Text
Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991. ISBN: 0130047635.
Subjects to be Covered
Groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups are discussed in great detail. Topics in groups include isomorphisms, homomorphisms, cosets, and quotient groups. Topics in vectors include real vector spaces, abstract fields, and infinite-dimension spaces. Topics in linear transformations include linear operators and eigenvectors, orthogonal matrices and rotations, and diagonalization. In symmetry the finite and discrete groups of motions, and the permutation representations are analyzed. Bilinear forms and linear groups include topics such as the Hermitian forms, the spectral Theorem, the skew-symmetric forms, the special unitary and linear groups, and the Lie algebra. I believe that all of the topics are interesting and useful. The table of contents for Chapters 2–8 of the text provides an overview, but time constraints will force us to skip some of the topics in those chapters.
Prerequisites
You should have some familiarity with proofs. Also, we will not go over the first chapter on matrix operations systematically. If you are not comfortable with matrix operations, please study Chapter 1.
Formal Course Requirements and Grading
There will be weekly problem sets, which will be graded, and three quizzes during the regular class time. There will be no final exam. Weighting in the final grade will be roughly 25% for the homework and 25% for each quiz. Please make a note of the quiz dates. They are sessions 12, 26 and 37.
Preparation
The readings section contains reading assignments and exercises on the topic of each lecture. Going through each topic systematically in class can get boring, and I will not do it. I rely on you to do the reading, if possible ahead of time. Even a few minutes before class will help your understanding a lot.
Work the exercises listed in the readings section, but do not turn them in. Most of them should not be too hard, once the material has been absorbed. If you have serious difficulties with these exercises, see me immediately. I've tried to keep the number of exercises that are pure drill to a minimum, so there will be times when you need additional practice. You can choose suitable ones yourself or consult me.
Homework
The problem sets are the most essential part of the course. They will contain some extensive and difficult problems, and require hard work. You will not be able to complete them in one sitting. I recommend that you get together with other students to work on these assignments. However, consulting solutions from previous years' problem sets is not permitted, and the solutions that you hand in must be written entirely by you.
In past years, a few students have had trouble finishing their assignments on time, and fall further behind each week. To avoid the stress that this causes, and for the sake of the graders, I've instituted a firm rule: Written assignments must be handed in on the day they are due. It is better to hand in an incomplete assignment than to put it off, hoping to finish the work in the future.