This is my textbook Elements of Abstract and Linear Algebra. It covers abstract algebra in general, but the focus is on linear algebra.   Here is a brief outline of the six chapters.

  1. Background:   Equivalence relations, injective, surjective, and bijective functions, product of sets, unique prime factorization in the integers (18 pages).

  2. Groups:   Normal subgroups, quotient groups, homomorphisms, permutations and symmetric groups, product of groups (18 pages).

  3. Rings:   Ideals and quotient rings, Zn, homomorphisms, polynomial rings, product of rings, Chinese remainder theorem, characteristic, Boolean rings (16 pages).

  4. Matrices and Matrix Rings:   Invertible matrices, elementary operations and elementary matrices, determinant, similarity, trace, characteristic polynomial (14 pages).

  5. Linear Algebra:   Modules, homomorphisms, quotient modules, products and coproducts, summands, independence, generating sets, free basis, uniqueness of dimension, change of basis, rank of a matrix, geometric interpretation of determinant, nilpotent homomorphisms, eigenvalues, characteristic roots, inner product spaces, orthogonal groups, diagonalization of symmetric matrices by orthogonal matrices (40 pages).

  6. Appendix:   Chinese remainder theorem, prime and maximal ideals, UFDs, splitting short exact sequences, finitely generated modules over Euclidean domains, Jordan form, determinants and multilinear forms, dual spaces (28 pages).

You may view any part of the book you wish. Although this is a copyrighted book, I am encouraging people to download and photocopy it, and use it as a textbook. This book evolved right out of the classroom, and I am looking forward to comments from users.

Here is how the book came about. In 1965 I taught abstract algebra at Rice University from the first issue of Herstein's book. Many of the students were freshmen, and five students from that class went on to receive PhD s in mathematics. I liked the text - some of the sections were really beautiful - but I thought it too advanced for a beginning course, and the linear transformations came too late in the book. Since then I have taught the course about a dozen times from various texts. My lecture notes evolved over the years and in 1986 I received a copyright on five chapters. They covered what I thought a year of undergraduate algebra should cover. They were about 100 pages, and I started using them as a base for my courses instead of standard texts. Here were some of my motives.

  1. To have something as short and inexpensive as possible. In my experience, most students prefer short books.

  2. To avoid all innovation. To organize the material in the most simple-minded, straightforward manner.

  3. To make the book linearly-ordered. To the extent possible, each section should use the previous sections and be used in the next sections. The book should flow like a river. If a topic is not used later, it should not be included. Teaching and learning the material should require as little work as possible. The only difficulties should be those inherent in the material.

  4. To omit as many topics as possible. This is a foundational course, not a topics course. The goal is to do the minimum amount of abstract algebra necessary to do the linear algebra, and to have material so basic that it is beneficial to students in computer science and the physical sciences.

  5. To offer an alternative to discrete mathematics. Most of the material in the first four chapters is covered in various discrete math courses. Computer science students might benefit from seeing this material organized from a purely mathematical viewpoint.

  6. To teach abstract and linear algebra as one coherent theory. Teaching abstract algebra and linear algebra as separate courses results in a loss of synergy and a loss of momentum.

In 1996 I had a sixth chapter typed, giving enough material for a full first year graduate course. A year later I revised the entire book and had it typed over in LaTeX. Still it is only about 135 pages. With this text, the professor does not extract the course from the text, but rather builds the course upon it. To me it is easier to build a course from a base than to extract it from a thick book. Because after you extract it, you still have to build it. The text has been edited by Professors John Zweibel, Huseyin Kocak, Dmitry Gokhman, and Professor Shulim Kaliman taught from it one year. Comments from these and other professors have resulted in over a hundred changes. A lower-level class with little background can do the first three chapters in a semester and then do chapters 4 and 5 the second semester. A better class can finish chapter 4 the first semester and then do 5 and 6 the second semester. I have made every effort to limit the material so that the abstract algebra fits into one semester. Still, you have to truck right along to finish chapter 4 in that time. Chapter 2 is by far and above, the most difficult part of the book. This is because groups are written in additive and multiplicative notation, and the concept of coset is confusing at first. After Chapter 2 it gets easier as you go along. Indeed, after the first four chapters, the linear algebra follows easily.

The present situation with college textbooks is a national disgrace. Textbooks are too big and too expensive. I was determined to speak right to the student and to have a short book, but keeping it under control was not easy. I worked at it off and on for fourteen years. It came out a little unusual, not by accident or oversight, but because I designed it that way. It is almost entirely the product of my will. Unfortunately mathematics is a difficult and heavy subject. The style and approach of this book is to make it a little lighter. The student has limited time during the semester for serious study, and this time should be allocated with care. The professor picks which topics to assign for serious study and which ones to "wave arms at". The focus is on going forward, because mathematics is learned in hindsight. This book works best when viewed lightly and read as a story, like a graduate student talking to an undergraduate over pizza.

One way to use this text is to have it photocopied double sided and then spiral bound. My students call it "the blue book" because I usually use solid blue covers. (But it has no used car prices). In photocopying double sided, make the odd pages on the right, else the page numbers will be in the center. Very embarrassing. Most machines, when reproducing from single to double sided, will offset the even pages to the left and the odd pages to the right, allowing for the binding. The spiral bindings can be comb or coil. Coil is better but it costs more. Kinko's and other stores give academic discounts, so it is quite inexpensive. I hope the professors and students who try it, enjoy it.

Edwin H. Connell

PS    Professors and students who use this book and wish to participate, are encouraged to send comments and suggestions. I have done about all I can do here without some input from outside.   Thanks.   Ed Connell

PPS    Join the revolution in education! Write a supplement to this book and put it online.   Ed C.

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