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| Matrices and Systems of Linear Equations | |
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| Introduction to Matrices and Systems of Linear Equations | |
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| Echelon Form and Gauss-Jordan Elimination | |
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| Consistent Systems of Linear Equations | |
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| Applications (Optional) | |
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| Matrix Operations | |
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| Algebraic Properties of Matrix Operations | |
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| Linear Independence and Nonsingular Matrices | |
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| Data Fitting, Numerical Integration, and Numerical Differentiation (Optional) | |
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| Matrix Inverses and Their Properties | |
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| Vectors in 2-Space and 3-Space | |
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| Vectors in the Plane | |
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| Vectors in Space | |
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| The Dot Product and the Cross Product | |
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| Lines and Planes in Space | |
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| The Vector Space Rn | |
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| Introduction | |
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| Vector Space Properties of Rn | |
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| Examples of Subspaces | |
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| Bases for Subspaces | |
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| Dimension | |
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| Orthogonal Bases for Subspaces | |
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| Linear Transformations from Rn to Rm | |
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| Least-Squares Solutions to Inconsistent Systems, with Applications to Data Fitting | |
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| Theory and Practice of Least Squares | |
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| The Eigenvalue Problem | |
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| The Eigenvalue Problem for (2 x 2) Matrices | |
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| Determinants and the Eigenvalue Problem | |
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| Elementary Operations and Determinants (Optional) | |
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| Eigenvalues and the Characteristic Polynomial | |
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| Eigenvectors and Eigenspaces | |
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| Complex Eigenvalues and Eigenvectors | |
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| Similarity Transformations and Diagonalization | |
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| Difference Equations | |
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| Markov Chains, Systems of Differential Equations (Optional) | |
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| Vector Spaces and Linear Transformations | |
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| Introduction | |
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| Vector Spaces | |
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| Subspaces | |
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| Linear Independence, Bases, and Coordinates | |
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| Dimension | |
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| Inner-Product Spaces, Orthogonal Bases, and Projections (Optional) | |
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| Linear Transformations | |
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| Operations with Linear Transformations | |
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| Matrix Representations for Linear Transformations | |
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| Change of Basis and Diagonalization | |
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| Determinants | |
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| Introduction | |
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| Cofactor Expansions of Determinants | |
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| Elementary Operations and Determinants | |
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| Cramer's Rule | |
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| Applications of Determinants: Inverses and Wronksians | |
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| Eigenvalues and Applications | |
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| Quadratic Forms | |
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| Systems of Differential Equations | |
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| Transformation to Hessenberg Form | |
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| Eigenvalues of Hessenberg Matrices | |
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| Householder Transformations | |
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| The QR Factorization and Least-Squares Solutions | |
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| Matrix Polynomials and the Cayley-Hamilton Theorem | |
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| Generalized Eigenvectors and Solutions of Systems of Differential Equations | |
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| Appendix: An Introduction to MATLAB | |
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| Answers to Selected Odd-Numbered Exercises | |
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| Index | |