Vectors | |

Introduction: The Racetrack Game | |

The Geometry and Algebra of Vectors | |

Length and Angle: The Dot Product | |

Exploration: Vectors and Geometry | |

Lines and Planes | |

Exploration: The Cross Product | |

Applications: Force Vectors | |

Code Vectors | |

Vignette: The Codabar System | |

Systems of Linear Equations | |

Introduction: Triviality | |

Introduction to Systems of Linear Equations | |

Direct Methods for Solving Linear Systems | |

Exploration: Lies My Computer Told Me | |

Exploration: Partial Pivoting | |

Exploration: Counting Operations: An Introduction to the Analysis of Algorithms | |

Spanning Sets and Linear Independence | |

Applications: Allocation of Resources | |

Balancing Chemical Equations | |

Network Analysis | |

Electrical Networks | |

Linear Economic Models | |

Finite Linear Games | |

Vignette: The Global Positioning System | |

Iterative Methods for Solving Linear Systems | |

Matrices | |

Introduction: Matrices in Action | |

Matrix Operations | |

Matrix Algebra | |

The Inverse of a Matrix | |

The LU Factorization | |

Subspaces, Basis, Dimension, and Rank | |

Introduction to Linear Transformations | |

Vignette: Robotics | |

Applications: Markov Chains | |

Linear Economic Models | |

Population Growth | |

Graphs and Digraphs | |

Error-Correcting Codes. | |

Eigenvalues and Eigenvectors | |

Introduction: A Dynamical System on Graphs | |

Introduction to Eigenvalues and Eigenvectors | |

Determinants | |

Vignette: Lewis Carroll's Condensation Method | |

Exploration: Geometric Applications of Determinants | |

Eigenvalues and Eigenvectors of n x n Matrices | |

Similarity and Diagonalization | |

Iterative Methods for Computing Eigenvalues | |

Applications and the Perron-Frobenius Theorem: Markov Chains | |

Population Growth | |

The Perron-Frobenius Theorem | |

Linear Recurrence Relations | |

Systems of Linear Differential Equations | |

Discrete Linear Dynamical Systems | |

Vignette: Ranking Sports Teams and Searching the Internet | |

Orthogonality | |

Introduction: Shadows on a Wall | |

Orthogonality in Rn | |

Orthogonal Complements and Orthogonal Projections | |

The Gram-Schmidt Process and the QR Factorization | |

Exploration: The Modified QR Factorization | |

Exploration: Approximating Eigenvalues with the QR Algorithm | |

Orthogonal Diagonalization of Symmetric Matrices | |

Applications: Dual Codes | |

Quadratic Forms | |

Graphing Quadratic Equations | |

Vector Spaces | |

Introduction: Fibonacci in (Vector) Space | |

Vector Spaces and Subspaces | |

Linear Independence, Basis, and Dimension | |

Exploration: Magic Squares | |

Change of Basis | |

Linear Transformations | |

The Kernel and Range of a Linear Transformation | |

The Matrix of a Linear Transformation | |

Exploration: Tilings, Lattices and the Crystallographic Restriction | |

Applications: Homogeneous Linear Differential Equations | |

Linear Codes | |

Distance and Approximation | |

Introduction: Taxicab Geometry | |

Inner Product Spaces | |

Exploration: Vectors and Matrices with Complex Entries | |

Exploration: Geometric Inequalities and Optimization Problems | |

Norms and Distance Functions | |

Least Squares Approximation | |

The Singular Value Decomposition | |

Vignette: Digital Image Compression | |

Applications: Approximation of Functions | |

Error-Correcting Codes | |

Mathematical Notation and Methods of Proof | |

Mathematical Induction | |

Complex Numbers | |

Polynomials | |

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