Linear Algebra

linear algebra,mathematics,textbook linear algebra Linear Systems 11 Solving Linear Systems 11 Gauss' Method 12 Describing the Solution Set 21 General = Particular + Homogeneous 30 Linear Geometry of n-Space 42 Vectors in Space 42 Length and Angle Measures* 48 Reduced Echelon Form 56 Gauss-Jordan Reduction 56 Row Equivalence 62 Topic: Computer Algebra Systems 72 Topic: Input-Output Analysis 74 Topic: Accuracy of Computations 78 Topic: Analyzing Networks 82 Vector Spaces 89 Definition of Vector Space 90 Definition and Examples 90 Subspaces and Spanning Sets 101 Linear Independence 112 Definition and Examples 112 Basis and Dimension 123 Basis 123 Dimension 129 Vector Spaces and Linear Systems 134 Combining Subspaces* 141 Topic: Fields 151 Topic: Crystals 153 Topic: Voting Paradoxes 157 Topic: Dimensional Analysis 162 Maps Between Spaces 169 Isomorphisms 169 Definition and Examples 169 Dimension Characterizes Isomorphism 178 Homomorphisms 186 Definition 186 Rangespace and Nullspace 193 Computing Linear Maps 205 Representing Linear Maps with Matrices 205 Any Matrix Represents a Linear Map* 215 Matrix Operations 222 Sums and Scalar Products 222 Matrix Multiplication 224 Mechanics of Matrix Multiplication 232 Inverses 241 Change of Basis 248 Changing Representations of Vectors 248 Changing Map Representations 252 Projection 260 Orthogonal Projection Into a Line* 260 Gram-Schmidt Orthogonalization* 264 Projection Into a Subspace* 270 Topic: Line of Best Fit 279 Topic: Geometry of Linear Maps 284 Topic: Markov Chains 291 Topic: Orthonormal Matrices 297 Determinants 303 Definition 304 Exploration* 304 Properties of Determinants 309 The Permutation Expansion 313 Determinants Exist* 322 Geometry of Determinants 329 Determinants as Size Functions 329 Other Formulas 336 Laplace's Expansion* 336 Topic: Cramer's Rule 341 Topic: Speed of Calculating Determinants 344 Topic: Projective Geometry 347 Similarity 359 Complex Vector Spaces 359 Factoring and Complex Numbers; A Review* 360 Complex Representations 361 Similarity 363 Definition and Examples 363 Diagonalizability 365 Eigenvalues and Eigenvectors 369 Nilpotence 377 Self-Composition* 377 Strings* 380 Jordan Form 391 Polynomials of Maps and Matrices* 391 Jordan Canonical Form* 398 Topic: Method of Powers 411 Topic: Stable Populations 415 Topic: Linear Recurrences 417 Appendix 425 Propositions 425 Quantifiers 427 Techniques of Proof 429 Sets, Functions, and Relations 430