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 48
 Gauss-Jordan Reduction......Page 56
 Row Equivalence......Page 62
Topic: Computer Algebra Systems......Page 72
Topic: Input-Output Analysis......Page 74
Topic: Accuracy of Computations......Page 78
Topic: Analyzing Networks......Page 82
Vector Spaces......Page 89
 Definition and Examples......Page 90
 Subspaces and Spanning Sets......Page 101
 Definition and Examples......Page 112
 Basis......Page 123
 Dimension......Page 129
 Vector Spaces and Linear Systems......Page 134
 Combining Subspaces*......Page 141
Topic: Fields......Page 151
Topic: Crystals......Page 153
Topic: Dimensional Analysis......Page 157
 Definition and Examples......Page 165
 Dimension Characterizes Isomorphism......Page 174
 Definition......Page 182
 Rangespace and Nullspace......Page 189
 Representing Linear Maps with Matrices......Page 201
 Any Matrix Represents a Linear Map*......Page 211
 Sums and Scalar Products......Page 218
 Matrix Multiplication......Page 220
 Mechanics of Matrix Multiplication......Page 228
 Inverses......Page 237
 Changing Representations of Vectors......Page 244
 Changing Map Representations......Page 248
 Orthogonal Projection Into a Line*......Page 256
 Gram-Schmidt Orthogonalization*......Page 260
 Projection Into a Subspace*......Page 266
Topic: Line of Best Fit......Page 275
Topic: Geometry of Linear Maps......Page 280
Topic: Markov Chains......Page 287
Topic: Orthonormal Matrices......Page 293
Determinants......Page 299
 Exploration*......Page 300
 Properties of Determinants......Page 305
 The Permutation Expansion......Page 309
 Determinants Exist*......Page 318
 Determinants as Size Functions......Page 325
 Laplace\'s Expansion*......Page 332
Topic: Cramer\'s Rule......Page 337
Topic: Speed of Calculating Determinants......Page 340
Topic: Projective Geometry......Page 343
Complex Vector Spaces......Page 355
 Factoring and Complex Numbers; A Review*......Page 356
 Complex Representations......Page 357
 Definition and Examples......Page 359
 Diagonalizability......Page 361
 Eigenvalues and Eigenvectors......Page 365
 Self-Composition*......Page 373
 Strings*......Page 376
 Polynomials of Maps and Matrices*......Page 387
 Jordan Canonical Form*......Page 394
Topic: Method of Powers......Page 407
Topic: Stable Populations......Page 411
Topic: Linear Recurrences......Page 413
Propositions......Page 421
Quantifiers......Page 423
Techniques of Proof......Page 425
Sets, Functions, and Relations......Page 426