Show Isomorphism Involving Dual Spaces Given finite-dimensional vector spaces $V$ and $W$ over af field $F$, show that there is an isomorphism $\phi: L(V,W) \rightarrow W \otimes_F V^*$ such that $\phi(STR)=(S \otimes R^*) \phi (T)$.
Here $L(V,W)$ denotes the space of linear maps from $V$ to $W$, $V^*$ denotes the dual space $L(V,F)$, and $R^*\in L(V^*,V^*)$ denotes the dual operator to R.
In order to show isomorphism, we have to show that $\phi$ is injective, surjective, and linear.
This is what I have so far:
Injectivity:
Let $v\in V$. And suppose $\phi(STR)(v)=0$.
Then $(S \otimes R^*) \phi (T)(v)=(S\phi (T)(v),R^*\phi (T)(v))=0$
I think my problem is that I'm not sure what to do with all the notations. Like from here, I'm already confused. And I'm not sure whether the above even makes since.
Thanks in advance!
 A: If I understand the problem correctly, a better way to phrase the statement of the exercise could be:
Given finite-dimensional vector spaces $V$ and $W$ over a field $F$, show that there exists an isomorphism $\phi \colon L(V,W) \to W \otimes_F V^*$ with the property that for every$R \in L(V,V)$ and $S \in L(W,W)$,
$$
\forall T \in L(V,W), \quad \phi(S \circ T \circ R) = (S \otimes R^*)(\phi(T)).
$$
In other words, for every $R \in L(V,V)$ and $S \in L(W,W)$ the following diagram commutes.
$$
\require{AMScd}
\begin{CD}
L(V,W) @>{\phi}>> W \otimes_F V^* \\
@V{S \,\circ\, (\_) \,\circ\, R}VV @VV{S \,\otimes\, R^*}V \\
L(V,W) @>>{\phi}> W \otimes_F V^*
\end{CD}
$$
Observe that the goal of the problem is to show the existence of an isomorphism.
First, let $n=\dim_F V$, and consider a basis $v_1,\dots,v_n$ for $V$.
Denote by $v_1^*,\dots,v_n^*$ to its dual basis.
Now, define a map $\phi \colon L(V,W) \to W \otimes_F V^*$ by the following rule:
$$
\forall T \in L(V,W), \quad \phi(T) = \sum_{i=1}^n T(v_i) \otimes v_i^*.
$$
I will leave to the reader to check that $\phi$ is an isomorphism (hint: define an inverse map using the universal property of the tensor product) and that the desired property is fullfilled.
If something about the notation or the exercise is still not clear, tell me. I hope this helps.
P.S. The assumption that $W$ is finite-dimensional is not necessary.
