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 $E$. In this subsection, $E$ is a space of finite dimension. I.D.1) Show that if $f$ is diagonalisable then every subspace of $E$ admits a complement in $E$ stable by $f$. One may start from a basis of $F$ and a basis of $E$ consisting of eigenvectors of $f$. I.D.2) Show that if $\mathbb{K} = \mathbb{C}$ and if every subspace of $E$ stable by $f$ admits a complement in $E$ stable by $f$, then $f$ is diagonalisable. What about the case if $\mathbb{K} = \mathbb{R}$?
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 $E$. In this subsection, $E$ is a space of finite dimension.
\textbf{I.D.1)} Show that if $f$ is diagonalisable then every subspace of $E$ admits a complement in $E$ stable by $f$.
One may start from a basis of $F$ and a basis of $E$ consisting of eigenvectors of $f$.
\textbf{I.D.2)} Show that if $\mathbb{K} = \mathbb{C}$ and if every subspace of $E$ stable by $f$ admits a complement in $E$ stable by $f$, then $f$ is diagonalisable.
What about the case if $\mathbb{K} = \mathbb{R}$?