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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