I have had trouble answering the following question which is from a study guide to a qualifying exam I will be taking later this summer. I am thinking this question has something to do with cyclic vectors but I have not been able to put the two definitions together.
Definition: If $\alpha$ is any vector in $V$, the $T$-cyclic subspace generated by $\alpha$ is the subspace $Z(\alpha;T)$ of all vectors of the form $g(T) \alpha$, $g \in F[x]$. If $Z(\alpha; T) = V$, then $\alpha$ is called a cyclic vector for $T$.
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 each $V$ the structure of a module over the polynomial ring $F[x]$ by defining $x \alpha = T(\alpha)$ for each $\alpha \in V$
In terms of the expression for $V$ as a direct sum of cyclic $F[x]$-modules, what are necessary and sufficient conditions in order that $V$ have only finitely many $T$-invariant $F$-subspaces?
Every linear operator I have encountered has finitely many $T$-invariant subspaces. Is there a good example of one that has infinitely $T$-invariant $F$-subspaces?
I was thinking that one direction might require $T$ not to have any cyclic vectors but I dont think this is the only hypothesis we need in order to answer even one direction for part 1.