# 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).

Read MoreWelcome! Math3ma is a math blog maintained by Tai-Danae, a PhD student at the CUNY Graduate Center.

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).

Read MoreContinuing 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).

Read MoreOver 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.

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.

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 MoreLast 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.*

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.

Welcome! Math3ma is a math blog maintained by Tai-Danae, a PhD student at the CUNY Graduate Center.

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