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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 | |