A Course in Linear Algebra

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ISBN: 9780486469089 | 0486469085
Cover: Paperback
Copyright: 9/14/2011

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Suitable for advanced undergraduates and graduate students, this text introduces basic concepts of linear algebra. Each chapter contains an introduction, definitions, and propositions, in addition to multiple examples, lemmas, theorems, corollaries, and proofs. Each chapter features numerous supplemental exercises, and solutions to selected problems appear at the end. 1988 edition.

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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A Course in Linear Algebra

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