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| Systems of Linear Equations | |
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| The Vector Space of m x n Matrices | |
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| The Space R[superscript n] | |
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| Linear Combinations and Linear Dependence | |
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| What Is a Vector Space? | |
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| Why Prove Anything? | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Applications to Graph Theory I | |
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| Self-Study Questions | |
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| Exercises | |
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| Systems | |
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| Rank: The Maximum Number of Linearly Independent Equations | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Applications to Circuit Theory | |
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| Self-Study Questions | |
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| Exercises | |
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| Gaussian Elimination | |
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| Spanning in Polynomial Spaces | |
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| Computational Issues: Pivoting | |
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| True-False Questions | |
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| Exercises | |
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| Computational Issues: Flops | |
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| Computer Projects | |
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| Exercises | |
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| Applications to Traffic Flow | |
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| Self-Study Questions | |
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| Exercises | |
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| Column Space and Nullspace | |
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| Subspaces | |
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| Subspaces of Functions | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Applications to Predator-Prey Problems | |
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| Self-Study Questions | |
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| Exercises | |
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| Chapter Summary | |
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| Linear Independence and Dimension | |
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| The Test for Linear Independence | |
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| Bases for the Column Space | |
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| Testing Functions for Independence | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Dimension | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Applications to Calculus | |
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| Self-Study Questions | |
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| Exercises | |
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| Applications to Differential Equations | |
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| Self-Study Questions | |
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| Exercises | |
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| Applications to the Harmonic Oscillator | |
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| Self-Study Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Row Space and the rank-nullity theorem | |
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| Bases for the Row Space | |
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| Rank-Nullity Theorem | |
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| Computational Issues: Computing Rank | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Chapter Summary | |
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| Linear Transformations | |
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| The Linearity Properties | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Applications to Control Theory | |
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| Self-Study Questions | |
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| Exercises | |
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| Matrix Multiplication (Composition) | |
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| Partitioned Matrices | |
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| Computational Issues: Parallel Computing | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Applications to Graph Theory II | |
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| Self-Study Questions | |
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| Exercises | |
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| Inverses | |
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| Computational Issues: Reduction versus Inverses | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Ill-Conditioned Systems | |
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| Exercises | |
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| Applications to Economics | |
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| Self-Study Questions | |
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| Exercises | |
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| The LU Factorization | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| The Matrix of a Linear Transformation | |
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| Coordinates | |
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| Application to Differential Equations | |
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| Isomorphism | |
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| Invertible Linear Transformations | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Chapter Summary | |
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| Determinants | |
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| Definition of the Determinant | |
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| The Rest of the Proofs | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Reduction and Determinants | |
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| Uniqueness of the Determinant | |
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| True-False Questions | |
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| Exercises | |
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| Application to Volume | |
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| Self-Study Questions | |
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| Exercises | |
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| A Formula for Inverses | |
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| Cramer's Rule | |
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| True-False Questions | |
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| Exercises | |
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| Chapter Summary | |
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| Eigenvectors and Eigenvalues | |
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| Eigenvectors | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Application to Markov Processes | |
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| Exercises | |
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| Diagonalization | |
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| Powers of Matrices | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Application to Systems of Differential Equations | |
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| Self-Study Questions | |
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| Exercises | |
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| Complex Eigenvectors | |
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| Complex Vector Spaces | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Chapter Summary | |
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| Orthogonality | |
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| The Scalar Product in R[superscript n] | |
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| Orthogonal/Orthonormal Bases and Coordinates | |
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| True-False Questions | |
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| Exercises | |
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| Application to Statistics | |
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| Self-Study Questions | |
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| Exercises | |
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| Projections: The Gram-Schmidt Process | |
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| The QR Decomposition | |
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| Uniqueness of the QR-factorization | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Fourier Series: Scalar Product Spaces | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Orthogonal Matrices | |
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| Householder Matrices | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Least Squares | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| Quadratic Forms: Orthogonal Diagonalization | |
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| The Spectral Theorem | |
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| The Principal Axis Theorem | |
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| True-False Questions | |
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| Exercises | |
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| Computer Projects | |
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| Exercises | |
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| The Singular Value Decomposition (SVD) | |
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| Application of the SVD to Least-Squares Problems | |
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| True-False Questions | |
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| Exercises | |
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| Computing the SVD Using Householder Matrices | |
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| Diagonalizing Symmetric Matrices Using Householder Matrices | |
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| Hermitian Symmetric and Unitary Matrices | |
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| True-False Questions | |
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| Exercises | |
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| Chapter Summary | |
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| Generalized Eigenvectors | |
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| Generalized Eigenvectors | |
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| Exercises | |
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| Chain Bases | |
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| Jordan Form | |
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| True-False Questions | |
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| Exercises | |
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| The Cayley-Hamilton Theorem | |
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| Chapter Summary | |
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| Numerical Techniques | |
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| Condition Number | |
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| Norms | |
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| Condition Number | |
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| Least Squares | |
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| Exercises | |
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| Computing Eigenvalues | |
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| Iteration | |
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| The QR Method | |
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| Proof of Theorem 3 | |
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| Exercises | |
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| Chapter Summary | |
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| Index | |