Let $\rho\colon G\to GL(V)$ be a linear representation of $G$ on a $k$-vector space $V$. The dual representation is
$$G\to GL(V^*),\quad g\mapsto(\varphi\mapsto\varphi\circ\rho(g^{-1})).$$
By the same rule, we get a linear representation on the coordinate ring $k[V]$ of $V$, which is the $k$-algebra generated by $V^*$, with pointwise multiplication. Now if $V$ is finite-dimensional, $e_1,\dots,e_n$ the standard basis, and $x_1,\dots,x_n$ the dual basis, then $k[V]\cong k[x_1,\dots,x_n]$. Hence there is a linear representation $\bar\rho$ of $G$ on $k[x_1,\dots,x_n]$.
My question is: Is there a way to go "back"? That is, given a linear action of a group $G$ on $k[x_1,\dots,x_n]$, does it come from a linear representation of $G$ on an $n$-dimensional vector space $V$?
More generally, given a linear representation $\bar\rho:G\to GL(k[V])$, does it come from a representation $G\to GL(V)$?
Edit: As an example, consider $G\subseteq GL_n(k)$ acting on $k[x_1,\dots,x_n]$ via $(A,f)\mapsto f(A\cdot x)$. Does this action come from a linear representation of $G$ on $k^n$? I could consider $G$ acting on $k^n$ by multiplication, but the action on the coordinate ring corresponding to this should be $(A,f)\mapsto f(A^{-1}\cdot x)$, or am I mistaken here?
This is motivated by the fact that in invariant theory, one studied actions of subgroups $G\subseteq GL_n(k)$ on the polynomial ring $k[x_1,\dots,x_n]$ as above. But many theorems or facts in invariant theory are formulated in the language of representations, and the invariant ring there is $k[V]^G$. Now if I know that under certain circumstances $k[V]^G$ is finitely generated, I wanted to "come back" to the initial problem of linear actions on $k[x_1,\dots,x_n]$. But I can't really connect these two yet. As written above, any representation of $G$ on some f.d. vector space $V$ induces an action on the coordinate ring $k[V]$, but why do these include the above actions on the polynomial ring? And if they do, how does the initial action on $k^n$ look, given $\rho:G\to GL(k[V])$?
Edit5: I finally found a way to hopefully make clearer what I'd like. Let $G\subseteq GL_n(k)$. Can every linear $G$-action on $k[x_1,\dots,x_n]$ be written as $\bar\rho:G\to GL(k[V])$, coming from some $\rho:G\to GL(V)$? So that the $k[V]^G$ really are a "generalization" of the invariant ring $k[x_1,\dots,x_n]^G$.