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