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. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$. I.C.1) Show that every subspace generated by a family of eigenvectors of $f$ is stable by $f$. Specify the endomorphism induced by $f$ on every eigenspace of $f$. I.C.2) Show that if $f$ admits an eigenspace of dimension at least equal to 2 then there exist infinitely many lines of $E$ stable by $f$. I.C.3) What can be said about $f$ if all subspaces of $E$ are stable by $f$?
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. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$.
\textbf{I.C.1)} Show that every subspace generated by a family of eigenvectors of $f$ is stable by $f$. Specify the endomorphism induced by $f$ on every eigenspace of $f$.
\textbf{I.C.2)} Show that if $f$ admits an eigenspace of dimension at least equal to 2 then there exist infinitely many lines of $E$ stable by $f$.
\textbf{I.C.3)} What can be said about $f$ if all subspaces of $E$ are stable by $f$?