Hi I am trying to find if there is a more simple way to prove the following theoreom without going through the two page proof in the Linear Algebra textbook I have been using.
Let $V$ be a finite-dimensional vector space over an infinite field $F$ and let $T:V \rightarrow V$ be a linear operator. Give to $V$ the structure of a module over the polynomial ring $F[x]$ by definiting $x \alpha = T(\alpha)$
Question: How do we sketch a proof that $V$ is a direct sum of cyclic $F[x]$-modules?
The proof I know for a vector space $V$ comes from Hoffman and Kunze's "Linear Algebra" and has four steps but is in the languae of cyclic subspaces $Z(\alpha,T)$. In particular I could reproduce the proof of the Cyclic decompostion theorem on p.233 which states if $W_0$ is a $T$-admissible subspace of $V$ there exists non-zero vectors $\alpha_1, \ldots, \alpha_r \in V$ with respective $T$-annihalators $p_1, \ldots, p_r$ such that $V = W_0\oplus Z(\alpha_1;T) \oplus \ldots \oplus Z(\alpha_r;T)$. I feel like there are some shortcuts to be made here.