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. 45 |

Code Vectors and Modular Arithmetic | p. 47 |

Chapter 1 Review | p. 53 |

Systems of Linear Equations | p. 61 |

Introduction to Systems of Linear Equations | p. 63 |

Exploration: Lies My Computer Told Me | p. 69 |

Direct Methods for Solving Linear Systems | p. 71 |

Exploration: Partial Pivoting | p. 87 |

Exploration: An Introduction to the Analysis of Algorithms | p. 89 |

Spanning Sets and Linear Independence | p. 91 |

Applications | p. 109 |

Iterative Methods for Solving Linear Systems | p. 121 |

Chapter 2 Review | p. 127 |

Matrices | p. 135 |

Matrix Operations | p. 137 |

Matrix Algebra | p. 143 |

The Inverse of a Matrix | p. 155 |

The LU Factorization | p. 163 |

Subspaces, Basis, Dimension, and Rank | p. 177 |

Introduction to Linear Transformations | p. 195 |

Applications | p. 209 |

Chapter 3 Review | p. 221 |

Eigenvalues and Eigenvectors | p. 231 |

Introduction to Eigenvalues and Eigenvectors | p. 233 |

Determinants | p. 247 |

Exploration: Geometric Applications of Determinants | p. 269 |

Eigenvalues and Eigenvectors of n x n Matrices | p. 275 |

Similarity and Diagonalization | p. 287 |

Iterative Methods for Computing Eigenvalues | p. 299 |

Applications and the Perron-Frobenius Theorem | p. 313 |

Chapter 4 Review | p. 327 |

Orthogonality | p. 337 |

Orthogonality in Rn | p. 339 |

Orthogonal Complements and Projections | p. 349 |

The Gram-Schmidt Process and the QR Factorization | p. 355 |

Exploration: The Modified QR Factorization | p. 359 |

Exploration: Approximating Eigenvalues with the QR Algorithm | p. 361 |

Orthogonal Diagonalization of Symmetric Matrices | p. 363 |

Applications | p. 369 |

Chapter 5 Review | p. 381 |

Vector Spaces | p. 395 |

Vector Spaces and Subspaces | p. 397 |

Linear Independence, Basis, and Dimension | p. 403 |

Exploration: Magic Squares | p. 413 |

Change of Basis | p. 415 |

Linear Transformations | p. 423 |

The Kernel and Range of a Linear Transformation | p. 429 |

The Matrix of a Linear Transformation | p. 439 |

Exploration: Tilings, Lattices, and Crystallographic Restriction | p. 449 |

Applications | p. 451 |

Chapter 6 Review | p. 457 |

Distance and Approximation | p. 471 |

Inner Product Spaces | p. 473 |

Exploration: Vectors and Matrices with Complex Entries | p. 481 |

Exploration: Geometric Inequalities and Optimization Problems | p. 485 |

Norms and Distance Functions | p. 489 |

Least Squares Approximation | p. 497 |

The Singular Value Decomposition | p. 505 |

Applications | p. 515 |

Chapter 7 Review | p. 521 |

Key Definitions and Concepts | p. 535 |

Theorems | p. 559 |

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