Lately at work I have found that I have a lot of time for reading, and ever since I got my new iPhone I have been able to download some math texts onto there in order that I might stay on top of my math game this summer. I have downloaded Commutative Algebra (Atiyah and Macdonald), Abstract Algebra (Dummit and Foote), and Categories for the Working Mathematician (MacLane) to name a few. I started out with Categories and then after about 40 pages got a little exhausted of the way of thinking so I switched over to doing some review for my Algebra qualifying exam at the University of Michigan.
One area of my Commutative Algebra class that we didn't really go into was Modules over PIDs so I decided to go through that chapter in Dummit and Foote. It also contains some seminal areas of Linear Algebra that have been missed in my education, namely the canonical forms (both rational and Jordan). Today for the first time I was reading through an explanation of the rational canonical form for transformations of vector spaces and I wanted to share some of the difficulties that I had in understanding the representation (on pg. 385 if you care).
Dummit and Foote does it as follows, they take the isomorphism from V <--> * F[x]/(a1(x)) x ... x F[x]/(an(x)). But when they say F[x] they really mean F[T] (where T is the tranformation which is the subject of the representation). And then they show that to each ak(x) there corresponds a "companion matrix" which represents the the transformation from *k to *k. But I am not trying to explain the proof. my confusion had to do with how this really differentiates between transformations. And the conclusions I came to was that it occurs when you select basis the elements bk in V which generate *k. (if this is incorrect I welcome comments)
And I guess this should obviously be the case as the whole point of the existence of the rational canonical form is that there exists an appropriate basis for V in which the transformation T can be represented as a certain form of matrix. It seems that Dummit and Foote did a poor job of emphasizing this fact in their book however and this is what I am going to blame my confusion on.
I tried looking at how a few other texts go about explaining it and I downloaded Hoffman and Kunze's Linear Algebra. But I got distracted by all the other good math in the book that I felt weak in that I never got to their explanation of the Rational Canonical Form. Hopefully I can get to that tomorrow...
TO BE CONTINUED
