Vector space/Flag/Invariant under endomorphism/Exercise

Let ${}V$ be an ${}n$-dimensional ${}K$-vector space over a field ${}K$. Let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

be a flag in ${}V$. Show that there exists a bijective linear mapping

$\varphi \colon V\longrightarrow V$

such that this flag is the only ${}\varphi$-invariant flag.

Create a solution