| 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 Rn | 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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