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Linear Systems and Matrices | |
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Linear Systems of Equations | |
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Elementary Operations and Gauss Elimination Method | |
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Homogeneous Linear Systems | |
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Introduction to Matrices and the Matrix of a Linear System | |
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Elementary Row Operations on a Matrix | |
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Proofs of Facts | |
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Chapter Review Questions and Project | |
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Algebra of Matrices | |
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Scalar Multiplication and Addition of Matrices | |
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Matrix Multiplication and Its Properties | |
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Transpose | |
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Proofs of Facts | |
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Chapter Review Questions and Projects | |
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Subspaces | |
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Linear Combination of Vectors | |
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Vector Subspaces | |
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Linear Dependence, Linear Independence, and Basis | |
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Proofs of Facts | |
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Chapter Review Questions and Project | |
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Rank | |
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Elementary Operations and Rank | |
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Null Space and Nullity of a Matrix | |
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Elementary Matrices | |
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Proofs of Facts | |
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Chapter Review Questions and Project | |
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Inverse, Rank Factorization, and LU-Decomposition | |
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Inverse of a Matrix and Its Properties | |
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Further Properties of Inverses | |
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Full-Rank Factorization | |
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LU-Decomposition of a Matrix | |
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Proofs of Facts | |
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Chapter Review Questions and Projects | |
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Determinants | |
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Determinant | |
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Properties of the Determinant | |
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Cofactors and Inverse of a Matrix | |
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Cramer's Rule | |
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Chapter Review Questions and Projects | |
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Eigenvalue Problems | |
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Eigenvalues and Eigenvectors | |
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Characteristic Polynomial | |
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Calculating Eigenvalues and Eigenvectors (Another Approach) and the Cayley-Hamilton Theorem | |
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Applications of the Cayley-Hamilton Theorem | |
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Properties of Eigenvalues, Diagonalizability, and Triangularizability | |
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Proofs of Facts | |
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Chapter Review Questions and Projects | |
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Inner Product Spaces | |
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Gram-Schmidt Orthogonalization Process | |
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Diagonalization of Symmetric Matrices | |
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Application of the Spectral Theorem | |
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Least-Squares Solution | |
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Generalized Inverse and Least-Squares Solution | |
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Proofs of Facts | |
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Chapter Review Questions and Projects | |
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Vector Spaces and Linear Mappings | |
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Vector Spaces | |
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Linear Dependence and Linear Independence | |
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Linear Mappings | |
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Some Properties of Linear Mappings: Image and Kernel | |
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Linear Mappings and Matrices | |
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Chapter Review Questions and Projects | |
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Determinants (Revisited) | |
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Permutations | |
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Determinants | |
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Cofactor Expansion | |
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Adjoint of a Matrix | |
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Cramer's Rule | |
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Product Theorem of Determinants | |
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Answers and Hints to Selected Exercises | |
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Drill Solutions Using Matlab | |
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Some Basic Matlab Operations | |
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Index | |