Prove that (V,T) is a cyclic $F[x]$-module

Question

I'm working on some problems on modules, and stumbled onto a problem that is closely related to linear algebra and came to a stall.

The problem is:

Suppose $V$ is a $n$-dimensional vector space over a field $F$, and $T:V \rightarrow V$ be a linear transformation.

Prove that $(V,T)$ is a cyclic $F[x]$-module iff there exists a vector $v_0 \in V$ such that the vectors $v_oT, v_0T^2...v_oT^{n-1}$ are linearly independent.

Could anyone shed some light on which direction to take this question?

Answer 1

$V$ is given the structure of an $F[x]$-module by taking the scalars in $F$ to act on vectors in $V$ by scalar multiplication, and by taking $x$ to act as the linear transformation $T$ on vectors in $V$.

For the "if" part, you know that $V$ is generated by $\{ v_0, \dots, T^{n-1}v_0 \}$ as a module over $F$, since this is a basis for $V$ over $F$. Can I suggest you show that $V$ is generated by the single element $v_0$ as a module over $F[x]$?

I'll leave you to work out the "only if" part.