In my current position as Director of Undergraduate Studies for the Georgia Tech School of Mathematics, I’ve been heavily involved with revamping our linear algebra curriculum. So I’ve spent a lot of time lately reading through various linear algebra books. The goal of this post is to give a self-contained proof of the existence and uniqueness of the Jordan Canonical Form which is somewhat different from the ‘usual’ proofs one finds in textbooks. I’m not claiming any novelty — I’m sure this approach has been discovered before — but I don’t know a good reference so I thought I’d record the details here.
The proof I give here does not use properties of polynomials (e.g. the Chinese Remainder Theorem), nor does it rely on the classification of finitely generated modules over a PID, so it might be of some pedagogical interest. The proof I give for the Generalized Eigenvector Decomposition is based on an auxiliary result — the Fitting Decomposition — which in my opinion ought to be better known. The proof I give of the structure theorem for nilpotent operators comes from these lecture notes of Curt McMullen (Theorem 5.19). It is particularly concise compared to some other arguments I’ve seen. Continue reading
Yesterday marked the second anniversary of my blog, and today is the 239th birthday of the U.S. In celebration of Independence Day, I want to explain what matroids are. Matroids were invented by Hassler Whitney (and independently by Takeo Nakasawa) to abstract the notion of linear independence from vector spaces to a much more general setting that includes acyclicity in graphs. Other pioneering early work on matroids was done by Garrett Birkhoff and Saunders MacLane. Matroid theory is a rich subject about which we will only scratch the surface here. In particular, there are many different (“cryptomorphic“) ways to present the matroid axioms which all turn out to be (non-obviously) equivalent to one another. We will focus on just a couple of ways of looking at matroids, emphasizing their connections to tropical geometry. Continue reading
The following post was originally published on the AMS Blog “On Teaching and Learning Mathematics”. I have reproduced it here with the permission of the AMS.
Last year, I began offering an online Number Theory and Cryptography course for gifted high school students through Georgia Tech. Fourteen high school seniors from metro Atlanta took the course in Fall 2014, and overall I would say it was a big success. We will be offering the course again in Fall 2015 and are expecting roughly double the number of students. After describing the structure of the course, I will relate some of my experiences and describe some of the things I learned along the way. I hope this article stimulates others to think outside the box about using technology in education without necessarily following the standard “MOOC” model.