Linear Algebra Book

Table of contents :
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