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| Vector Spaces. | |
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| Introduction | |
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| Vector Spaces | |
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| Subspaces | |
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| Linear Combinations and Systems of Linear Equations | |
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| Linear Dependence and Linear Independence | |
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| Bases and Dimension | |
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| Maximal Linearly Independent Subsets | |
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| Linear Transformations and Matrices | |
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| Linear Transformations, Null Spaces, and Ranges | |
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| The Matrix Representation of a Linear Transformation | |
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| Composition of Linear Transformations and Matrix Multiplication | |
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| Invertibility and Isomorphisms | |
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| The Change of Coordinate Matrix | |
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| Dual Spaces | |
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| Homogeneous Linear Differential Equations with Constant Coefficients | |
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| Elementary Matrix Operations and Systems of Linear Equations | |
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| Elementary Matrix Operations and Elementary Matrices | |
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| The Rank of a Matrix and Matrix Inverses | |
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| Systems of Linear EquationsTheoretical Aspects | |
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| Systems of Linear EquationsComputational Aspects | |
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| Determinants | |
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| Determinants of Order 2 | |
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| Determinants of Order n | |
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| Properties of Determinants | |
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| SummaryImportant Facts about Determinants | |
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| A Characterization of the Determinant | |
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| Diagonalization | |
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| Eigenvalues and Eigenvectors | |
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| Diagonalizability | |
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| Matrix Limits and Markov Chains | |
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| Invariant Subspaces and the Cayley-Hamilton Theorem | |
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| Inner Product Spaces | |
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| Inner Products and Norms | |
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| The Gram-Schmidt Orthogonalization Process and Orthogonal Complements | |
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| The Adjoint of a Linear Operator | |
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| Normal and Self-Adjoint Operators | |
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| Unitary and Orthogonal Operators and Their Matrices | |
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| Orthogonal Projections and the Spectral Theorem | |
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| The Singular Value Decomposition and the Pseudoinverse | |
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| Bilinear and Quadratic Forms | |
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| Einstein's Special Theory of Relativity | |
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| Conditioning and the Rayleigh Quotient | |
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| The Geometry of Orthogonal Operators | |
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| Canonical Forms | |
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| The Jordan Canonical Form I | |
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| The Jordan Canonical Form II | |
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| The Minimal Polynomial | |
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| Rational Canonical Form | |
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| Appendices | |
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| Sets | |
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| Functions | |
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| Fields | |
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| Complex Numbers | |
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| Polynomials | |
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| Answers to Selected Exercises | |
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| Index | |