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Visual Linear Algebra

Visual Linear Algebra

ISBN: 978-0-471-68299-8

Mar 2005

576 pages

$218.95

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Description

Visual Linear Algebra is a new kind of textbook—a blend of interactive computer tutorials and traditional text. The computer tutorials provide a lively learning environment in which students are introduced to concepts and methods and where they develop their intuition. The traditional sections provide the backbone whose core is the development of theory and where students’ understanding is solidified. Although the design of Visual Linear Algebra is novel, the goals for the book are quite traditional. Foremost among these is to provide a rich set of materials that help students achieve a thorough understanding of the core topics of linear algebra and genuine competence in using them.

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Chapter 1 Systems of Linear Equations 1

1.1 Solving Linear Systems 2

1.2 Geometric Perspectives on Linear Systems 18

1.3A Solving Linear Systems Using Maple 26

1.3B Solving Linear Systems Using Mathematics 36

1.4 Curve Fitting and Temperature Distribution-Application 45

Chapter 2 Vectors 56

2.1 Geometry of Vectors 57

2.2 Linear Combinations of Vectors 71

2.3 Decomposing the Solution of a Linear System 84

2.4 Linear Independence of Vectors 92

2.5 Theory of Vector Concepts 106

Chapter 3 Matrix Algebra 113

3.1 Product of a Matrix and a Vector 114

3.2 Matrix Multiplication 125

3.3 Rules of Matrix Algebra 140

3.4 Markov Chains-Application 147

3.5 Inverse of a Matrix 161

3.6 Theory of Matrix Inverses 175

3.7 Cryptology-Application 180

Chapter 4 Linear Transformations 193

4.1 Introduction to Matrix Transformations 194

4.2 Geometry of Matrix Transformations of the Plane 198

4.3 Geometry of Matrix Transformations of 3-Space 220

4.4 Linear Transformations 232

4.5 Computer Graphics-Applications 237

Chapter 5 Vector Spaces 255

5.1 Subspaces of Rᴺ 256

5.2 Basis and Dimension 262

5.3 Theory of Basis and Dimension 272

5.4 Subspaces Associated with a Matrix 275

5.5 Theory of Subspaces Associated with a Matrix 285

5.6 Loops and Spanning Trees-Application 289

5.7 Abstract Vector Spaces 297

Chapter 6 Determinants 306

6.1 Determinants and Cofactors 307

6.2 Properties of Determinants 311

6.3 Theory of Determination 322

Chapter 7 Eigenvalues and Eigenvectors 328

7.1 Introduction to Eigenvalues and Eigenvectors 329

7.2 The Characteristics Polynomial 343

7.3 Discrete Dynamical Systems-Application 356

7.4 Diagonalization and Similar Matrices 373

7.5 Theory of Eigenvalues and Eigenvectors 389

7.6 Systems of Linear Differential Equations-Application 395

7.7 Complex Numbers and Complex Vectors 407

7.8 Complex Eigenvalue and Eigenvectors 411

Chapter 8 Orthogonality 431

8.1 Dot Product and Orthogonal Vectors 432

8.2 Orthogonal Projections in R² and R³ 437

8.3 Orthogonal Projections and Orthogonal Bases in Rᴺ 447

8.4 Theory of Orthogonality 456

8.5 Least-Squares Solution-Application 463

8.6 Weighted Least-Squares and Inner Products on Rᴺ 475

8.7 Approximation of Functions and Integral Inner Products 483

8.8 Inner Product Spaces 496

Appendix A Glossary of Linear Algebra Definitions 503

Appendix B Linear Algebra Theorems 510

Appendix C Advice for Using Maple with Visual Linear Algebra 519

Appendix D Commands Used in Maple Tutorials 521

Appendix E Advice for Using Mathematica with Visual Linear Algebra 527

Appendix F Commands Used in Mathematica Tutorials 530

Appendix G Answers and Hints for Selected Pencil and Paper Problems 537

Index 545

  • Tutorials and traditional text.  Visual Linear Algebra covers the topics in a standard one-semester introductory linear algebra course in forty-seven sections arranged in eight chapters. In each chapter, some sections are written in a traditional textbook style and some are tutorials designed to be worked through using either Maple or Mathematica.
  • About the tutorials.    Each tutorial is a self-contained treatment of a core topic or application of linear algebra that a student can work through with minimal assistance from an instructor.  The thirty tutorials are provided on the accompanying CD both as Maple worksheets and as Mathematica notebooks. They also appear in print as sections of the textbook.
  • Geometry is used extensively to help students develop their intuition about the concepts of linear algebra.
  • Applications.  Students benefit greatly from working through an application, if the application captures their interest and the materials give them substantial activities that yield worthwhile results. Ten carefully selected applications have been developed and an entire tutorial is devoted to each of them.
  • Active Learning. To encourage students to be active learners, the tutorials have been designed to engage and retain their interest. The exercises, demonstrations, explorations, visualizations, and animations are designed to stimulate students’ interest, encourage them to think clearly about the mathematics they are working through, and help them check their comprehension.