ELEMENTS OF ABSTRACT AND L.pdf

Elements of
Abstract and Linear Algebra
E. H. Connell
ii
. Connell
Department of Mathematics
University of Miami
. Box 249085
Coral Gables, Florida 33124 USA
******@
Mathematical Subject Classifications (1991): 12-01, 13-01, 15-01, 16-01, 20-01
°c 1999 . Connell
October 5, 2000 [.edu/∼ec/book/]
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 . 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. anize the material in the most simple-minded
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 anize 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 sc