| Author: |
David Poole
|
| Release at: | 2015 |
| Pages: | 722 |
| Edition: |
First Edition
|
| File Size: | 68 MB |
| File Type: | |
| Language: | English |
Content of Linear Algebra A Modern Introduction
Chapter 1 Vectors 1
1.0 Introduction: The Racetrack Game 1
1.1 The Geometry and Algebra of Vectors 3
1.2 Length and Angle: The Dot Product 18
1.3 Exploration: Vectors and Geometry 32
Lines and Planes 34
Exploration: The Cross Product 48
Writing Project: The Origins of the Dot Product and Cross Product 49
1.4 Applications 50
Force Vectors 50
Chapter Review 55
Chapter 2 Systems of Linear Equations 57
2.0 Introduction: Triviality 57
2.1 Introduction to Systems of Linear Equations 58
2.2 Direct Methods for Solving Linear Systems 64
Writing Project: A History of Gaussian Elimination 82
Explorations: Lies My Computer Told Me 83
Partial Pivoting 84
Counting Operations: An Introduction to the
Analysis of Algorithms 85
2.3 Spanning Sets and Linear Independence 88
2.4 Applications 99
Allocation of Resources 99
Balancing Chemical Equations 101
Network Analysis 102
Electrical Networks 104
Linear Economic Models 107
Finite Linear Games 109
Vignette: The Global Positioning System 121
2.5 Iterative Methods for Solving Linear Systems 124
Chapter Review 134
Chapter 3 Matrices 136
3.0 Introduction: Matrices in Action 136
3. 1 Matrix Operations 138
3.2 Matrix Algebra 154
3.3 The Inverse of a Matrix 163
3.4 The LU Factorization 180
3.5 Subspaces, Basis, Dimension, and Rank
3.6 Introduction to Linear Transformations
Vignette: Robotics 226
3.7 Applications 230
Markov Chains 230
Linear Economic Models 235
Population Growth 239
Graphs and Digraphs 241
Chapter Review 251
Chapter 4 Eigenvalues and Eigenvectors 253
4.0 Introduction: A Dynamical System on Graphs 253
4.1 Introduction to Eigenvalues and Eigenvectors 254
4.2 Determinants 263
Writing Project: Which Came First: The Matrix or the Determinant?
Vignette: Lewis Carroll's Condensation Method 284
Exploration: Geometric Applications of Determinants 286
4.3 Eigenvalues and Eigenvectors of n X n Matrices 292
Writing Project: The History of Eigenvalues 301
3.4 Similarity and Diagonalization 301
3.5 Iterative Methods for Computing Eigenvalues 31 1
3.6 Applications and the Perron-Frobenius Theorem 325
Markov Chains 325
Population Growth 330
The Perron-Frobenius Theorem 332
Linear Recurrence Relations 335
Systems of Linear Differential Equations 340
Discrete Linear Dynamical Systems 348
Vignette: Ranking Sports Teams and Searching the Internet 356
Chapter Review 364
Chapter 5 Orthogonality 366
5.0 Introduction: Shadows on a Wall 366
5.1 Orthogonality in IR" 368
5.2 Orthogonal Complements and Orthogonal Projections 378
5.3 The Gram-Schmidt Process and the QR Factorization 388
Explorations: The Modified QR Factorization 396
Approximating Eigenvalues with the QR Algorithm 398
5.4 Orthogonal Diagonalization of Symmetric Matrices 400
5.5 Applications 408
Quadratic Forms 408
Graphing Quadratic Equations 415
Chapter Review 425
Chapter 6 Vector Spaces 427
6.0 Introduction: Fibonacci in (Vector) Space 427
6. 1 Vector Spaces and Subspaces 429
6.2 Writing Project: The Rise of Vector Spaces 443
6.3 Linear Independence, Basis, and Dimension 443
6.4 Exploration: Magic Squares 460
6.5 Change of Basis 463
6.6 Linear Transformations 472
The Kernel and Range of a Linear Transformation 481
The Matrix of a Linear Transformation 497
Exploration: Tilings, Lattices, and the Crystallographic Restriction
6.7 Applications 518
Homogeneous Linear Differential Equations 518
Chapter Review 527
Chapter 7 Distance and Approximation 529
7.0 Introduction: Taxicab Geometry 529
7. 1 Inner Product Spaces 531
Explorations: Vectors and Matrices with Complex Entries 543
Geometric Inequalities and Optimization Problems 547
7.2 Norms and Distance Functions 552
7.3 Least Squares Approximation 568
7.4 The Singular Value Decomposition 590
Vignette: Digital Image Compression 607
7.5 Applications 610
Approximation of Functions 610
Chapter Review 618
Chapter 8 Codes Online only 620
8. 1 Code Vectors 620
Vignette: The Codabar System 626
8.2 Error-Correcting Codes 627
8.3 Dual Codes 632
8.4 Linear Codes 639
8.5 The Minimum Distance of a Code 644
APPENDIX A Mathematical Notation and Methods of Proof Al
APPENDIX B Mathematical Induction B 1
APPENDIX C Complex Numbers Cl
APPENDIX D Polynomials D 1
APPENDIX E Technology Bytes Online only
Answers to Selected Odd-Numbered Exercises ANSI
Index II
1.0 Introduction: The Racetrack Game 1
1.1 The Geometry and Algebra of Vectors 3
1.2 Length and Angle: The Dot Product 18
1.3 Exploration: Vectors and Geometry 32
Lines and Planes 34
Exploration: The Cross Product 48
Writing Project: The Origins of the Dot Product and Cross Product 49
1.4 Applications 50
Force Vectors 50
Chapter Review 55
Chapter 2 Systems of Linear Equations 57
2.0 Introduction: Triviality 57
2.1 Introduction to Systems of Linear Equations 58
2.2 Direct Methods for Solving Linear Systems 64
Writing Project: A History of Gaussian Elimination 82
Explorations: Lies My Computer Told Me 83
Partial Pivoting 84
Counting Operations: An Introduction to the
Analysis of Algorithms 85
2.3 Spanning Sets and Linear Independence 88
2.4 Applications 99
Allocation of Resources 99
Balancing Chemical Equations 101
Network Analysis 102
Electrical Networks 104
Linear Economic Models 107
Finite Linear Games 109
Vignette: The Global Positioning System 121
2.5 Iterative Methods for Solving Linear Systems 124
Chapter Review 134
Chapter 3 Matrices 136
3.0 Introduction: Matrices in Action 136
3. 1 Matrix Operations 138
3.2 Matrix Algebra 154
3.3 The Inverse of a Matrix 163
3.4 The LU Factorization 180
3.5 Subspaces, Basis, Dimension, and Rank
3.6 Introduction to Linear Transformations
Vignette: Robotics 226
3.7 Applications 230
Markov Chains 230
Linear Economic Models 235
Population Growth 239
Graphs and Digraphs 241
Chapter Review 251
Chapter 4 Eigenvalues and Eigenvectors 253
4.0 Introduction: A Dynamical System on Graphs 253
4.1 Introduction to Eigenvalues and Eigenvectors 254
4.2 Determinants 263
Writing Project: Which Came First: The Matrix or the Determinant?
Vignette: Lewis Carroll's Condensation Method 284
Exploration: Geometric Applications of Determinants 286
4.3 Eigenvalues and Eigenvectors of n X n Matrices 292
Writing Project: The History of Eigenvalues 301
3.4 Similarity and Diagonalization 301
3.5 Iterative Methods for Computing Eigenvalues 31 1
3.6 Applications and the Perron-Frobenius Theorem 325
Markov Chains 325
Population Growth 330
The Perron-Frobenius Theorem 332
Linear Recurrence Relations 335
Systems of Linear Differential Equations 340
Discrete Linear Dynamical Systems 348
Vignette: Ranking Sports Teams and Searching the Internet 356
Chapter Review 364
Chapter 5 Orthogonality 366
5.0 Introduction: Shadows on a Wall 366
5.1 Orthogonality in IR" 368
5.2 Orthogonal Complements and Orthogonal Projections 378
5.3 The Gram-Schmidt Process and the QR Factorization 388
Explorations: The Modified QR Factorization 396
Approximating Eigenvalues with the QR Algorithm 398
5.4 Orthogonal Diagonalization of Symmetric Matrices 400
5.5 Applications 408
Quadratic Forms 408
Graphing Quadratic Equations 415
Chapter Review 425
Chapter 6 Vector Spaces 427
6.0 Introduction: Fibonacci in (Vector) Space 427
6. 1 Vector Spaces and Subspaces 429
6.2 Writing Project: The Rise of Vector Spaces 443
6.3 Linear Independence, Basis, and Dimension 443
6.4 Exploration: Magic Squares 460
6.5 Change of Basis 463
6.6 Linear Transformations 472
The Kernel and Range of a Linear Transformation 481
The Matrix of a Linear Transformation 497
Exploration: Tilings, Lattices, and the Crystallographic Restriction
6.7 Applications 518
Homogeneous Linear Differential Equations 518
Chapter Review 527
Chapter 7 Distance and Approximation 529
7.0 Introduction: Taxicab Geometry 529
7. 1 Inner Product Spaces 531
Explorations: Vectors and Matrices with Complex Entries 543
Geometric Inequalities and Optimization Problems 547
7.2 Norms and Distance Functions 552
7.3 Least Squares Approximation 568
7.4 The Singular Value Decomposition 590
Vignette: Digital Image Compression 607
7.5 Applications 610
Approximation of Functions 610
Chapter Review 618
Chapter 8 Codes Online only 620
8. 1 Code Vectors 620
Vignette: The Codabar System 626
8.2 Error-Correcting Codes 627
8.3 Dual Codes 632
8.4 Linear Codes 639
8.5 The Minimum Distance of a Code 644
APPENDIX A Mathematical Notation and Methods of Proof Al
APPENDIX B Mathematical Induction B 1
APPENDIX C Complex Numbers Cl
APPENDIX D Polynomials D 1
APPENDIX E Technology Bytes Online only
Answers to Selected Odd-Numbered Exercises ANSI
Index II
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