We consider the differentiation endomorphism $D$ on $\mathbb{K}[X]$ defined by $D(P) = P'$ for all $P$ in $\mathbb{K}[X]$. III.A.1) Verify that for all $n$ in $\mathbb{N}$, $\mathbb{K}_n[X]$ is stable by $D$ and give the matrix $A_n$ of the endomorphism induced by $D$ on $\mathbb{K}_n[X]$ in the canonical basis of $\mathbb{K}_n[X]$. III.A.2) Let $F$ be a subspace of $\mathbb{K}[X]$, of finite non-zero dimension, stable by $D$. a) Justify the existence of a natural integer $n$ and a polynomial $R$ of degree $n$ such that $R \in F$ and $F \subset \mathbb{K}_n[X]$. b) Show that the family $(D^i(R))_{0 \leqslant i \leqslant n}$ is a free family of $F$. c) Deduce that $F = \mathbb{K}_n[X]$. III.A.3) Give all subspaces of $\mathbb{K}[X]$ stable by $D$.
We consider the differentiation endomorphism $D$ on $\mathbb{K}[X]$ defined by $D(P) = P'$ for all $P$ in $\mathbb{K}[X]$.
\textbf{III.A.1)} Verify that for all $n$ in $\mathbb{N}$, $\mathbb{K}_n[X]$ is stable by $D$ and give the matrix $A_n$ of the endomorphism induced by $D$ on $\mathbb{K}_n[X]$ in the canonical basis of $\mathbb{K}_n[X]$.
\textbf{III.A.2)} Let $F$ be a subspace of $\mathbb{K}[X]$, of finite non-zero dimension, stable by $D$.
a) Justify the existence of a natural integer $n$ and a polynomial $R$ of degree $n$ such that $R \in F$ and $F \subset \mathbb{K}_n[X]$.
b) Show that the family $(D^i(R))_{0 \leqslant i \leqslant n}$ is a free family of $F$.
c) Deduce that $F = \mathbb{K}_n[X]$.
\textbf{III.A.3)} Give all subspaces of $\mathbb{K}[X]$ stable by $D$.