$f$-invariant subspaces for $\chi_f(X)=\mu_f(X)=X^n$ Let $n\geq 1\in\mathbb{N}$, let $V$ be an $n$-dimensional $\mathbb{F}$-vector space and let $f:V\to V$ be a linear endomorphism such that 
$$\chi_f(X)=\mu_f(X)=X^n,$$ where $\chi_f$ denotes the characteristic polynomial of $f$ and $\mu_f$ denotes the according minimal polynomial.
Assignment: Find all $f$-invariant subspaces.

My work so far: I can see that $f$ is tridiagonalizable since $\chi_f$ splits so that gives us an $f$-invariant flag but what now? Are there more $f$-invariant subspaces of $V$?

Thank you very much in advance.

Hint: There's a vector $v$, such that $v,f(v), \dots, f^{n-1}(v)$ form a basis.
If a vector in an invariant subspace with coordinates $(a_1,a_2,\dots, a_n)$ in this basis has $a_i$ as the first nonzero coordinate, then all vectors with $0$ coordinates for indices $<i$ will belong to the subspace.
