Linear algebra

Solving Linear Systems......Page 11 Gauss' Method......Page 12 Describing the Solution Set......Page 21 General = Particular + Homogeneous......Page 30 Vectors in Space......Page 42 Length and Angle Measures*......Page 49 Gauss-Jordan Reduction......Page 56 Row Equivalence......Page 62 Topic: Computer Algebra Systems......Page 71 Topic: Input-Output Analysis......Page 73 Topic: Accuracy of Computations......Page 77 Topic: Analyzing Networks......Page 81 Vector Spaces......Page 87 Definition and Examples......Page 88 Subspaces and Spanning Sets......Page 99 Definition and Examples......Page 109 Basis......Page 120 Dimension......Page 126 Vector Spaces and Linear Systems......Page 132 Combining Subspaces*......Page 139 Topic: Fields......Page 148 Topic: Crystals......Page 150 Topic: Voting Paradoxes......Page 154 Topic: Dimensional Analysis......Page 160 Definition and Examples......Page 167 Dimension Characterizes Isomorphism......Page 176 Definition......Page 184 Rangespace and Nullspace......Page 191 Representing Linear Maps with Matrices......Page 203 Any Matrix Represents a Linear Map*......Page 213 Sums and Scalar Products......Page 220 Matrix Multiplication......Page 223 Mechanics of Matrix Multiplication......Page 230 Inverses......Page 239 Changing Representations of Vectors......Page 246 Changing Map Representations......Page 250 Orthogonal Projection Into a Line*......Page 258 Gram-Schmidt Orthogonalization*......Page 262 Projection Into a Subspace*......Page 268 Topic: Line of Best Fit......Page 277 Topic: Geometry of Linear Maps......Page 282 Topic: Markov Chains......Page 289 Topic: Orthonormal Matrices......Page 295 Determinants......Page 301 Exploration*......Page 302 Properties of Determinants......Page 307 The Permutation Expansion......Page 311 Determinants Exist*......Page 319 Determinants as Size Functions......Page 327 Laplace's Expansion*......Page 334 Topic: Cramer's Rule......Page 339 Topic: Speed of Calculating Determinants......Page 342 Topic: Projective Geometry......Page 345 Complex Vector Spaces......Page 357 Factoring and Complex Numbers; A Review*......Page 358 Complex Representations......Page 359 Definition and Examples......Page 361 Diagonalizability......Page 363 Eigenvalues and Eigenvectors......Page 367 Self-Composition*......Page 375 Strings*......Page 378 Polynomials of Maps and Matrices*......Page 389 Jordan Canonical Form*......Page 396 Topic: Method of Powers......Page 409 Topic: Stable Populations......Page 413 Topic: Linear Recurrences......Page 415 Propositions......Page 423 Quantifiers......Page 425 Techniques of Proof......Page 427 Sets, Functions, and Relations......Page 428