In Dummit and Foote, it states: "We will see in Chapter 12 that the .. ideal structure of the ring $F[x]$ forces the $F[x]$-module structure of $V$ to be correspondingly uncomplicated, and this in turn provides a great deal of information about the linear transformation $T$..".
I understand how to view $V$ as an $F[x]$-module (including the rational canonical form and the Jordan canonical form), but I seem to have some disconnect between that and the source of this construction because I often have trouble understanding what the isomorphism: $$V \cong F[x]/\langle f_1 \rangle \oplus F[x]/ \langle f_2 \rangle\oplus \dots \oplus F[x]/ \langle f_k \rangle$$ (that results from this view of $V$ followed by the application of the Fundamental Theorem) looks like -- i.e. the translation to the right-hand-side.
For example, how can I see (hopefully through an explicit and simple example) that:
$$\langle v \rangle \cong F[x]/\langle f_v \rangle$$ (what would $v$ look like on the right-hand-side? I would guess that it corresponds to $f_v$, but why? and how would this relate to the rational canonical form? Why does each $v$ have such an $f_v$ etc... These are the type of questions that pop into mind that I cannot seem to answer and that maybe a simple example or two could help clarify).
$V$ needs to be finite dimensional.
$$V \cong F[x]/\langle f_1 \rangle \oplus F[x]/ \langle f_2 \rangle\oplus \dots \oplus F[x]/ \langle f_k \rangle$$ is an obfuscated way to say that there are some $v_1,\ldots,v_k\in V$ such that $$V = \bigoplus_{j=1}^k span(v_j,Tv_j,T^2v_j,\ldots)$$
$span(v_j,Tv_j,T^2v_j,\ldots)$ is better written as $F[T] v_j$ which is isomorphic to $F[x]/(f_{v_j}(x))$ as a $F[x]$-module, the isomorphism being to send $g(x)+(f_{v_j}(x))$ to $g(T)v_j$.
$f_{v_j}(x)\in F[x]$ is the minimal polynomial of $T|_{F[T] v_j}$.