A Course in Linear Algebra

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  • Format: Paperback
  • Copyright: 2011-09-14
  • Publisher: Dover Publications

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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.

Author Biography

The authors are Professors of Mathematics at College of the Holy Cross.

Table of Contents

Prefacep. iii
Erratap. vii
A Guide to the Exercisesp. xi
Vector Spacesp. 1
Introductionp. 1
Vector Spacesp. 2
Subspacesp. 12
Linear Combinationsp. 21
Linear Dependence and Linear Independencep. 26
Interlude on Solving Systems of Linear Equationsp. 32
Bases and Dimensionp. 47
Chapter Summaryp. 58
Supplementary Exercisesp. 59
Linear Transformationsp. 62
Introductionp. 62
Linear Transformationsp. 63
Linear Transformations between Finite-Dimensional Spacesp. 73
Kernel and Imagep. 84
Applications of the Dimension Theoremp. 95
Composition of Linear Transformationsp. 106
The Inverse of a Linear Transformationp. 114
Change of Basisp. 122
Chapter Summaryp. 129
Supplementary Exercisesp. 130
The Determinant Functionp. 133
Introductionp. 133
The Determinant as Areap. 134
The Determinant of an n x n Matrixp. 140
Further Properties of the Determinantp. 153
Chapter Summaryp. 160
Supplementary Exercisesp. 160
Eigenvalues, Eigenvectors, Diagonalization, and the Spectral Theorem in Rnp. 162
Introductionp. 162
Eigenvalues and Eigenvectorsp. 163
Diagonalizabilityp. 175
Geometry in Rnp. 184
Orthogonal Projections and the Gram-Schmidt Processp. 190
Symmetric Matricesp. 200
The Spectral Theoremp. 206
Chapter Summaryp. 217
Supplementary Exercisesp. 218
Complex; Numbers and Complex Vector Spacesp. 224
Introductionp. 224
Complex Numbersp. 225
Vector Spaces Over a Fieldp. 234
Geometry in a Complex Vector Spacep. 241
Chapter Summaryp. 249
Supplementary Exercisesp. 251
Jordan Canonical Formp. 253
Introductionp. 253
Triangular Formp. 254
A Canonical Form for Nilpotent Mappingsp. 263
Jordan Canonical Formp. 273
Computing Jordan Formp. 281
The Characteristic Polynomial and the Minimal Polynomialp. 287
Chapter Summaryp. 294
Supplementary Exercisesp. 295
Differential Equationsp. 299
Introductionp. 299
Two Motivating Examplesp. 300
Constant Coefficient Linear Differential Equations The Diagonalizable Casep. 305
Constant (Coefficient Linear Differential Equations: The General Casep. 312
One Ordinary Differential Equation with Constant Coefficientsp. 323
An Eigenvalue Problemp. 332
Chapter Summaryp. 340
Supplementary Exercisesp. 341
Some Basic Logic and Set Theoryp. 344
Setsp. 344
Statements and Logical Operatorsp. 345
Statements with Quantifiersp. 348
Further Notions from Set Theoryp. 349
Relations and Functionsp. 351
Injectivity, Surjectivity, and Bijectivityp. 354
Composites and Inverse Mappingsp. 354
Some (Optional) Remarks on Mathematics and Logicp. 355
Mathematical Inductionp. 359
Solutionsp. 367
Indexp. 429
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