# June 2015

# Stone Weierstrass Theorem

The Stone Weierstrass Theorem is a generalization of the familiar Weierstrass Approximation Theorem. In this post, we introduce the Stone Weierstrass Theorem and, by looking at counterexamples, discover why each of the hypotheses of the theorem are necessary.

# Rational Canonical Form: Example #2 (with Galois Theory)

Last week we saw an example of how to use the rational canonical form (RCF) to classify matrices of a given order in $GL_2(\mathbb{Q})$. 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 $\mathbb{Q}$ 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.

# Rational Canonical Form: Example #1

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.*

# Rational Canonical Form: A Summary

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*.

In sum, the RCF is important because it allows us to classify linear transformations on a vector space *up to conjugation.* 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.

# Finitely Generated Modules Over a PID

We know what it means to have a module $M$ over a (commutative, say) ring $R$. We also know that if our ring $R$ is actually a field, our module becomes a vector space. But what happens if $R$ is "merely" a PID? Answer: A lot. Today we'll look at a proposition, which, thanks to the language of exact sequences, is quite simple and from which the Fundamental Theorem of Finitely Generated Modules over a PID follows almost immediately. The information below is loosely based on section 12.1 of Dummit and Foote' *Abstract Algebra*.