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.B.1) Show that there exist at least two subspaces of $E$ stable by $f$ and give an example of an endomorphism of $\mathbb{R}^2$ which admits only two stable subspaces. I.B.2) Show that if $E$ is of finite dimension $n \geqslant 2$ and if $f$ is non-zero and non-injective, then there exist at least three subspaces of $E$ stable by $f$ and at least four when $n$ is odd. Give an example of an endomorphism of $\mathbb{R}^2$ which admits only three stable subspaces.
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.B.1)} Show that there exist at least two subspaces of $E$ stable by $f$ and give an example of an endomorphism of $\mathbb{R}^2$ which admits only two stable subspaces.
\textbf{I.B.2)} Show that if $E$ is of finite dimension $n \geqslant 2$ and if $f$ is non-zero and non-injective, then there exist at least three subspaces of $E$ stable by $f$ and at least four when $n$ is odd.
Give an example of an endomorphism of $\mathbb{R}^2$ which admits only three stable subspaces.