Let $G$ be a closed subgroup (algebraic group) of $GL(n,K)$. If $G$ leaves stable a subspace $W$ of $K^n$, prove that $\mathfrak{g} \subseteq \mathfrak{gl}(n,K)$ does likewise. Converse?
Here, $K$ is an algebraically closed field. $\mathfrak{g}$, $\mathfrak{gl}(n,K)$ are the Lie algebras of $G$ and $GL(n,K)$ respectively.
I think $\mathfrak{g}$ acts on $K^n$ by matrix production. But I find it difficult to corresponds $G$ with $\mathfrak{g}$, especially in the matrix condition. Thanks.