In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. If $f$ is an endomorphism of $E$, for every subspace $F$ of $E$ stable by $f$ we denote by $f_F$ the endomorphism of $F$ induced by $f$.
Show that a line $F$ generated by a vector $u$ is stable by $f$ if and only if $u$ is an eigenvector of $f$.