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 |