I have a question related with the notion of representation of algebraic groups. Let $G$ an algebraic group over $k$ and let $V$ a finite dimensional $k$-vector space. We have that $\mathbb{V}=\rm{Spec}(\rm{S}^{\bullet}_{k}(V^{\ast}))$ is an algebraic variety whose rational points are the points of $V$. I read that a representation of an algebraic group is to give a morphism of algebraic groups
$$\rho:G\rightarrow\rm{Gl}(V)\text{,}$$ but I dont understand how I should interpret $\rm{Gl}(V)$, I think it should be $\rm{Gl}(\mathbb{V})$, because to give a representation of a group over a vector space is equivalent to give an action of $G$ over $V$.
Thank you for your time.