| |
| |
| 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 | |