The Rational Canonical Form over F and over K (Algebra 3: Lecture 2 Video 4)

Lecture 2: We started this lecture by recalling the process described at the end of the previous lecture that showed that a linear transformation T has a rational canonical form.  We then proved that this rational canonical form is unique.  We used this to answer a question about when different linear transformations S and T on the same vector space V give rise to isomorphic F[x]-modules.  We showed how starting from an n x n matrix A with entries in F, we can define a linear transformation from it, which allows us to define the rational canonical form of a matrix.  At the end of the lecture we discussed the situation where A is an n x n matrix with entries in a field F that is a subfield of a field K.  We proved that the rational canonical form of A is the same whether we compute it over the smaller field F or over the larger field K.

Reading: In this lecture we followed part of Section 12.2, pages 475-478, very closely.