Vectors | p. 1 |

The Geometry and Algebra of Vectors | p. 3 |

Length and Angle: The Dot Product | p. 11 |

Exploration: Vectors and Geometry | p. 25 |

Lines and Planes | p. 27 |

Exploration: The Cross Product | p. 43 |

Applications | p. 45 |

Review | p. 51 |

Systems of Linear Equations | p. 59 |

Introduction to Systems of Linear Equations | p. 61 |

Exploration: Lies My Computer Told Me | p. 67 |

Direct Methods for Solving Linear Systems | p. 69 |

Exploration: Partial Pivoting | p. 85 |

Exploration: An Introduction to the Analysis of Algorithms | p. 87 |

Spanning Sets and Linear Independence | p. 89 |

Applications | p. 107 |

Iterative Methods for Solving Linear Systems | p. 119 |

Review | p. 125 |

Matrices | p. 133 |

Matrix Operations' | p. 135 |

Matrix Algebra | p. 141 |

The Inverse of a Matrix | p. 153 |

The LU Factorization | p. 161 |

Subspaces, Basis, Dimension, and Rank | p. 175 |

Introduction to Linear Transformations | p. 193 |

Applications | p. 207 |

Review | p. 223 |

Eigenvalues and Eigenvectors | p. 233 |

Introduction to Eigenvalues and Eigenvectors | p. 235 |

Determinants | p. 249 |

Exploration: Geometric Applications of Determinants | p. 271 |

Eigenvalues and Eigenvectors of n × n Matrices | p. 277 |

Similarity and Diagonalization | p. 289 |

Iterative Methods for Computing Eigenvalues | p. 301 |

Applications and the Perron-Frobenius Theorem | p. 315 |

Review | p. 331 |

Orthogonality | p. 341 |

Orthogonality in Rn | p. 343 |

Orthogonal Complements and Projections | p. 353 |

The Gram-Schmidt Process and the QR Factorization | p. 359 |

Exploration: The Modified QR Factorization | p. 363 |

Exploration: Approximating Eigenvalues with the QR Algorithm | p. 365 |

Orthogonal Diagonalization of Symmetric Matrices | p. 367 |

Applications | p. 373 |

Review | p. 387 |

Vector Spaces | p. 401 |

Vector Spaces and Subspaces | p. 403 |

Linear Independence, Basis, and Dimension | p. 409 |

Exploration: Magic Squares | p. 419 |

Change of Basis | p. 421 |

Linear Transformations | p. 429 |

The Kernel and Range of a Linear Transformation | p. 435 |

The Matrix of a Linear Transformation | p. 445 |

Exploration: Tilings, Lattices, and Crystallographic Restriction | p. 455 |

Applications | p. 457 |

Review463 | |

Distance and Approximation | p. 477 |

Inner Product Spaces | p. 479 |

Exploration: Vectors and Matrices with Complex Entries | p. 491 |

Norms and Distance Functions | p. 495 |

Least Squares Approximation | p. 503 |

The Singular Value Decomposition | p. 511 |

Applications | p. 521 |

Review | p. 527 |

Key Definitions and Concepts | p. 541 |

Theorems | p. 565 |

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