Rational canonical form from the structure theorem

Rational canonical form from the structure theorem: Let $V$ be a finite-dimensional vector space over a field $F$ and let $T:V\to V$ be linear. Then there exists a basis of $V$ for which the matrix of $T$ is in rational canonical form.

This is an application of the structure theorem over a PID to the polynomial ring $F[x]$ (with $F$ a field), after applying the polynomial-module construction to a linear map on a vector space; see rational canonical form theorem.

Equivalent characterizations

Equivalently, viewing $V$ as an $F[x]$ -module via $x\cdot v = T(v)$ , there is an isomorphism

V \cong \bigoplus_{j=1}^k F[x]/(f_j(x))

with monic polynomials $f_1\mid f_2\mid\cdots\mid f_k$ (the invariant factors of $T$ ).