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Linear Equations in Linear Algebra | |
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Introductory Example: Linear Models in Economics and Engineering | |
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Systems of Linear Equations | |
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Row Reduction and Echelon Forms | |
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Vector Equations | |
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The Matrix Equation Ax = b | |
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Solution Sets of Linear Systems | |
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Applications of Linear Systems | |
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Linear Independence | |
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Introduction to Linear Transformations | |
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The Matrix of a Linear Transformation | |
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Linear Models in Business, Science, and Engineering | |
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Supplementary Exercises | |
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Matrix Algebra | |
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Introductory Example: Computer Models in Aircraft Design | |
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Matrix Operations | |
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The Inverse of a Matrix | |
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Characterizations of Invertible Matrices | |
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Partitioned Matrices | |
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Matrix Factorizations | |
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The Leontief Input=Output Model | |
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Applications to Computer Graphics | |
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Subspaces of Rn | |
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Dimension and Rank | |
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Supplementary Exercises | |
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Determinants | |
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Introductory Example: Determinants in Analytic Geometry | |
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Introduction to Determinants | |
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Properties of Determinants | |
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Cramer's Rule, Volume, and Linear Transformations | |
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Supplementary Exercises | |
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Vector Spaces | |
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Introductory Example: Space Flight and Control Systems | |
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Vector Spaces and Subspaces | |
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Null Spaces, Column Spaces, and Linear Transformations | |
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Linearly Independent Sets Bases | |
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Coordinate Systems | |
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The Dimension of a Vector Space | |
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Rank | |
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Change of Basis | |
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Applications to Difference Equations | |
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Applications to Markov Chains | |
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Supplementary Exercises | |
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Eigenvalues and Eigenvectors | |
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Introductory Example: Dynamical Systems and Spotted Owls | |
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Eigenvectors and Eigenvalues | |
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The Characteristic Equation | |
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Diagonalization | |
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Eigenvectors and Linear Transformations | |
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Complex Eigenvalues | |
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Discrete Dynamical Systems | |
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Applications to Differential Equations | |
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Iterative Estimates for Eigenvalues | |
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Supplementary Exercises | |
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Orthogonality and Least Squares | |
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Introductory Example: Readjusting the North American Datum | |
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Inner Product, Length, and Orthogonality | |
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Orthogonal Sets | |
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Orthogonal Projections | |
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The Gram-Schmidt Process | |
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Least-Squares Problems | |
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Applications to Linear Models | |
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Inner Product Spaces | |
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Applications of Inner Product Spaces | |
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Supplementary Exercises | |
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Symmetric Matrices and Quadratic Forms | |
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Introductory Example: Multichannel Image Processing | |
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Diagonalization of Symmetric Matrices | |
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Quadratic Forms | |
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Constrained Optimization | |
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The Singular Value Decomposition | |
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Applications to Image Processing and Statistics | |
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Supplementary Exercises (ONLINE ONLY) | |
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The Geometry of Vector Spaces | |
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Introductory Example: The Platonic Solids | |
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Affine Combinations | |
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Affine Independence | |
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Convex Combinations | |
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Hyperplanes | |
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Polytopes | |
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Curves and Surfaces | |
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Supplementary Exercises (ONLINE ONLY) | |
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Optimization | |
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Introductory Example: The Berlin Airlift | |
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Matrix Games | |
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Linear Programming - Geometric Method | |
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Linear Programming - Simplex Method | |
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Duality | |
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Supplementary Exercises | |
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Appendices | |
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Uniqueness of the Reduced Echelon Form | |
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Complex Numbers | |
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Glossary | |
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Answers to Odd-Numbered Exercises | |
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Index | |