What are the invariant subspaces of $f$ an endomorphism on $V$ over $\mathbb{C}$ with basis $e_1, \ldots, e_n$ such that $f(e_1)=e_1, f(e_i)=e_i+e_{i-1}$ for $i\gt 1$?
I am thinking the invariant subspaces must just be $\langle e_1, \ldots, e_k\rangle$ for $1\le k\le n$. But I don't know how to prove or disprove that there are no other invariant subspaces.