Linear Algebra Book

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: Voting Paradoxes......Page 157
Topic: Dimensional Analysis......Page 162
 Definition and Examples......Page 169
 Dimension Characterizes Isomorphism......Page 178
 Definition......Page 186
 Rangespace and Nullspace......Page 193
 Representing Linear Maps with Matrices......Page 205
 Any Matrix Represents a Linear Map*......Page 215
 Sums and Scalar Products......Page 222
 Matrix Multiplication......Page 224
 Mechanics of Matrix Multiplication......Page 232
 Inverses......Page 241
 Changing Representations of Vectors......Page 248
 Changing Map Representations......Page 252
 Orthogonal Projection Into a Line*......Page 260
 Gram-Schmidt Orthogonalization*......Page 264
 Projection Into a Subspace*......Page 270
Topic: Line of Best Fit......Page 279
Topic: Geometry of Linear Maps......Page 284
Topic: Markov Chains......Page 291
Topic: Orthonormal Matrices......Page 297
Determinants......Page 303
 Exploration*......Page 304
 Properties of Determinants......Page 309
 The Permutation Expansion......Page 313
 Determinants Exist*......Page 322
 Determinants as Size Functions......Page 329
 Laplace's Expansion*......Page 336
Topic: Cramer's Rule......Page 341
Topic: Speed of Calculating Determinants......Page 344
Topic: Projective Geometry......Page 347
Complex Vector Spaces......Page 359
 Factoring and Complex Numbers; A Review*......Page 360
 Complex Representations......Page 361
 Definition and Examples......Page 363
 Diagonalizability......Page 365
 Eigenvalues and Eigenvectors......Page 369
 Self-Composition*......Page 377
 Strings*......Page 380
 Polynomials of Maps and Matrices*......Page 391
 Jordan Canonical Form*......Page 398
Topic: Method of Powers......Page 411
Topic: Stable Populations......Page 415
Topic: Linear Recurrences......Page 417
Propositions......Page 425
Quantifiers......Page 427
Techniques of Proof......Page 429
Sets, Functions, and Relations......Page 430