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Unformatted text preview: Elements of Abstract and Linear Algebra E. H. Connell ii E.H. Connell Department of Mathematics University of Miami P.O. Box 249085 Coral Gables, Florida 33124 USA [email protected] Mathematical Subject Classifications (1991): 1201, 1301, 1501, 1601, 2001 c 1999 E.H. Connell March 20, 2004 iii Introduction In 1965 I first taught an undergraduate course in abstract algebra. It was fun to teach because the material was interesting and the class was outstanding. Five of those students later earned a Ph.D. in mathematics. Since then I have taught the course about a dozen times from various texts. Over the years I developed a set of lecture notes and in 1985 I had them typed so they could be used as a text. They now appear (in modified form) as the first five chapters of this book. Here were some of my motives at the time. 1) To have something as short and inexpensive as possible. In my experience, students like short books. 2) To avoid all innovation. To organize the material in the most simpleminded straightforward manner. 3) To order the material linearly. To the extent possible, each section should use the previous sections and be used in the following sections. 4) To omit as many topics as possible. This is a foundational course, not a topics course. If a topic is not used later, it should not be included. There are three good reasons for this. First, linear algebra has top priority. It is better to go forward and do more linear algebra than to stop and do more group and ring theory. Second, it is more important that students learn to organize and write proofs themselves than to cover more subject matter. Algebra is a perfect place to get started because there are many “easy” theorems to prove. There are many routine theorems stated here without proofs, and they may be considered as exercises for the students. Third, the material should be so fundamental that it be appropriate for students in the physical sciences and in computer science. Zillions of students take calculus and cookbook linear algebra, but few take abstract algebra courses. Something is wrong here, and one thing wrong is that the courses try to do too much group and ring theory and not enough matrix theory and linear algebra. 5) To offer an alternative for computer science majors to the standard discrete mathematics courses. Most of the material in the first four chapters of this text is covered in various discrete mathematics courses. Computer science majors might benefit by seeing this material organized from a purely mathematical viewpoint. iv Over the years I used the five chapters that were typed as a base for my algebra courses, supplementing them as I saw fit. In 1996 I wrote a sixth chapter, giving enough material for a full first year graduate course. This chapter was written in the same “style” as the previous chapters, i.e., everything was right down to the nub. It hung together pretty well except for the last two sections on determinants and dual spaces. These were independent topics stuck on at the end.These were independent topics stuck on at the end....
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 Spring '09
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