| Chapter One: Linear Systems |
1 |
| I Solving Linear Systems . . . . . . . . . |
1 |
| 1 Gauss' Method . . . . . . . . . . . . . |
2 |
| 2 Describing the Solution Set . . . . . . |
11 |
| 3 General = Particular + Homogeneous . . . |
20 |
| II Linear Geometry of n-Space . . . . . . |
32 |
| 1 Vectors in Space . . . . . . . . . . . . |
32 |
| 2 Length and Angle Measures_ . . . . . . . |
38 |
| III Reduced Echelon Form . . . . . . . . . |
46 |
| 1 Gauss-Jordan Reduction . . . . . . . . . |
46 |
| 2 Row Equivalence . . . . . . . . . . . . |
52 |
| Topic: Computer Algebra Systems . . . . . |
62 |
| Topic: Input-Output Analysis . . . . . . . |
64 |
| Topic: Accuracy of Computations . . . . . |
68 |
| Topic: Analyzing Networks . . . . . . . . |
72 |
| Chapter Two: Vector Spaces |
79 |
| I De_nition of Vector Space . . . . . . . |
80 |
| 1 De_nition and Examples . . . . . . . . . |
80 |
| 2 Subspaces and Spanning Sets . . . . . . |
91 |
| II Linear Independence . . . . . . . . . . |
101 |
| 1 De_nition and Examples . . . . . . . . . |
101 |
| III Basis and Dimension . . . . . . . . . |
112 |
| 1 Basis . . . . . . . . . . . . . . . . . |
112 |
| 2 Dimension . . . . . . . . . . . . . . . |
118 |
| 3 Vector Spaces and Linear Systems . . . . |
123 |
| 4 Combining Subspaces_ . . . . . . . . . . |
130 |
| Topic: Fields . . . . . . . . . . . . . . |
140 |
| Topic: Crystals . . . . . . . . . . . . . |
142 |
| Topic: Voting Paradoxes . . . . . . . . . |
146 |
| Topic: Dimensional Analysis . . . . . . . |
152 |
| Chapter Three: Maps Between Spaces |
159 |
| I Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
159 |
| 1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . |
159 |
| 2 Dimension Characterizes Isomorphism . . . . . . . . . . . . . |
168 |
| II Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . |
176 |
| 1 De_nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
176 |
| 2 Rangespace and Nullspace . . . . . . . . . . . . . . . . . . . . . |
183 |
| III Computing Linear Maps . . . . . . . . . . . . . . . . . . . . . . . |
195 |
| 1 Representing Linear Maps with Matrices . . . . . . . . . . . . |
195 |
| 2 Any Matrix Represents a Linear Map_ . . . . . . . . . . . . . . |
205 |
| IV Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . |
212 |
| 1 Sums and Scalar Products . . . . . . . . . . . . . . . . . . . . . |
212 |
| 2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . |
214 |
| 3 Mechanics of Matrix Multiplication . . . . . . . . . . . . . . . . |
222 |
| 4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
231 |
| V Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
238 |
| 1 Changing Representations of Vectors . . . . . . . . . . . . . . |
238 |
| 2 Changing Map Representations . . . . . . . . . . . . . . . . . . |
242 |
| VI Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
250 |
| 1 Orthogonal Projection Into a Line_ . . . . . . . . . . . . . . . . |
250 |
| 2 Gram-Schmidt Orthogonalization_ . . . . . . . . . . . . . . . . |
254 |
| 3 Projection Into a Subspace_ . . . . . . . . . . . . . . . . . . . . |
260 |
| Topic: Line of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . |
269 |
| Topic: Geometry of Linear Maps . . . . . . . . . . . . . . . . . . . |
274 |
| Topic: Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . |
281 |
| Topic: Orthonormal Matrices . . . . . . . . . . . . . . . . . . . . . . |
287 |
| Chapter Four: Determinants |
293 |
| I Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
294 |
| 1 Exploration_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
294 |
| 2 Properties of Determinants . . . . . . . . . . . . . . . . . . . . |
299 |
| 3 The Permutation Expansion . . . . . . . . . . . . . . . . . . . . |
303 |
| 4 Determinants Exist_ . . . . . . . . . . . . . . . . . . . . . . . . |
312 |
| II Geometry of Determinants . . . . . . . . . . . . . . . . . . . . . . |
319 |
| 1 Determinants as Size Functions . . . . . . . . . . . . . . . . . . |
319 |
| III Other Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
326 |
| 1 Laplace's Expansion_ . . . . . . . . . . . . . . . . . . . . . . . . |
326 |
| Topic: Cramer's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . |
331 |
| Topic: Speed of Calculating Determinants . . . . . . . . . . . . |
334 |
| Topic: Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . |
337 |
| Chapter Five: Similarity |
349 |
| I Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . |
349 |
| 1 Factoring and Complex Numbers; A Review_ . . . . . . . . |
350 |
| 2 Complex Representations . . . . . . . . . . . . . . . . . . . . . |
351 |
| II Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
353 |
| 1 De_nition and Examples . . . . . . . . . . . . . . . . . . . |
353 |
| 2 Diagonalizability . . . . . . . . . . . . . . . . . . . . . |
355 |
| 3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . |
359 |
| III Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . |
367 |
| 1 Self-Composition_ . . . . . . . . . . . . . . . . . . . . . |
367 |
| 2 Strings_ . . . . . . . . . . . . . . . . . . . . . . . . . . |
370 |
| IV Jordan Form . . . . . . . . . . . . . . . . . . . . . . . . |
381 |
| 1 Polynomials of Maps and Matrices_ . . . . . . . . . . . . . |
381 |
| 2 Jordan Canonical Form_ . . . . . . . . . . . . . . . . . . . |
388 |
| Topic: Method of Powers . . . . . . . . . . . . . . . . . . . |
401 |
| Topic: Stable Populations . . . . . . . . . . . . . . . . . . |
405 |
| Topic: Linear Recurrences . . . . . . . . . . . . . . . . . . |
407 |
| Appendix |
A-1 |
| Propositions . . . . . . . . . . . . . . . . . . . . . . . . . |
A-1 |
| Quanti_ers . . . . . . . . . . . . . . . . . . . . . . . . . . |
A-3 |
| Techniques of Proof . . . . . . . . . . . . . . . . . . . . . |
A-5 |
| Sets, Functions, and Relations . . . . . . . . . . . . . . . . |
A-7 |