Preface xiii of the results and references to Axler [1996] for the proofs. If Chapter 6 is covered, then the student will see Jordan forms, which are a special case of the triangular forms. It is inevitable that some who look at this material will think I have misjudged the situation. I still hope they will be willing to present the mathematics in a way that makes it accessible to their students. In fact, that is what I think teachers are for. Take, for example, linear algebra. What does the typical undergraduate know? If they only had a single course on the subject, the answer is likely that there are huge gaps. That is a course for which, because it is taken by such a large, varied audience, few texts do a job that mathematicians find exciting. I hope most students facing this book will have had a second course in that subject as Chapters 3 through 6 depend heavily on it. This is why I included Chapter 3 to establish a starting point. In fact, Chapters 3, 4, and 6 together with the relevant parts of the Appendix could constitute a second course in linear algebra. It misses a few things one ordinarily covers there, but in the end the important theorems from that course (for example, the Spectral Theorem and Jordan forms) are covered, though from a more advanced point of view. In fact, this points out something else about this book. There is a lack of balance in that linear algebra surfaces more than any other topic. There are three reasons for this: linear algebra is a subject that every undergradu- ate has seen, it affords many opportunities for connections, and it is a topic that hardly any undergraduate understands. Of all the courses in the un- dergraduate curriculum, linear algebra seems to have the shortest half-life. Even good students seem to forget it in less time than they took to learn it. That’s a fact. It was true when I was an undergraduate and it remains so today. I think the problem is the sterility with which linear algebra is usually presented. I hope that by covering at least one of the chapters that uses linear algebra, the reader will be prompted to review the subject, ap- preciate it more fully, and retain the material longer. In fact, a lot of this book might be subtitled, “Variations on a Theme of Linear Algebra.” Some Advice to the Teacher The material in the book falls into different categories. Some chapters, such as Chapter 5 on Matrices and Topology, seek to show directly how material from two different courses in the undergraduate curriculum are related. Topological questions are asked about sets of matrices, and these questions need linear algebra for the answer. I like this chapter, but it may be that if you teach it, you will have to ask your students to accept some mathematical results, like putting matrices in triangular form, without their ever having seen a proof. That is somewhat counter to the prevailing

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