Action of an algebraic group is closed.

Question

I am studying geometric invariant theory and I came to the following question. Let us consider an algebraic subgroup $G$ of $SL(V)$ with $V$ a $n$-dimensional $k$-vector space. Let us consider the scheme $$\text{Hom}(W,V)$$ where $W$ is another $k$-vector space of the same dimension. We have a natural $G$-action on the above scheme given by (on closed points) $$g\cdot f(w):=g(f(w))$$ Let us consider a closed point an the set of isomorphisms: $$\text{Isom}(W_{k'},V_{k'})\subset \text{Hom}(W_{k'},V_{k'})$$ All the isotropy groups are just the identity as can be trivially checked. But, is it every orbit closed? I am stuck with this question for days.