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.