Linear algebra and its applications 6th edition

【实例简介】Linear algebra and its applications 6th edition

Contents About the Authors 3 Preface 12 A Note to Students 22 Chapter 1 Linear Equations in Linear Algebra 25 INTRODUCTORY EXAMPLE: Linear Models in Economics and Engineering 25 1.1 Systems of Linear Equations 26 1.2 Row Reduction and Echelon Forms 37 1.3 Vector Equations 50 1.4 The Matrix Equation Ax D b 61 1.5 Solution Sets of Linear Systems 69 1.6 Applications of Linear Systems 77 1.7 Linear Independence 84 1.8 Introduction to Linear Transformations 91 1.9 The Matrix of a Linear Transformation 99 1.10 Linear Models in Business, Science, and Engineering 109 Projects 117 Supplementary Exercises 117 Chapter 2 Matrix Algebra 121 INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design 121 2.1 Matrix Operations 122 2.2 The Inverse of a Matrix 135 2.3 Characterizations of Invertible Matrices 145 2.4 Partitioned Matrices 150 2.5 Matrix Factorizations 156 2.6 The Leontief Input–Output Model 165 2.7 Applications to Computer Graphics 171 5 6 Contents 2.8 Subspaces of Rn 179 2.9 Dimension and Rank 186 Projects 193 Supplementary Exercises 193 Chapter 3 Determinants 195 INTRODUCTORY EXAMPLE: Weighing Diamonds 195 3.1 Introduction to Determinants 196 3.2 Properties of Determinants 203 3.3 Cramer’s Rule, Volume, and Linear Transformations 212 Projects 221 Supplementary Exercises 221 Chapter 4 Vector Spaces 225 INTRODUCTORY EXAMPLE: Discrete-Time Signals and Digital Signal Processing 225 4.1 Vector Spaces and Subspaces 226 4.2 Null Spaces, Column Spaces, Row Spaces, and Linear Transformations 235 4.3 Linearly Independent Sets; Bases 246 4.4 Coordinate Systems 255 4.5 The Dimension of a Vector Space 265 4.6 Change of Basis 273 4.7 Digital Signal Processing 279 4.8 Applications to Difference Equations 286 Projects 295 Supplementary Exercises 295 Chapter 5 Eigenvalues and Eigenvectors 297 INTRODUCTORY EXAMPLE: Dynamical Systems and Spotted Owls 297 5.1 Eigenvectors and Eigenvalues 298 5.2 The Characteristic Equation 306 5.3 Diagonalization 314 5.4 Eigenvectors and Linear Transformations 321 5.5 Complex Eigenvalues 328 5.6 Discrete Dynamical Systems 335 5.7 Applications to Differential Equations 345 5.8 Iterative Estimates for Eigenvalues 353 5.9 Applications to Markov Chains 359 Projects 369 Supplementary Exercises 369 Contents 7 Chapter 6 Orthogonality and Least Squares 373 INTRODUCTORY EXAMPLE: Artificial Intelligence and Machine Learning 373 6.1 Inner Product, Length, and Orthogonality 374 6.2 Orthogonal Sets 382 6.3 Orthogonal Projections 391 6.4 The Gram–Schmidt Process 400 6.5 Least-Squares Problems 406 6.6 Machine Learning and Linear Models 414 6.7 Inner Product Spaces 423 6.8 Applications of Inner Product Spaces 431 Projects 437 Supplementary Exercises 438 Chapter 7 Symmetric Matrices and Quadratic Forms 441 INTRODUCTORY EXAMPLE: Multichannel Image Processing 441 7.1 Diagonalization of Symmetric Matrices 443 7.2 Quadratic Forms 449 7.3 Constrained Optimization 456 7.4 The Singular Value Decomposition 463 7.5 Applications to Image Processing and Statistics 473 Projects 481 Supplementary Exercises 481 Chapter 8 The Geometry of Vector Spaces 483 INTRODUCTORY EXAMPLE: The Platonic Solids 483 8.1 Affine Combinations 484 8.2 Affine Independence 493 8.3 Convex Combinations 503 8.4 Hyperplanes 510 8.5 Polytopes 519 8.6 Curves and Surfaces 531 Project 542 Supplementary Exercises 543 Chapter 9 Optimization 545 INTRODUCTORY EXAMPLE: The Berlin Airlift 545 9.1 Matrix Games 546 9.2 Linear Programming Geometric Method 560 9.3 Linear Programming Simplex Method 570 9.4 Duality 585 Project 594 Supplementary Exercises 594 8 Contents Chapter 10 Finite-State Markov Chains C-1 (Available Online) INTRODUCTORY EXAMPLE: Googling Markov Chains C-1 10.1 Introduction and Examples C-2 10.2 The Steady-State Vector and Google’s PageRank C-13 10.3 Communication Classes C-25 10.4 Classification of States and Periodicity C-33 10.5 The Fundamental Matrix C-42 10.6 Markov Chains and Baseball Statistics C-54 Appendixes A Uniqueness of the Reduced Echelon Form 597 B Complex Numbers 599 Credits 604 Glossary 605 Answers to Odd-Numbered Exercises A-1 Index I-1 

【实例截图】

【核心代码】

Contents
About the Authors 3
Preface 12
A Note to Students 22
Chapter 1 Linear Equations in Linear Algebra 25
INTRODUCTORY EXAMPLE: Linear Models in Economics
and Engineering 25
1.1 Systems of Linear Equations 26
1.2 Row Reduction and Echelon Forms 37
1.3 Vector Equations 50
1.4 The Matrix Equation A x D b 61
1.5 Solution Sets of Linear Systems 69
1.6 Applications of Linear Systems 77
1.7 Linear Independence 84
1.8 Introduction to Linear Transformations 91
1.9 The Matrix of a Linear Transformation 99
1.10 Linear Models in Business, Science, and Engineering 109
Projects 117
Supplementary Exercises 117
Chapter 2 Matrix Algebra 121
INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design 121
2.1 Matrix Operations 122
2.2 The Inverse of a Matrix 135
2.3 Characterizations of Invertible Matrices 145
2.4 Partitioned Matrices 150
2.5 Matrix Factorizations 156
2.6 The Leontief Input–Output Model 165
2.7 Applications to Computer Graphics 171
5
6 Contents
2.8 Subspaces of R n 179
2.9 Dimension and Rank 186
Projects 193
Supplementary Exercises 193
Chapter 3 Determinants 195
INTRODUCTORY EXAMPLE: Weighing Diamonds 195
3.1 Introduction to Determinants 196
3.2 Properties of Determinants 203
3.3 Cramer’s Rule, Volume, and Linear Transformations 212
Projects 221
Supplementary Exercises 221
Chapter 4 Vector Spaces 225
INTRODUCTORY EXAMPLE: Discrete-Time Signals and Digital
Signal Processing 225
4.1 Vector Spaces and Subspaces 226
4.2 Null Spaces, Column Spaces, Row Spaces, and Linear
Transformations 235
4.3 Linearly Independent Sets; Bases 246
4.4 Coordinate Systems 255
4.5 The Dimension of a Vector Space 265
4.6 Change of Basis 273
4.7 Digital Signal Processing 279
4.8 Applications to Difference Equations 286
Projects 295
Supplementary Exercises 295
Chapter 5 Eigenvalues and Eigenvectors 297
INTRODUCTORY EXAMPLE: Dynamical Systems and Spotted Owls 297
5.1 Eigenvectors and Eigenvalues 298
5.2 The Characteristic Equation 306
5.3 Diagonalization 314
5.4 Eigenvectors and Linear Transformations 321
5.5 Complex Eigenvalues 328
5.6 Discrete Dynamical Systems 335
5.7 Applications to Differential Equations 345
5.8 Iterative Estimates for Eigenvalues 353
5.9 Applications to Markov Chains 359
Projects 369
Supplementary Exercises 369
Contents 7
Chapter 6 Orthogonality and Least Squares 373
INTRODUCTORY EXAMPLE: Artificial Intelligence and Machine
Learning 373
6.1 Inner Product, Length, and Orthogonality 374
6.2 Orthogonal Sets 382
6.3 Orthogonal Projections 391
6.4 The Gram–Schmidt Process 400
6.5 Least-Squares Problems 406
6.6 Machine Learning and Linear Models 414
6.7 Inner Product Spaces 423
6.8 Applications of Inner Product Spaces 431
Projects 437
Supplementary Exercises 438
Chapter 7 Symmetric Matrices and Quadratic Forms 441
INTRODUCTORY EXAMPLE: Multichannel Image Processing 441
7.1 Diagonalization of Symmetric Matrices 443
7.2 Quadratic Forms 449
7.3 Constrained Optimization 456
7.4 The Singular Value Decomposition 463
7.5 Applications to Image Processing and Statistics 473
Projects 481
Supplementary Exercises 481
Chapter 8 The Geometry of Vector Spaces 483
INTRODUCTORY EXAMPLE: The Platonic Solids 483
8.1 Affine Combinations 484
8.2 Affine Independence 493
8.3 Convex Combinations 503
8.4 Hyperplanes 510
8.5 Polytopes 519
8.6 Curves and Surfaces 531
Project 542
Supplementary Exercises 543
Chapter 9 Optimization 545
INTRODUCTORY EXAMPLE: The Berlin Airlift 545
9.1 Matrix Games 546
9.2 Linear Programming Geometric Method 560
9.3 Linear Programming Simplex Method 570
9.4 Duality 585
Project 594
Supplementary Exercises 594
8 Contents
Chapter 10 Finite-State Markov Chains C-1
(Available Online)
INTRODUCTORY EXAMPLE: Googling Markov Chains C-1
10.1 Introduction and Examples C-2
10.2 The Steady-State Vector and Google’s PageRank C-13
10.3 Communication Classes C-25
10.4 Classification of States and Periodicity C-33
10.5 The Fundamental Matrix C-42
10.6 Markov Chains and Baseball Statistics C-54
Appendixes
A Uniqueness of the Reduced Echelon Form 597
B Complex Numbers 599
Credits 604
Glossary 605
Answers to Odd-Numbered Exercises A-1
Index I-1