Klappentext

Elementary Linear Algebra: Applications Version, 12th Edition, gives an elementary treatment of linear algebra that is suitable for a first course for undergraduate students. The classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation. In this edition, a new section has been added to describe the applications of linear algebra in emerging fields such as data science, machine learning, climate science, geomatics, and biological modeling. New exercises have been added with special attention to the expanded early introduction

to linear transformations and new examples have been added, where needed, to support the exercise sets. Calculus is not a prerequisite, but there are clearly labeled exercises and examples (which can be omitted without loss of continuity) for students who have studied calculus.

Inhalt

1 Systems of Linear Equations and Matrices 1

1.1 Introduction to Systems of Linear Equations 2

1.2 Gaussian Elimination 11

1.3 Matrices and Matrix Operations 25

1.4 Inverses; Algebraic Properties of Matrices 40

1.5 Elementary Matrices and a Method for Finding A - 1 53

1.6 More on Linear Systems and Invertible Matrices 62

1.7 Diagonal, Triangular, and Symmetric Matrices 69

1.8 Introduction to Linear Transformations 76

1.9 Compositions of Matrix Transformations 90

1.10 Applications of Linear Systems 98

Network Analysis 98

Electrical Circuits 100

Balancing Chemical Equations 103

Polynomial Interpolation 105

1.11 Leontief Input-Output Models 110

2 Determinants 119

2.1 Determinants by Cofactor Expansion 119

2.2 Evaluating Determinants by Row Reduction 127

2.3 Properties of Determinants; Cramer's Rule 134

3 Euclidean Vector Spaces 149

3.1 Vectors in 2-Space, 3-Space, and n-Space 149

3.2 Norm, Dot Product, and Distance in R n 161

3.3 Orthogonality 175

3.4 The Geometry of Linear Systems 186

3.5 Cross Product 193

4 General Vector Spaces 205

4.1 Real Vector Spaces 205

4.2 Subspaces 214

4.3 Spanning Sets 223

4.4 Linear Independence 231

4.5 Coordinates and Basis 241

4.6 Dimension 251

4.7 Change of Basis 259

4.8 Row Space, Column Space, and Null Space 266

4.9 Rank, Nullity, and the Fundamental Matrix Spaces 279

5 Eigenvalues and Eigenvectors 295

5.1 Eigenvalues and Eigenvectors 295

5.2 Diagonalization 305

5.3 Complex Vector Spaces 315

5.4 Differential Equations 327

5.5 Dynamical Systems and Markov Chains 333

6 Inner Product Spaces 347

6.1 Inner Products 347

6.2 Angle and Orthogonality in Inner Product Spaces 358

6.3 Gram-Schmidt Process; QR-Decomposition 367

6.4 Best Approximation; Least Squares 382

6.5 Mathematical Modeling Using Least Squares 391

6.6 Function Approximation; Fourier Series 398

7 Diagonalization and Quadratic Forms 407

7.1 Orthogonal Matrices 407

7.2 Orthogonal Diagonalization 416

7.3 Quadratic Forms 424

7.4 Optimization Using Quadratic Forms 437

7.5 Hermitian, Unitary, and Normal Matrices 444

8 General Linear Transformations 455

8.1 General Linear Transformations 455

8.2 Compositions and Inverse Transformations 468

8.3 Isomorphism 480

8.4 Matrices for General Linear Transformations 486

8.5 Similarity 496

8.6 Geometry of Matrix Operators 502

9 Numerical Methods 519

9.1 LU-Decompositions 519

9.2 The Power Method 529

9.3 Comparison of Procedures for Solving Linear Systems 538

9.4 Singular Value Decomposition 542

9.5 Data Compression Using Singular Value Decomposition 550

10 Applications of Linear Algebra 555

10.1 Constructing Curves and Surfaces Through Specified Points 555

10.2 The Earliest Applications of Linear Algebra 561

10.3 Cubic Spline Interpolation 568

10.4 Markov Chains 578

10.5 Graph Theory 587

10.6 Games of Strategy 597

10.7 Forest Management 605

10.8 Computer Graphics 612

10.9 Equilibrium Temperature Distributions 620

10.10 Computed Tomography 629

10.11 Fractals 639

10.12 Chaos 655

10.13 Cryptography 668

10.14 Genetics 679

10.15 Age-Specific Population Growth 688

10.16 Harvesting of Animal Populations 697

10.17 A Least Squares Model for Human Hearing 705

10.18 Warps and Morphs 711

10.19 Internet Search Engines 720

10.20 Facial Recognition 726

Appendix A Working with Proofs A1

Appendix B Complex Numbers A7

Answers to Exercises A15

Index i1