I am having difficulty understanding the following sentence, located in section 1.4.1:
Suppose that $S = k[x_1,\dots,x_r ]$ is a polynomial ring, and that $G$ is represented as a group of linear transformations of the vector space of linear forms of $S$ -that is, we are given a homomorphism of groups $G\to GL_r(k)$, where we regard the latter group as the group of invertible linear transformations of the vector space with basis $x_1,\dots,x_r$.
When he writes that $G$ being represented as a group of linear maps of the linear forms of $S$ means a group homomorphism $G\to GL_r(k)$, I don't see how this makes sense. When I saw the words "represented as a group", I thought immediately of a group representation, so a group homomorphism $G\to GL(V)$; in this case it would appear that $V$ is the vector space of linear forms of $S$. But then $GL(V)$ is not isomorphic to $GL_r(k)$ as $V$ is not finite dimensional over $k$. So I don't really understand how the two ideas are equivalent (even if he didn't mean a representation in this sense, I don't see how the notions are the same).
Would anyone mind explaining this? And if somehow this sentence doesn't make sense, should I just interpret "$G$ is represented as a group of linear transformations of the vector space of linear forms of $S$" to mean a group homomorphism $G\to GL_r(k)$ for the rest of the section/book?
I think "linear forms of $S$"
might be a typo, and if it isn't then itmeans the homogeneous linear polynomials in $S$ (note that he writes "of" and not "on"). From context it's clear that the intended meaning is that Eisenbud wants to consider a linear action of $G$ on a finite-dimensional vector space $V = \text{span}(x_1, \dots x_n)$ which is then extended to a $k$-algebra action on the symmetric algebra $S(V) \cong k[x_1, \dots x_n]$.