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Linear Algebra: Ideas and Applications, 3rd Edition

ISBN: 978-0-470-17884-3
504 pages
July 2008, ©2008
Linear Algebra: Ideas and Applications, 3rd Edition (0470178841) cover image

Description

This expanded new edition presents a thorough and up-to-date introduction to the study of linear algebra

Linear Algebra, Third Edition provides a unified introduction to linear algebra while reinforcing and emphasizing a conceptual and hands-on understanding of the essential ideas. Promoting the development of intuition rather than the simple application of methods, the book successfully helps readers to understand not only how to implement a technique, but why its use is important.

The book 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 Third Edition also features:
  • A new chapter on generalized eigenvectors and chain bases with coverage of the Jordan form and the Cayley-Hamilton theorem
  • A new chapter on numerical techniques, including a discussion of the condition number
  • A new section on Hermitian symmetric and unitary matrices
  • An exploration of computational approaches to finding eigenvalues, such as the forward iteration, reverse iteration, and the QR method
  • Additional exercises that consist of application, numerical, and conceptual questions as well as true-false questions

Illuminating applications of linear algebra are provided throughout most parts of the book along with self-study questions that allow the reader to replicate the treatments independently of the book. Each chapter concludes with a summary of key points, and most topics are accompanied by a "Computer Projects" section, which contains worked-out exercises that utilize the most up-to-date version of MATLAB(r). A related Web site features Maple translations of these exercises as well as additional supplemental material.

Linear Algebra, Third Edition is an excellent undergraduate-level textbook for courses in linear algebra. It is also a valuable self-study guide for professionals and researchers who would like a basic introduction to linear algebra with applications in science, engineering, and computer science.

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Table of Contents

Preface.

Features of the Text.

1. Systems of Linear Equations 1

1.1 The Vector Space of m x n Matrices 1

The Space Rn.

Linear Combinations and Linear Dependence.

What Is a Vector Space?

Why Prove Anything?

True-False Questions.

Exercises.

1.1.1 Computer Projects 21

Exercises.

1.1.2 Applications to Graph Theory I 24

Self-Study Questions.

Exercises.

1.2 Systems 27

Rank: The Maximum Number of Linearly Independent Equations.

True-False Questions.

Exercises.

1.2.1 Computer Projects 39

Exercises.

1.2.2 Applications to Circuit Theory 40

Self-Study Questions.

Exercises.

1.3 Gaussian Elimination 45

Spanning in Polynomial Spaces.

Computational Issues: Pivoting.

True-False Questions.

Exercises.

Computational Issues: Flops.

1.3.1 Computer Projects 67

Exercises.

1.3.2 Applications to Traffic Flow 69

Self-Study Questions.

Exercises.

1.4 Column Space and Nullspace 71

Subspaces.

Subspaces of Functions.

True-False Questions.

Exercises.

1.4.1 Computer Projects 90

Exercises.

1.4.2 Applications to Predator-Prey Problems 92

Self-Study Questions.

Exercises.

Chapter Summary.

2. Linear Independence and Dimension 97

2.1 The Test for Linear Independence 97

Bases for the Column Space.

Testing Functions for Independence.

True-False Questions.

Exercises.

2.1.1 Computer Projects 112

2.2 Dimension 113

True-False Questions.

Exercises.

2.2.1 Computer Projects 126

Exercises.

2.2.2 Applications to Calculus 128

Self-Study Questions.

Exercises.

2.2.3 Applications to Differential Equations 130

Self-Study Questions.

Exercises.

2.2.4 Applications to the Harmonic Oscillator 133

Self-Study Questions.

Exercises.

2.2.5 Computer Projects 136

Exercises.

2.3 Row Space and the Rank-Nullity Theorem 139

Bases for the Row Space.

Rank-Nullity Theorem.

Computational Issues: Computing Rank.

True-False Questions.

Exercises.

2.3.1 Computer Projects 152

Chapter Summary.

3. Linear Transformations 155

3.1 The Linearity Properties 155

True-False Questions.

Exercises.

3.1.1 Computer Projects 168

3.1.2 Applications to Control Theory 170

Self-Study Questions.

Exercises.

3.2 Matrix Multiplication (Composition) 174

Partitioned Matrices.

Computational Issues: Parallel Computing.

True-False Questions.

Exercises.

3.2.1 Computer Projects 187

3.2.2 Applications to Graph Theory II 188

Self-Study Questions.

Exercises.

3.3 Inverses 190

Computational Issues: Reduction vs. Inverses.

True-False Questions.

Exercises.

Ill Conditioned Systems.

3.3.1 Computer Projects 203

Exercises.

3.3.2 Applications to Economics 205

Self-Study Questions.

Exercises.

3.4 The LU Factorization 211

Exercises.

3.4.1 Computer Projects 220

Exercises.

3.5 The Matrix of a Linear Transformation 221

Coordinates.

Application to Differential Equations.

Isomorphism.

Invertible Linear Transformations.

True-False Questions.

Exercises.

3.5.1 Computer Projects 239

Chapter Summary.

4. Determinants 243

4.1 Definition of the Determinant 243

4.1.1 The Rest of the Proofs 249

True-False Questions.

Exercises.

4.1.2 Computer Projects 255

4.2 Reduction and Determinants 255

Uniqueness of the Determinant.

True-False Questions.

Exercises.

4.2.1 Application to Volume 264

Self-Study Questions.

Exercises.

4.3 A Formula for Inverses 267

Cramer's Rule.

True-False Questions.

Exercises.

Chapter Summary.

5. Eigenvectors and Eigenvalues 275

5.1 Eigenvectors 275

True-False Questions.

Exercises.

5.1.1 Computer Projects 286

5.1.2 Application to Markov Processes 286

Exercises.

5.2 Diagonalization 292

Powers of Matrices.

True-False Questions.

Exercises.

5.2.1 Computer Projects 298

5.2.2 Application to Systems of Differential Equations 299

Self-Study Questions.

Exercises.

5.3 Complex Eigenvectors 302

Complex Vector Spaces.

Exercises.

5.3.1 Computer Projects 311

Exercises.

Chapter Summary.

6. Orthogonality 313

6.1 The Scalar Product in Rn 313

Orthogonal/Orthonormal Bases and Coordinates.

True-False Questions.

Exercises.

6.1.1 Application to Statistics 324

Self-Study Questions.

Exercises.

6.2 Projections: The Gram-Schmidt Process 326

The QR Decomposition 334

Uniqueness of the QR-factoriaition.

True-False Questions.

Exercises.

6.2.1 Computer Projects 338

Exercises.

6.3 Fourier Series: Scalar Product Spaces 340

Exercises.

6.3.1 Computer Projects 352

Exercises.

6.4 Orthogonal Matrices 354

Householder Matrices.

True-False Questions.

Exercises.

6.4.1 Computer Projects 366

Exercises.

6.5 Least Squares 367

Exercises.

6.5.1 Computer Projects 376

Exercises.

6.6 Quadratic Forms: Orthogonal Diagonalization 377

The Spectral Theorem.

The Principal Axis Theorem.

True-False Questions.

Exercises.

6.6.1 Computer Projects 390

Exercises.

6.7 The Singular Value Decomposition (SVD) 392

Application of the SVD to Least-Squares Problems.

True-False Questions.

Exercises.

Computing the SVD Using Householder Matrices.

Diagonalizing Symmetric Matrices Using Householder Matrices.

6.8 Hermitian Symmetric and Unitary Matrices 404

True-False Questions.

Exercises.

Chapter Summary.

7. Generalized Eigenvectors 415

7.1 Generalized Eigenvectors 415

Exercises.

7.2 Chain Bases 424

Jordan Form.

True-False Questions.

Exercises.

The Cayley-Hamilton Theorem.

Chapter Summary.

8. Numerical Techniques 439

8.1 Condition Number 439

Norms.

Condition Number.

Least Squares.

Exercises.

8.2 Computing Eigenvalues 445

Iteration.

The QR Method.

Exercises.

Chapter Summary.

Answers and Hints.

Index 477

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Author Information

Richard C. Penney, PhD, is Professor in the Department of Mathematics and Director of the Mathematics/Statistics Actuarial Science Program at Purdue University. Dr. Penney is the author of numerous journal articles and has received several major teaching awards.

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New to This Edition

  • A new section has been added on computational linear algebra to reflect the increase in the power, speed, and availability of computers.  More extensive discussions of the condition number as well as the numerical computation of eigenvalues are included.
  • Two new chapters are provided on numerical linear algebra and the Jordan normal form
  • Full sections on both Hermitian and Unitary matrices have been added, as well as sections on computational linear algebra and wavelets. 
  • An increase in the amount of exercises and the True-False questions is evident in this new edition.  The exercises continue to be "conceptual but not theoretical" in nature, and they compliment the approach and style of the book.
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The Wiley Advantage

  • A student solutions manual is available, which contains worked out examples, hints for some of the exercises, and translations of the computer exercises into various platforms, i.e. MapleTM, MATLAB®, and TI calculators. 
  • All exercises have been updated to reflect the most up-to-date versions of Maple and MATLAB.
  •  A sample syllabus for a two semester class has been added to the front matter.
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Reviews

"Linear Algebra (third edition) is an excellent undergraduate-level textbook for courses in linear algebra.  It is also valuable self-study guide for professionals and researches who would like a basic introduction to linear algebra with applications in science, engineering, and computer science." (Mathematical Review, Issue 2009e)

"This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications." (Electric Review, November 2008)

"This book should make a good text for introductory courses." (Computing Reviews, September 30, 2008)

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Professor Reviews

"Linear Algebra (third edition) is an excellent undergraduate-level textbook for courses in linear algebra.  It is also valuable self-study guide for professionals and researches who would like a basic introduction to linear algebra with applications in science, engineering, and computer science." (Mathematical Review, Issue 2009e)

"This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications." (Electric Review, Nov 2008)

"This book should make a good text for introductory courses." (Computing Reviews, September 30, 2008)

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