Symmetric power of vector space Let $V$ be a vector space over a field $k$ of char. zero and denote by $Sym^n_k V$ its $n$-th symmetric power over $k$.
Now I simply want to know what
$Hom_k(V,Sym^n_k V)$
is for $n \geq 2$. To be more precise: might it be zero? Can one say anything more about this $Hom$?

As Dylan Moreland points out, this is a strange question. Unless you are asking for some kind of natural isomorphisms we are in vector space and so you should do as the vector space theorists do--count dimensions. Namely, if $\dim_k V=m$ then $\dim_k \text{Sym}^n(V)={m+n-1 \choose n}$. Thus, $\dim_k \text{Hom}_k(V,\text{Sym}^n(V))=m{m+n-1 \choose n}$.
You should note that one of the things that makes vector spaces so incredibly nice is that the statement $\text{Hom}_k(V,W)$ is zero is almost absurd. Vector space theory is just studying finite sets and maps between those finite sets. Thus, the statement "Do there exist vector space maps $V\to W$?" is the same thing as asking "Do there exist SET maps $X\to Y$?" where $X$ and $Y$ are the bases of $V$ and $W$.
