Infinite Series |
| An Approximation Problem |
| Hilbert Bases |
| Fourier Expansions |
| A Characterization of Hilbert Bases |
| Hilbert Dimension |
| A Characterization of Hilbert Spaces |
| Tensor Products / 14： |
| Universality |
| Bilinear Maps |
| 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 Tensor Algebra |
| The Antisymmetric Tensor Algebra: The Exterior Product Space |
| The Determinant |
| Positive Solutions to Linear Systems: Convexity and Separation / 15： |
| Convex, Closed and Compact Sets |
| Convex Hulls |
| Linear and Affine Hyperplanes |
| Separation |
| Affine Geometry / 16： |
| Affine Combinations |
| Affine Hulls |
| The Lattice of Flats |
| Affine Independence |
| Affine Transformations |
| Projective Geometry |
| Operator Factorizations: QR and Singular Value / 17： |
| The QR Decomposition |
| Singular Values |
| The Moore-Penrose Generalized Inverse |
| Least Squares Approximation |
| The Umbral Calculus / 18： |
| 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 |
| Umbral Operators and Automorphisms of the Umbral Algebra |
| Umbral Shifts and Derivations of the Umbral Algebra |
| The Transfer Formulas |
| A Final Remark |
| References |
| Index |
| Preface to the Second Edition |
| Preface to the First Edition |
| Preliminaries |
| Algebraic Structures / Part 1： |
| Basic Linear Algebra / Part I： |
| Vector Spaces / 1： |
