Let $k$ be a field. Let $F$ be a covariant functor on the category of $k$-algebras to the category of sets. Assume that the opposite functor $F^{op}$ on the category of affine $k$-schemes is a sheaf. (This last assumption might not be necessary.)

Now, suppose that $F(R)$ is a finitely generated $R$-module for all $k$-algebras $R$ and that

$F(R) = F(k)\otimes_k R$.

It seems that in this case $F$ is "representable" by the vector space $$F(k).$$

I don't know what this means and I'm probably missing some very standard construction here. It should mean that $F^{op}$ is representable by a finite type $k$-scheme.

I know one can associate to a finite-dimensional vector space $E$ over $k$ the affine variety $\mathrm{Spec} \ \mathrm{Sym}(E).$ Is it clear that in this case the functor $F$ is representable by the affine variety $\mathrm{Spec} \ \mathrm{Sym}(E)$?