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