**Preface**

**for**

**The Keys to Linear
Algebra**

This book is designed as a text for a one-semester introductory undergraduate course in linear algebra. Driven by applications, each chapter begins with a realistic problem to motivate the need for learning the subsequent material. Appropriate theory and technology are then used at the end of the chapter to solve the opening problem. Related project exercises involve the student actively in technology-based problem solving---a feature students have responded to enthusiastically in class testing. Numerous other applications are drawn from physics, statistics, business, and computer science to illustrate where and how the techniques of linear algebra apply.

What makes this book unique, however, is that,
in addition to teaching the standard topics of linear algebra, a
primary objective is to equip students with general
problem-solving skills that are needed in their subsequent
mathematics and related courses. This goal is accomplished by
identifying and explaining the underlying mathematical*
thinking processes* that arise in linear algebra. These
thinking processes--- hereafter referred to collectively as the *why*
mathematics---include unification, generalization, abstraction,
identifying similarities and differences, converting visual
images to symbolic form and vice versa, understanding and
creating definitions and proofs, and developing and working with
axiomatic systems. Teaching these ideas explicitly is designed to
reduce the time and frustration involved in learning linear
algebra and to provide the student with a deeper and more-lasting
appreciation of the subject ... and of mathematics in general.

**Teaching Mathematical Thinking
Processes**

To teach the *why *mathematics, a
discussion of such concepts as unification, generalization,
abstraction, and so on, is presented when those ideas arise
naturally in the context of linear algebra. Each such concept is
named and then referred to whenever the idea appears subsequently
in the book.

To understand and to use the techniques of unification, generalization, abstraction, and developing axiomatic systems, students need to acquire certain basic mathematical skills. These skills include doing proofs, identifying similarities and differences, converting visual images to symbolic form (and vice versa), and understanding and creating definitions.

The concept of a proof is fundamental to all
advanced mathematics courses. The approach used here, as
presented in Appendix A, is adapted from *How to Read and Do
Proof*, second edition, by Daniel Solow and is reprinted with
permission from John Wiley & Sons, Inc.

The ability to identify similarities and differences among various mathematical concepts is essential to unification in that like properties from different problems must be isolated, identified, and then brought together in a single framework. This skill is also used in creating definitions, where it is necessary to identify a common property of all objects being defined.

Much effort is also devoted to teaching students how to convert visual images to symbolic form. When students learn mathematics, they typically develop their own ways of imagining and picturing specific concepts---such as the projection of one vector onto another. However, it is one thing to visualize such concepts; it is another thing entirely to translate such an image to symbolic form, especially when quantifiers are involved. The art of doing so is illustrated and explained with many examples. Students are also shown how to check for syntax and logic errors that can arise in the translation process. The reverse technique of converting symbolic mathematics to visual form is also taught.

One area in which the student is given special help is in understanding definitions. Although a student may be able to visualize objects with a desirable property---such as linearly independent vectors---it is quite a challenge to create (or even to understand) the symbolic definition. Merely presenting the definition is inadequate because doing so fails to explain how the definition was arrived at and why the definition is correct. The approach taken here is to teach the student to identify similarities shared by all items having the property being defined, and then to translate those observations to symbolic form. They are also taught to verify that the definition includes all objects having the desirable property while excluding all other objects.

**Pedagogical Features**

The pedagogical features of this book include a realistic introductory problem at the beginning of each chapter to motivate the need for learning the techniques presented in the chapter. Those techniques are then used at the end of the chapter to solve the problem.

Each mathematical thinking process is set off in the text and is easily identified by an icon. These thinking processes appear not only in the appropriate chapter summaries but also in their entirety in Appendix B.

Each chapter contains many numbered and titled examples to illustrate the topic under discussion. Definitions and theorems have special design features for easy reference. Extensive use of figures provides the student with visual images of mathematical concepts.

Each numbered section is followed by numerous
exercises, some of which give the student practice in performing
mechanical computations while others test their understanding of
the *why *mathematics. The last numbered section in each
chapter has exercises that are designed to be solved with MATLAB,
Maple, Mathematica, or a graphing calculator. Solutions to
exercises whose numbers are in blue are given in the back of the
book. The instructor may therefore want to assign for homework
those exercises whose numbers are in black.