Proving that every invariant subspace of $\mathbb{F}^n$ is of the form $span(e_1,...,e_k)$

Question

Let $T_{J_n(0)}:\mathbb{F}^n \to \mathbb{F}^n$. I need to prove that every invariant subspace $W$ of $\mathbb{F}^n$ is of the form $Span(e_1,...,e_k)$ for some $0\leq k \leq n$.

This is what I've got so far:

for $k=0$ we get $span( \phi )=\{ 0\} \leq W$. Now let $k$ be the maximal value such that $span(e_1,...,e_k) \leq W$. We assume, by way of contradiction, that $span(e_1,...,e_k) \neq W$. Let $w\in W$. So $w = \sum a_i e_i$. We know that there exists some $l>k$ such that $a_l e_l \neq 0$.

From here I can't figure out a way to find a solid contradiction. I know that this $l$ value can contradict the maximality of $k$ but I don't know how to show it.

Answer 1

Suppose $l$ is the largest such that $a_l \ne 0$. If $l > k$ then $T^{l - k - 1}w = v + a_l e_{k + 1}$ for some $v \in \operatorname{span}(e_1, \dots, e_k)$. Since $W$ is an invariant subspace, this shows that $a_l e_{k + 1} \in W \implies e_{k + 1} \in W$.