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APPLIED LINEAR ALGEBRA
AND
MATRIX ANALYSIS
Thomas S. Shores
Author address:
C
OPYRIGHT
c
MAY 2000 ALL RIGHTS RESERVED
Contents
Preface i
Chapter 1. LINEAR SYSTEMS OF EQUATIONS 1
1.1. Some Examples 1
1.2. Notations and a Review of Numbers 9
1.3. Gaussian Elimination: Basic Ideas 18
1.4. Gaussian Elimination: General Procedure 29
1.5. *Computational Notes and Projects 39
Review 47
Chapter 2. MATRIX ALGEBRA 49
2.1. Matrix Addition and Scalar Multiplication 49
2.2. Matrix Multiplication 55
2.3. Applications of Matrix Arithmetic 62
2.4. Special Matrices and Transposes 72
2.5. Matrix Inverses 85
2.6. Basic Properties of Determinants 96
2.7. *Applications and Proofs for Determinants 106
2.8. *Tensor Products 114
2.9. *Computational Notes and Projects 118
Review 123
Chapter 3. VECTOR SPACES 125
3.1. Definitions and Basic Concepts 125
3.2. Subspaces 135
3.3. Linear Combinations 142
3.4. Subspaces Associated with Matrices and Operators 152
3.5. Bases and Dimension 160
3.6. Linear Systems Revisited 166
3.7. *Change of Basis and Linear Operators 174
3
4 CONTENTS
3.8. *Computational Notes and Projects 178
Review 182
Chapter 4. GEOMETRICAL ASPECTS OF STANDARD SPACES 185
4.1. Standard Norm and Inner Product 185
4.2. Applications of Norms and Inner Products 192
4.3. Unitary and Orthogonal Matrices 202
4.4. *Computational Notes and Projects 210
Review 212
Chapter 5. THE EIGENVALUE PROBLEM 213
5.1. Definitions and Basic Properties 213
5.2. Similarity and Diagonalization 223
5.3. Applications to Discrete Dynamical Systems 232
5.4. Orthogonal Diagonalization 240
5.5. *Schur Form and Applications 244
5.6. *The Singular Value Decomposition 247
5.7. *Computational Notes and Projects 250
Review 259
Chapter 6. GEOMETRICAL ASPECTS OF ABSTRACT SPACES 261
6.1. Normed Linear Spaces 261
6.2. Inner Product Spaces 266
6.3. Gram-Schmidt Algorithm 276
6.4. Linear Systems Revisited 286
6.5. *Operator Norms 295
6.6. *Computational Notes and Projects 299
Review 306
Appendix A. Table of Symbols 307
Appendix B. Solutions to Selected Exercises 309
Bibliography 323
Index 325
Preface
This book is about matrix and linear algebra, and their applications. For many students
the tools of matrix and linear algebra will be as fundamental in their professional work
as the tools of calculus; thus it is important to ensure that students appreciate the utility
and beautyof these subjects, as well as understand the mechanics. One way to do so is to
show how concepts of matrix and linear algebra make concrete problems workable. To
this end, applied mathematics and mathematical modeling ought to have an important
role in an introductory treatment of linear algebra.
One of the features of this book is that we weave significant motivating examples into
the fabric of the text. Needless to say, I hope that instructors will not omit this ma-
terial; that would be a missed opportunity for linear algebra! The text has a strong
orientation towards numerical computation and applied mathematics, which means that
matrix analysis plays a central role. All three of the basic components of linear algebra
– theory, computation and applications – receive their due. The proper balance of these
components will give a diverse audience of physical science, social science, statistics,
engineering and math students the tools they need as well as the motivation to acquire
these tools. Another feature of this text is an emphasis on linear algebra as an exper-
imental science; this emphasis is to be found in certain examples, computer exercises
and projects. Contemporary mathematical software makes an ideal “lab” for mathemat-
ical experimentation. At the same time, this text is independent of specific hardware
and software platforms. Applications and ideas should play center stage, not software.
This book is designed for an introductory course in matrix and linear algebra. It is
assumed that the student has had some exposure to calculus. Here are some of its main
goals:
To provide a balanced blend of applications, theory and computation which em-
phasizes their interdependence.
To assist those who wish to incorporate mathematical experimentation through
computer technology into the class. Each chapter has an optional section on
computational notes and projects and computer exercises sprinkled throughout.
The student should use the locally available tools to carry out the experiments
suggested in the project and use the word processing capabilities of the com-
puter system to create small reports on his/her results. In this way they gain
experience in the use of the computer as a mathematical tool. One can also en-
vision reports on a grander scale as mathematical “term papers.” I have made
such assignments in some of my own classes with delightful results. A few
major report topics are included in the text.
i
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