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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