Over the past few weeks, we've been chatting about quotient groups in hopes of answering the question, "What's a quotient group, really?" In short, we noted that the quotient of a group G by a normal subgroup N is a means of organizing the group elements according to how they fail---or don't fail---to satisfy the property required to belong to N. The key point was that there's only one way to belong to N, but generally there may be several ways to fail to belong.Read More
In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses.Read More
Last week we saw an example of how to use the rational canonical form (RCF) to classify matrices of a given order in GL_2(Q). Today we have a similar example (taken from CUNY's spring 2015 qualifying exam) where now our matrices have entires in the finite field F_13. The fact that our field is F_13 instead of Q actually makes little difference in how to approach the solution, but I think this problem is particularly nice because part of it calls on some Galois Theory.Read More
Last time we discussed the rational canonical form (RCF) of a linear transformation, and we mentioned that any two similar linear transformations have the same RCF. It's this fact which allows us to classify distinct linear transformations on a given F-vector space V for some field F. Today, to illustrate this, we'll work through a concrete example:
Find representatives for the distinct conjugacy classes of matrices of finite order in the multiplicative group of 2x2 matrices with rational entries.Read More
This post is intended to be a hopefully-not-too-intimidating summary of the rational canonical form (RCF) of a linear transformation. Of course, anything which involves the word "canonical" is probably intimidating no matter what. But even so, I've attempted to write a distilled version of the material found in (the first half of) section 12.2 from Dummit and Foote's Abstract Algebra.
Below we'll set up some background, then define the rational canonical form, and close by discussing why the RCF looks the way it does. Next week we'll go through an explicit example to see exactly how the RCF can be used to classify linear transformations.Read More