Linear Algebra: Ideas and Applications, 4th Edition
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- Published: 18 December 2015
- ISBN: 9781118909584
- Author(s): Richard C. Penney
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Praise for the Third Edition
“This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications.”
– Electric Review
A comprehensive introduction, Linear Algebra: Ideas and Applications, Fourth Edition provides a discussion of the theory and applications of linear algebra that blends abstract and computational concepts. With a focus on the development of mathematical intuition, the book emphasizes the need to understand both the applications of a particular technique and the mathematical ideas underlying the technique.
The book introduces each new concept in the context of an explicit numerical example, which allows the abstract concepts to grow organically out of the necessity to solve specific problems. The intuitive discussions are consistently followed by rigorous statements of results and proofs.
Linear Algebra: Ideas and Applications, Fourth Edition also features:
- Two new and independent sections on the rapidly developing subject of wavelets
- A thoroughly updated section on electrical circuit theory
- Illuminating applications of linear algebra with self-study questions for additional study
- End-of-chapter summaries and sections with true-false questions to aid readers with further comprehension of the presented material
- Numerous computer exercises throughout using MATLAB code
Linear Algebra: Ideas and Applications, Fourth Edition is an excellent undergraduate-level textbook for one or two semester courses for students majoring in mathematics, science, computer science, and engineering. With an emphasis on intuition development, the book is also an ideal self-study reference.
Preface xi
Features of the Text xiii
Acknowledgments xvii
About the Companion Website xix
1 Systems of Linear Equations 1
1.1 The Vector Space of m × n Matrices 1
The Space R^{n} 4
Linear Combinations and Linear Dependence 6
What is a Vector Space? 11
Exercises 17
1.1.1 Computer Projects 22
1.1.2 Applications to Graph Theory I 25
Exercises 27
1.2 Systems 28
Rank: The Maximum Number of Linearly Independent Equations 35
Exercises 38
1.2.1 Computer Projects 41
1.2.2 Applications to Circuit Theory 41
Exercises 46
1.3 Gaussian Elimination 47
Spanning in Polynomial Spaces 58
Computational Issues: Pivoting 61
Exercises 63
Computational Issues: Counting Flops 68
1.3.1 Computer Projects 69
1.3.2 Applications to Traffic Flow 72
1.4 Column Space and Nullspace 74
Subspaces 77
Exercises 86
1.4.1 Computer Projects 94
Chapter Summary 95
2 Linear Independence and Dimension 97
2.1 The Test for Linear Independence 97
Bases for the Column Space 104
Testing Functions for Independence 106
Exercises 108
2.1.1 Computer Projects 113
2.2 Dimension 114
Exercises 123
2.2.1 Computer Projects 127
2.2.2 Applications to Differential Equations 128
Exercises 131
2.3 Row Space and the rank-nullity theorem 132
Bases for the Row Space 134
Computational Issues: Computing Rank 142
Exercises 143
2.3.1 Computer Projects 146
Chapter Summary 147
3 Linear Transformations 149
3.1 The Linearity Properties 149
Exercises 157
3.1.1 Computer Projects 162
3.2 Matrix Multiplication (Composition) 164
Partitioned Matrices 171
Computational Issues: Parallel Computing 172
Exercises 173
3.2.1 Computer Projects 178
3.2.2 Applications to Graph Theory II 180
Exercises 181
3.3 Inverses 182
Computational Issues: Reduction versus Inverses 188
Exercises 190
3.3.1 Computer Projects 195
3.3.2 Applications to Economics 197
Exercises 202
3.4 The LU Factorization 203
Exercises 212
3.4.1 Computer Projects 214
3.5 The Matrix of a Linear Transformation 215
Coordinates 215
Isomorphism 228
Invertible Linear Transformations 229
Exercises 230
3.5.1 Computer Projects 235
Chapter Summary 236
4 Determinants 238
4.1 Definition of the Determinant 238
4.1.1 The Rest of the Proofs 246
Exercises 249
4.1.2 Computer Projects 251
4.2 Reduction and Determinants 252
Uniqueness of the Determinant 256
Exercises 258
4.2.1 Volume 261
Exercises 263
4.3 A Formula for Inverses 264
Exercises 268
Chapter Summary 269
5 Eigenvectors and Eigenvalues 271
5.1 Eigenvectors 271
Exercises 279
5.1.1 Computer Projects 282
5.1.2 Application to Markov Processes 283
Exercises 285
5.2 Diagonalization 287
Powers of Matrices 288
Exercises 290
5.2.1 Computer Projects 292
5.2.2 Application to Systems of Differential Equations 293
Exercises 295
5.3 Complex Eigenvectors 296
Complex Vector Spaces 303
Exercises 304
5.3.1 Computer Projects 305
Chapter Summary 306
6 Orthogonality 308
6.1 The Scalar Product in R^{N} 308
Orthogonal/Orthonormal Bases and Coordinates 312
Exercises 316
6.2 Projections: The Gram-Schmidt Process 318
The QR Decomposition 325
Uniqueness of the QR Factorization 327
Exercises 328
6.2.1 Computer Projects 331
6.3 Fourier Series: Scalar Product Spaces 333
Exercises 341
6.3.1 Application to Data Compression: Wavelets 344
Exercises 352
6.3.2 Computer Projects 353
6.4 Orthogonal Matrices 355
Householder Matrices 361
Exercises 364
Discrete Wavelet Transform 367
6.4.1 Computer Projects 369
6.5 Least Squares 370
Exercises 377
6.5.1 Computer Projects 380
6.6 Quadratic Forms: Orthogonal Diagonalization 381
The Spectral Theorem 385
The Principal Axis Theorem 386
Exercises 392
6.6.1 Computer Projects 395
6.7 The Singular Value Decomposition (SVD) 396
Application of the SVD to Least-Squares Problems 402
Exercises 404
Computing the SVD Using Householder Matrices 406
Diagonalizing Matrices Using Householder Matrices 408
6.8 Hermitian Symmetric and Unitary Matrices 410
Exercises 417
Chapter Summary 419
7 Generalized Eigenvectors 421
7.1 Generalized Eigenvectors 421
Exercises 429
7.2 Chain Bases 431
Jordan Form 438
Exercises 443
The Cayley-Hamilton Theorem 445
Chapter Summary 445
8 Numerical Techniques 446
8.1 Condition Number 446
Norms 446
Condition Number 448
Least Squares 451
Exercises 451
8.2 Computing Eigenvalues 452
Iteration 453
The QR Method 457
Exercises 462
Chapter Summary 464
Answers and Hints 465
Index 487
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