Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. For every map $\lambda$ from $\mathcal { E }$ to $\mathbb { K }$, we set: $$\mathcal { A } _ { \lambda } = \left\{ M \in \mathcal { A } \mid \Phi _ { H } ( M ) = \lambda ( H ) M \text { for all } H \in \mathcal { E } \right\}$$ Let $\lambda$ be a map from $\mathcal { E }$ to $\mathbb { K }$. a) Show that $\mathcal { A } _ { \lambda }$ is a vector subspace of $\mathcal { A }$. b) Show that if $\mathcal { A } _ { \lambda }$ is not reduced to $\{ 0 \}$, then $\lambda$ is a linear form on $\mathcal { E }$.
Let $\mathcal { A }$ be a non-zero vector subspace of $\mathcal { M } ( n , \mathbb { K } )$ stable by bracket, and let $\mathcal { E }$ be the intersection of $\mathcal { A }$ and $\mathcal { D } ( n , \mathbb { K } )$. For every map $\lambda$ from $\mathcal { E }$ to $\mathbb { K }$, we set:
$$\mathcal { A } _ { \lambda } = \left\{ M \in \mathcal { A } \mid \Phi _ { H } ( M ) = \lambda ( H ) M \text { for all } H \in \mathcal { E } \right\}$$
Let $\lambda$ be a map from $\mathcal { E }$ to $\mathbb { K }$.
a) Show that $\mathcal { A } _ { \lambda }$ is a vector subspace of $\mathcal { A }$.
b) Show that if $\mathcal { A } _ { \lambda }$ is not reduced to $\{ 0 \}$, then $\lambda$ is a linear form on $\mathcal { E }$.