In teaching a beginning course in abstract algebra, one must suppress the urge to cover a lot of material and to be as general as possible. The di¢ culties in teaching such a course are pedagogical, not mathematical. The subject matter is abstract, yet it must be kept meaningful for students meeting abstractness for perhaps the -rst time. The mathematical framework which ties these questions together is called abstract algebra. Not surpris-ingly, given the name, the course is going to be about abstract algebra. Exercise 1.1. What does abstract mean? Note 1.2. The exercises given in the course notes are practice problems with the exception of this particular introduction. all the mathematics that is currently known, but that you understand all the mathematics that might ever be known. This, to many, would be one de nition of God, and we may safely assume that we are all mortal here. The good news lies in what constitutes a proof. A proof is simply a step-by-step revelation of some mathematical truth. This book is a survey of abstract algebra with emphasis on linear algebra. It is intended for students in mathematics, computer science, and the physical sciences. The rst three or four chapters can stand alone as a one semester course in abstract algebra. However they are structured to provide the background for the chapter on linear algebra. 2) fundamental concepts of abstract mathematics. We start with the language of Propositional Logic, where the rules for proofs are very straightforward. Adding sets and quanti ers to this yields First-Order Logic, which is the language of modern mathematics. Being able to do proofs in this setting is the main skill