Linear Algebra: Ideas and Applications Set, 4th Edition
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- Published: 14 April 2016
- ISBN: 9781118911884
- Author(s): Richard C. Penney
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A unified introduction to linear algebra that reinforces and emphasizes a conceptual and hands-on understanding of the essential ideas. Promoting the development of intuition rather than the simple application of methods, this book successfully helps readers to understand not only how to implement a technique, but why its use is important. In addition, the author outlines an analytical, algebraic, and geometric discussion of the provided definitions, theorems, and proofs. For each concept, an abstract foundation is presented together with its computational output, and this parallel structure clearly and immediately illustrates the relationship between the theory and its appropriate applications. The Fourth Edition features new coverage on orthogonal wavelets, which is a cutting edge application of linear algebra that has only become prominent within the last 10 years.
The Student Solutions Manual contains solutions to the odd numbered problems and is available to further aid in reader comprehension, and an Instructor's Solutions Manual (inclusive of suggested syllabi) is available via written request to the Publisher. Both the Student and Instructor Manuals also have been enhanced with further discussions of the applications sections, which is ideal for readers who wish to obtain a deeper knowledge than that provided by pure algorithmic approaches. A related website houses the referenced MATLAB code as well as full-color images of select figures.
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 Rn 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 RN 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
SOLUTIONS MANUAL
STUDENT MANUAL 1
1 SYSTEMS OF LINEAR EQUATIONS 3
1.1 The Vector Space of m × n Matrices 3
1.1.2 Applications to Graph Theory I 7
1.2 Systems 8
1.2.2 Applications to Circuit Theory 11
1.3 Gaussian Elimination 13
1.3.2 Applications to Traffic Flow 18
1.4 Column Space and Nullspace 19
2 LINEAR INDEPENDENCE AND DIMENSION 26
2.1 The Test for Linear Independence 26
2.2 Dimension 33
2.2.2 Applications to Differential Equations 37
2.3 Row Space and the Rank-Nullity Theorem 38
3 LINEAR TRANSFORMATIONS 43
3.1 The Linearity Properties 43
3.2 Matrix Multiplication (Composition) 49
3.2.2 Applications to Graph Theory II 55
3.3 Inverses 55
3.3.2 Applications to Economics 60
3.4 The LU Factorization 61
3.5 The Matrix of a Linear Transformation 62
4 DETERMINANTS 67
4.1 Definition of the Determinant 67
4.2 Reduction and Determinants 69
4.2.1 Volume 72
4.3 A Formula for Inverses 74
5 EIGENVECTORS AND EIGENVALUES 76
5.1 Eigenvectors 76
5.1.2 Application to Markov Processes 79
5.2 Diagonalization 80
5.2.1 Application to Systems of Differential Equations 82
5.3 Complex Eigenvectors 83
6 ORTHOGONALITY 85
6.1 The Scalar Product in ℝn 85
6.2 Projections: The Gram-Schmidt Process 87
6.3 Fourier Series: Scalar Product Spaces 89
6.4 Orthogonal Matrices 92
6.5 Least Squares 93
6.6 Quadratic Forms: Orthogonal Diagonalization 94
6.7 The Singular Value Decomposition (SVD) 97
6.8 Hermitian Symmetric and Unitary Matrices 98
7 GENERALIZED EIGENVECTORS 100
7.1 Generalized Eigenvectors 100
7.2 Chain Bases 104
8 NUMERICAL TECHNIQUES 107
8.1 Condition Number 107
8.2 Computing Eigenvalues 108
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