Math 20810: Honors Linear Algebra

Fall `11

Weekly schedule

Instructor: Jeffrey Diller (click for contact info, general policies, etc.)

Official Time and place: MWF 8:30-9:20 AM, Hayes-Healy 229.

Textbook: Linear Algebra Done Wrong by Sergei Treil. The good news is that the book is free. The bad news is that it's available only online. There are lots of linear algebra textbooks to choose from, and none of them seems perfect to me. This one covers approximately the material I'd like to cover at approximately the level I'd like to work at, and the price is right. If you find what you think is a typo or have some particular idea for how the book might be improved, please bring it to my attention. One of the ways we can show our appreciation to Treil is to help him make his book better.

I will very freely use other sources for the material in this class.  I recommend looking at the following books that I've put on the reserve shelf in the math library:

Linear Algebra Done Right by Axler

Linear Algebra with Applications by Bretscher

Linear Algebra by Hoffman and Kunze

Linear Algebra and its Applications by Lax.

A Terse Introduction to Linear Algebra by Katznelson and Katznelson.

Axler's book is particularly well-written and has a very appealing point of view, though it more or less assumes some previous exposure to linear algebra. Bretscher's book is one of the best books out there written for a `standard' sophomore course in linear algebra. Refer to it, especially when you need more examples or when our textbook seems too difficult. The last three books, listed in roughly increasing order of difficulty, are more comprehensive. Hoffman and Kunze would be a plausible candidate for a textbook for this course. The other two are a little too hard for a first time around, but they'd be fantastic to own once you're a little more comfortable with linear algebra. Lax has all sorts of cool extra topics in it and approaches the subject with the attitude of an analyst and an applied mathematician.  Katznelson x 2 is more of a pure algebra take on the subject, and as its title suggests, it doesn't waste words.

Finally, I should note that Apostol's Calculus (vol 1), which most of you used in the honors sequence last year, has its own treatment of linear algebra (chaps 12-16).  It does other useful things like treat the field axioms and discuss complex numbers.  So assuming you still own the book, please continue to hang on to it.

What we'll cover: My plan for this semester (tentative, as always) is to cover Chapters 1-5. I won't necessarily take the material in the order it appears in the book. For instance, I'd like to say something about fields (not really discussed in Treil) and solving linear systems (not discussed til chapter 2 of Treil) right away. The broad topics to cover are vector spaces, linear transformations, determinants, diagonalization, and inner product spaces.

In the second semester, I plan to take on more sophisticated topics and applications.  Certainly these include ordinary differential equations, operators on inner product spaces and Jordan canonical form.  Beyond that, I'm not sure.  Quadratic forms, multilinear algebra, and convexity are all topics worthy of attention.  So are many of the applications of linear algebra: e.g. network theory, finite element method, simplex method, numerical analysis, platonic solids, game theory, coding theory, etc.

How you will be evaluated:

    Midterm Exam: Tuesday 10/11, 6:20-8:20PM  in  Hayes-Healy 125, with take home portion due in my mailbox at noon on Friday 10/14. Worth 20% of your final grade.

Help sessions: every Monday 7:30-8:30 PM in the partitioned off area in the math library.

Further comments:  homework is key to this course.  I highly encourage you to work with your classmates on homework problems.  In lieu of office hours, I'd like to set up a weekly help session on Monday afternoon or evening where we can all meet and discuss questions about the homework together.  On the other hand, since you have a week to work on each assignment, and I want to discourage putting off starting til the last minute, I plan to be really unavailable for help on Tuesdays. 

Please take a lot of care in writing up solutions to homework problems.  The english as well as the math should be good, and your first draft of a solution to a problem is probably not the one you should turn in for grading.  By now some of you have discovered the joys and sorrows of TeX, a software package most mathematicians use for writing papers, etc.  A really good way to learn and become proficient with TeX is to use it to write up solutions to homework problems.  If you'd like to use it in this course, I'd be glad to help.  In fact, if there's enough interest, I could arrange a tutorial session for getting started with TeX.