Consider $V$ a vector space over a field $F$ with dim$_F V = n \geq 1$ and let $T : V \rightarrow V$ be a linear map. Now suppose that $W$ is a cyclic submodule of the $F[x]$-module $V_T$, which is generated by some $v \in W$, and suppose that Ann$(v) = (f)$ where $f \in F[x]$ is monic of degree $k \geq 1$.
How might I go about proving that $v, T(v), \ldots, T^{k-1}(v)$ is an $F$-linear basis for $W$? I'm kind of stuck to get started on this problem - would appreciate any help.
Well, if you had a linear relation $$a_0v+a_1T(v)+\dots +a_{k-1}T^{k-1}(v)=0,$$ it would mean the polynomial $\;g(x)=a_0+a_1x+\dots +a_{k-1}x^{k-1}$ belongs to $\operatorname{Ann}(v)$ and as it has degree less than the generator degree of this ideal, $g(x)$ is the $0$ polynomial.