Lemma 1

Let $\Pi$ be a subgroup of $V^*$ that generates the vector space $V^*$, and $m$ an integer $\geq 0$. Then pr$_m$(exp $\Pi$ ) generates the vector space $\mathbf{S}^m(V^*)$.

By Algebra, Chap. I, §8, no. 2, Prop. 2, any product of $m$ elements of $V^*$ is a $k$-linear combination of elements of the form $x^m$ where $x\in \Pi$. But $x^m=m$! pr$_m$(exp $x$). Q.E.D.

By transport of structure, every automorphism of V defines automorphisms of the algebras $\mathbf{S}(V)$ and $\mathbf{S}(V)^*$; this gives linear representations of $\mathbf{G}\mathbf{L}(V)$ on $\mathbf{S}(V)$ and $\mathbf{S}(V)^*$.