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More About This Title Linear Algebra: Ideas and Applications, Third Edition
English
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 theoremA new chapter on numerical techniques, including a discussion of the condition numberA new section on Hermitian symmetric and unitary matricesAn exploration of computational approaches to finding eigenvalues, such as the forward iteration, reverse iteration, and the QR methodAdditional 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.
English
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.
English
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
English
"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)
