Kenneth M. Shiskowski - Principles of Linear Algebra with Mathematica

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Library of Congress Cataloging-in-Publication Data:

Shiskowski, Kenneth, 1954
Principles of linear algebra with Mathematica / Kenneth Shiskowski, Karl Frinkle.
p. cm. (Pure and applied mathematics)
Includes index.
ISBN 978-0-470-63795-1 (hardback)
1. Algebras, LinearData processing. 2. Mathematica (Computer file) I. Frinkle, Karl, 1977 II. Title.
QA185.D37S454 2011

512.5028553dc22
2011006420

With all textbooks, one should attempt to be consistent with notation, not only within the text, but also within the field of mathematics on which it is based. For the most part, we have done this.

Preface

This book is an attempt to cross the gap between beginning linear algebra and the computational linear algebra that one encounters more frequently in applied settings. The underlying theory behind many topics in the field of linear algebra is relatively simple to grasp; however, to actually apply this knowledge to nontrivial problems becomes computationally intensive. To do these computations by hand would be tedious at best, and many times simply unrealistic. Furthermore, attempting to solve such problems by the old pencil-and-paper method does not give the average reader any extra insight into the problem. Mathematica allows readers to overcome these obstacles, giving them the power to perform complex computations that would take hours by hand, and can help to visualize many of the geometric interpretations of linear algebra topics in two and three dimensions in a very intuitive fashion. We hope that this book will challenge the reader to become proficient in both theoretical and computational aspects of linear algebra.

Chapter 1 of this book is a brief introduction to Mathematica and will help the reader become more comfortable with the program. This chapter focuses on the commands and programming most commonly used when studying linear algebra and its applications. Mathematica commands will always be in bold, with output (if any) displayed in a left-justified fashion below each Mathematica command. Readers can enter these commands and obtain the same results, assuming that they have entered the commands correctly. Note also that all of the images in this book were produced with Mathematica. The overall intent of this book is to use Mathematica to enhance the concepts of linear algebra, and therefore Mathematica is integrated into this book in a very casual manner. Where one normally explains how to perform some operation by hand in a standard text, we often simply use Mathematica commands to perform the same task. Thus, the reader should attempt to become familiar with the Mathematica syntax as quickly as possible.

At the end of each section, you will find two types of problems: Homework problems and Mathematica problems. The former consists of strictly pen-and-pencil computation problems, inquiries into theory, and questions about concepts discussed in the section. The idea behind these problems is to ensure that the reader has an understanding of the concepts introduced and can put them to use in problems that can be worked out by hand. For example, Mathematica can multiply matrices together much faster than any person can and without any algebraic mistakes, so why should the reader ever perform these tasks by hand? The answer is simple: In order to fully grasp the mechanics of matrix multiplication, simple problems must be worked out by hand. This manual labor, although usually deemed tedious, is an important tool in learning reinforcement. The Mathematica problems portion of the homework typically involves problems that would take too long, or would be too computationally complex, to solve by hand. There are many problems in the Mathematica problems portion that simply ask you to verify your answers to questions from the preceding Homework problems, implying that you can think of Mathematica as a solutions manual for a large percentage of this text. You will also notice that several sections are missing the Homework problems section. These sections correspond to special topics that are discussed because they can be explored in detail only with Mathematica.

We suggest that students and instructors alike visit the books companion Website, which can be found at either of the following addresses:

http://carmine.se.edu/kfrinkle/PrinciplesOfLinearAlgebraWithMathematica

http://people.emich.edu/kshiskows/PrinciplesOfLinearAlgebraWithMathematica

At these locations, you can download Mathematica notebooks, corresponding to each sections Mathematica commands, along with many other resources. These files can be used with the book to enable the reader to do problems or practice the material without retyping the entire