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A Course in Linear Algebra
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Overview
Suitable for advanced undergraduates and graduate students, this text offers a complete introduction to the basic concepts of linear algebra. Interesting and inspiring in its approach, it imparts an understanding of the subject's logical structure as well as the ways in which linear algebra provides solutions to problems in many branches of mathematics.
The authors define general vector spaces and linear mappings at the outset and base all subsequent developments on these concepts. This approach provides a readymade context, motivation, and geometric interpretation for each new computational technique. Proofs and abstract problemsolving are introduced from the start, offering students an immediate opportunity to practice applying what they've learned. Each chapter contains an introduction, summary, and supplementary exercises. The text concludes with a pair of helpful appendixes and solutions to selected exercises.
Product Details
ISBN13:  9780486469089 

Publisher:  Dover Publications 
Publication date:  09/14/2011 
Series:  Dover Books on Mathematics Series 
Pages:  464 
Sales rank:  1,211,223 
Product dimensions:  6.10(w) x 9.10(h) x 1.00(d) 
About the Author
The authors are Professors of Mathematics at College of the Holy Cross.
Read an Excerpt
The authors are Professors of Mathematics at College of the Holy Cross.
First Chapter
The authors are Professors of Mathematics at College of the Holy Cross.
Table of Contents
Preface iii
Errata vii
A Guide to the Exercises xi
Chapter 1 Vector Spaces 1
Introduction 1
1.1 Vector Spaces 2
1.2 Subspaces 12
1.3 Linear Combinations 21
1.4 Linear Dependence and Linear Independence 26
1.5 Interlude on Solving Systems of Linear Equations 32
1.6 Bases and Dimension 47
Chapter Summary 58
Supplementary Exercises 59
Chapter 2 Linear Transformations 62
Introduction 62
2.1 Linear Transformations 63
2.2 Linear Transformations between FiniteDimensional Spaces 73
2.3 Kernel and Image 84
2.4 Applications of the Dimension Theorem 95
2.5 Composition of Linear Transformations 106
2.6 The Inverse of a Linear Transformation 114
2.7 Change of Basis 122
Chapter Summary 129
Supplementary Exercises 130
Chapter 3 The Determinant Function 133
Introduction 133
3.1 The Determinant as Area 134
3.2 The Determinant of an n x n Matrix 140
3.3 Further Properties of the Determinant 153
Chapter Summary 160
Supplementary Exercises 160
Chapter 4 Eigenvalues, Eigenvectors, Diagonalization, and the Spectral Theorem in R^{n} 162
Introduction 162
4.1 Eigenvalues and Eigenvectors 163
4.2 Diagonalizability 175
4.3 Geometry in Rn 184
4.4 Orthogonal Projections and the GramSchmidt Process 190
4.5 Symmetric Matrices 200
4.6 The Spectral Theorem 206
Chapter Summary 217
Supplementary Exercises 218
Chapter 5 Complex; Numbers and Complex Vector Spaces 224
Introduction 224
5.1 Complex Numbers 225
5.2 Vector Spaces Over a Field 234
5.3 Geometry in a Complex Vector Space 241
Chapter Summary 249
Supplementary Exercises 251
Chapter 6 Jordan Canonical Form 253
Introduction 253
6.1 Triangular Form 254
6.2 A Canonical Form for Nilpotent Mappings 263
6.3 Jordan Canonical Form 273
6.4 Computing Jordan Form 281
6.5 The Characteristic Polynomial and the Minimal Polynomial 287
Chapter Summary 294
Supplementary Exercises 295
Chapter 7 Differential Equations 299
Introduction 299
7.1 Two Motivating Examples 300
7.2 Constant Coefficient Linear Differential Equations The Diagonalizable Case 305
7.3 Constant (Coefficient Linear Differential Equations: The General Case 312
7.4 One Ordinary Differential Equation with Constant Coefficients 323
7.5 An Eigenvalue Problem 332
Chapter Summary 340
Supplementary Exercises 341
Appendix 1 Some Basic Logic and Set Theory 344
A1.1 Sets 344
A1.2 Statements and Logical Operators 345
A1.3 Statements with Quantifiers 348
A1.4 Further Notions from Set Theory 349
A1.5 Relations and Functions 351
A1.6 Injectivity, Surjectivity, and Bijectivity 354
A1.7 Composites and Inverse Mappings 354
A1.8 Some (Optional) Remarks on Mathematics and Logic 355
Appendix 2 Mathematical Induction 359
Solutions 367
Index 429
Reading Group Guide
Preface iii
Errata vii
A Guide to the Exercises xi
Chapter 1 Vector Spaces 1
Introduction 1
1.1 Vector Spaces 2
1.2 Subspaces 12
1.3 Linear Combinations 21
1.4 Linear Dependence and Linear Independence 26
1.5 Interlude on Solving Systems of Linear Equations 32
1.6 Bases and Dimension 47
Chapter Summary 58
Supplementary Exercises 59
Chapter 2 Linear Transformations 62
Introduction 62
2.1 Linear Transformations 63
2.2 Linear Transformations between FiniteDimensional Spaces 73
2.3 Kernel and Image 84
2.4 Applications of the Dimension Theorem 95
2.5 Composition of Linear Transformations 106
2.6 The Inverse of a Linear Transformation 114
2.7 Change of Basis 122
Chapter Summary 129
Supplementary Exercises 130
Chapter 3 The Determinant Function 133
Introduction 133
3.1 The Determinant as Area 134
3.2 The Determinant of an n x n Matrix 140
3.3 Further Properties of the Determinant 153
Chapter Summary 160
Supplementary Exercises 160
Chapter 4 Eigenvalues, Eigenvectors, Diagonalization, and the Spectral Theorem in R^{n} 162
Introduction 162
4.1 Eigenvalues and Eigenvectors 163
4.2 Diagonalizability 175
4.3 Geometry in Rn 184
4.4 Orthogonal Projections and the GramSchmidt Process 190
4.5 Symmetric Matrices 200
4.6 The Spectral Theorem 206
Chapter Summary 217
Supplementary Exercises 218
Chapter 5 Complex; Numbers and Complex Vector Spaces 224
Introduction 224
5.1 Complex Numbers 225
5.2 Vector Spaces Over a Field 234
5.3 Geometry in a Complex Vector Space 241
Chapter Summary 249
Supplementary Exercises 251
Chapter 6 Jordan Canonical Form 253
Introduction 253
6.1 Triangular Form 254
6.2 A Canonical Form for Nilpotent Mappings 263
6.3 Jordan Canonical Form 273
6.4 Computing Jordan Form 281
6.5 The Characteristic Polynomial and the Minimal Polynomial 287
Chapter Summary 294
Supplementary Exercises 295
Chapter 7 Differential Equations 299
Introduction 299
7.1 Two Motivating Examples 300
7.2 Constant Coefficient Linear Differential Equations The Diagonalizable Case 305
7.3 Constant (Coefficient Linear Differential Equations: The General Case 312
7.4 One Ordinary Differential Equation with Constant Coefficients 323
7.5 An Eigenvalue Problem 332
Chapter Summary 340
Supplementary Exercises 341
Appendix 1 Some Basic Logic and Set Theory 344
A1.1 Sets 344
A1.2 Statements and Logical Operators 345
A1.3 Statements with Quantifiers 348
A1.4 Further Notions from Set Theory 349
A1.5 Relations and Functions 351
A1.6 Injectivity, Surjectivity, and Bijectivity 354
A1.7 Composites and Inverse Mappings 354
A1.8 Some (Optional) Remarks on Mathematics and Logic 355
Appendix 2 Mathematical Induction 359
Solutions 367
Index 429
Interviews
Preface iii
Errata vii
A Guide to the Exercises xi
Chapter 1 Vector Spaces 1
Introduction 1
1.1 Vector Spaces 2
1.2 Subspaces 12
1.3 Linear Combinations 21
1.4 Linear Dependence and Linear Independence 26
1.5 Interlude on Solving Systems of Linear Equations 32
1.6 Bases and Dimension 47
Chapter Summary 58
Supplementary Exercises 59
Chapter 2 Linear Transformations 62
Introduction 62
2.1 Linear Transformations 63
2.2 Linear Transformations between FiniteDimensional Spaces 73
2.3 Kernel and Image 84
2.4 Applications of the Dimension Theorem 95
2.5 Composition of Linear Transformations 106
2.6 The Inverse of a Linear Transformation 114
2.7 Change of Basis 122
Chapter Summary 129
Supplementary Exercises 130
Chapter 3 The Determinant Function 133
Introduction 133
3.1 The Determinant as Area 134
3.2 The Determinant of an n x n Matrix 140
3.3 Further Properties of the Determinant 153
Chapter Summary 160
Supplementary Exercises 160
Chapter 4 Eigenvalues, Eigenvectors, Diagonalization, and the Spectral Theorem in R^{n} 162
Introduction 162
4.1 Eigenvalues and Eigenvectors 163
4.2 Diagonalizability 175
4.3 Geometry in Rn 184
4.4 Orthogonal Projections and the GramSchmidt Process 190
4.5 Symmetric Matrices 200
4.6 The Spectral Theorem 206
Chapter Summary 217
Supplementary Exercises 218
Chapter 5 Complex; Numbers and Complex Vector Spaces 224
Introduction 224
5.1 Complex Numbers 225
5.2 Vector Spaces Over a Field 234
5.3 Geometry in a Complex Vector Space 241
Chapter Summary 249
Supplementary Exercises 251
Chapter 6 Jordan Canonical Form 253
Introduction 253
6.1 Triangular Form 254
6.2 A Canonical Form for Nilpotent Mappings 263
6.3 Jordan Canonical Form 273
6.4 Computing Jordan Form 281
6.5 The Characteristic Polynomial and the Minimal Polynomial 287
Chapter Summary 294
Supplementary Exercises 295
Chapter 7 Differential Equations 299
Introduction 299
7.1 Two Motivating Examples 300
7.2 Constant Coefficient Linear Differential Equations The Diagonalizable Case 305
7.3 Constant (Coefficient Linear Differential Equations: The General Case 312
7.4 One Ordinary Differential Equation with Constant Coefficients 323
7.5 An Eigenvalue Problem 332
Chapter Summary 340
Supplementary Exercises 341
Appendix 1 Some Basic Logic and Set Theory 344
A1.1 Sets 344
A1.2 Statements and Logical Operators 345
A1.3 Statements with Quantifiers 348
A1.4 Further Notions from Set Theory 349
A1.5 Relations and Functions 351
A1.6 Injectivity, Surjectivity, and Bijectivity 354
A1.7 Composites and Inverse Mappings 354
A1.8 Some (Optional) Remarks on Mathematics and Logic 355
Appendix 2 Mathematical Induction 359
Solutions 367
Index 429
Recipe
Preface iii
Errata vii
A Guide to the Exercises xi
Chapter 1 Vector Spaces 1
Introduction 1
1.1 Vector Spaces 2
1.2 Subspaces 12
1.3 Linear Combinations 21
1.4 Linear Dependence and Linear Independence 26
1.5 Interlude on Solving Systems of Linear Equations 32
1.6 Bases and Dimension 47
Chapter Summary 58
Supplementary Exercises 59
Chapter 2 Linear Transformations 62
Introduction 62
2.1 Linear Transformations 63
2.2 Linear Transformations between FiniteDimensional Spaces 73
2.3 Kernel and Image 84
2.4 Applications of the Dimension Theorem 95
2.5 Composition of Linear Transformations 106
2.6 The Inverse of a Linear Transformation 114
2.7 Change of Basis 122
Chapter Summary 129
Supplementary Exercises 130
Chapter 3 The Determinant Function 133
Introduction 133
3.1 The Determinant as Area 134
3.2 The Determinant of an n x n Matrix 140
3.3 Further Properties of the Determinant 153
Chapter Summary 160
Supplementary Exercises 160
Chapter 4 Eigenvalues, Eigenvectors, Diagonalization, and the Spectral Theorem in R^{n} 162
Introduction 162
4.1 Eigenvalues and Eigenvectors 163
4.2 Diagonalizability 175
4.3 Geometry in Rn 184
4.4 Orthogonal Projections and the GramSchmidt Process 190
4.5 Symmetric Matrices 200
4.6 The Spectral Theorem 206
Chapter Summary 217
Supplementary Exercises 218
Chapter 5 Complex; Numbers and Complex Vector Spaces 224
Introduction 224
5.1 Complex Numbers 225
5.2 Vector Spaces Over a Field 234
5.3 Geometry in a Complex Vector Space 241
Chapter Summary 249
Supplementary Exercises 251
Chapter 6 Jordan Canonical Form 253
Introduction 253
6.1 Triangular Form 254
6.2 A Canonical Form for Nilpotent Mappings 263
6.3 Jordan Canonical Form 273
6.4 Computing Jordan Form 281
6.5 The Characteristic Polynomial and the Minimal Polynomial 287
Chapter Summary 294
Supplementary Exercises 295
Chapter 7 Differential Equations 299
Introduction 299
7.1 Two Motivating Examples 300
7.2 Constant Coefficient Linear Differential Equations The Diagonalizable Case 305
7.3 Constant (Coefficient Linear Differential Equations: The General Case 312
7.4 One Ordinary Differential Equation with Constant Coefficients 323
7.5 An Eigenvalue Problem 332
Chapter Summary 340
Supplementary Exercises 341
Appendix 1 Some Basic Logic and Set Theory 344
A1.1 Sets 344
A1.2 Statements and Logical Operators 345
A1.3 Statements with Quantifiers 348
A1.4 Further Notions from Set Theory 349
A1.5 Relations and Functions 351
A1.6 Injectivity, Surjectivity, and Bijectivity 354
A1.7 Composites and Inverse Mappings 354
A1.8 Some (Optional) Remarks on Mathematics and Logic 355
Appendix 2 Mathematical Induction 359
Solutions 367
Index 429