# What is a Natural Transformation? Definition and Examples, Part 2

Continuing our list of examples of natural transformations, here is: Example #2 (double dual space of a vector space) and Example #3 (representability and Yoneda's lemma).

# What is a Functor? Definitions and Examples, Part 2

Continuing yesterday's list of examples of functors, here is Example #3 (the chain rule from multivariable calculus), Example #4 (contravariant functors), and Example #5 (representable functors).

# A Quotient of the General Linear Group, Intuitively

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.

# Resources for Intro-Level Graduate Courses

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.

# 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(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.

# 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.