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Preface | p. iii |
Errata | p. vii |
A Guide to the Exercises | p. xi |
Vector Spaces | p. 1 |
Introduction | p. 1 |
Vector Spaces | p. 2 |
Subspaces | p. 12 |
Linear Combinations | p. 21 |
Linear Dependence and Linear Independence | p. 26 |
Interlude on Solving Systems of Linear Equations | p. 32 |
Bases and Dimension | p. 47 |
Chapter Summary | p. 58 |
Supplementary Exercises | p. 59 |
Linear Transformations | p. 62 |
Introduction | p. 62 |
Linear Transformations | p. 63 |
Linear Transformations between Finite-Dimensional Spaces | p. 73 |
Kernel and Image | p. 84 |
Applications of the Dimension Theorem | p. 95 |
Composition of Linear Transformations | p. 106 |
The Inverse of a Linear Transformation | p. 114 |
Change of Basis | p. 122 |
Chapter Summary | p. 129 |
Supplementary Exercises | p. 130 |
The Determinant Function | p. 133 |
Introduction | p. 133 |
The Determinant as Area | p. 134 |
The Determinant of an n x n Matrix | p. 140 |
Further Properties of the Determinant | p. 153 |
Chapter Summary | p. 160 |
Supplementary Exercises | p. 160 |
Eigenvalues, Eigenvectors, Diagonalization, and the Spectral Theorem in R^{n} | p. 162 |
Introduction | p. 162 |
Eigenvalues and Eigenvectors | p. 163 |
Diagonalizability | p. 175 |
Geometry in Rn | p. 184 |
Orthogonal Projections and the Gram-Schmidt Process | p. 190 |
Symmetric Matrices | p. 200 |
The Spectral Theorem | p. 206 |
Chapter Summary | p. 217 |
Supplementary Exercises | p. 218 |
Complex; Numbers and Complex Vector Spaces | p. 224 |
Introduction | p. 224 |
Complex Numbers | p. 225 |
Vector Spaces Over a Field | p. 234 |
Geometry in a Complex Vector Space | p. 241 |
Chapter Summary | p. 249 |
Supplementary Exercises | p. 251 |
Jordan Canonical Form | p. 253 |
Introduction | p. 253 |
Triangular Form | p. 254 |
A Canonical Form for Nilpotent Mappings | p. 263 |
Jordan Canonical Form | p. 273 |
Computing Jordan Form | p. 281 |
The Characteristic Polynomial and the Minimal Polynomial | p. 287 |
Chapter Summary | p. 294 |
Supplementary Exercises | p. 295 |
Differential Equations | p. 299 |
Introduction | p. 299 |
Two Motivating Examples | p. 300 |
Constant Coefficient Linear Differential Equations The Diagonalizable Case | p. 305 |
Constant (Coefficient Linear Differential Equations: The General Case | p. 312 |
One Ordinary Differential Equation with Constant Coefficients | p. 323 |
An Eigenvalue Problem | p. 332 |
Chapter Summary | p. 340 |
Supplementary Exercises | p. 341 |
Some Basic Logic and Set Theory | p. 344 |
Sets | p. 344 |
Statements and Logical Operators | p. 345 |
Statements with Quantifiers | p. 348 |
Further Notions from Set Theory | p. 349 |
Relations and Functions | p. 351 |
Injectivity, Surjectivity, and Bijectivity | p. 354 |
Composites and Inverse Mappings | p. 354 |
Some (Optional) Remarks on Mathematics and Logic | p. 355 |
Mathematical Induction | p. 359 |
Solutions | p. 367 |
Index | p. 429 |
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