Let $V,W$ be finite-dimensional vector spaces over the same field $\mathbb{F}$ and let $L(V,W)$ be the vector space of $\mathbb{F}$-linear transformations from $V$ to $W$.
Prove that the mapping $\psi : L(V,W) → L(W^*, V^*)$ given by $\psi(T) = T^t$ is an isomorphism.
notation and definition:
$V^*$ is the dual space of $V$.
$T^t$ is the transpose of $T$.
Suppose $\psi(T)=0=T^t$. Then $T=(T^t)^t=(0)^t=0$, so that ker$(\psi)={0}$.
ok, it needs to be mentioned that the function $\psi$, in the way you have defined it, is actually acting on the matrix representations of the linear transformations, and not the linear transformations themselves, so supposing dim$(V)=n$ and dim$(W)=m$ we actually have a composition of isomorphisms $\phi_1 \circ \psi \circ \phi_2^{-1}$, where $\phi:L(V,W) \rightarrow M_{m \times n}(\mathbb{F})$ sends each linear transformation to its matrix representation and similarly $\phi_2 :L(W^*,V^*) \rightarrow M_{n \times m}(\mathbb{F})$ sends each linear transformation between the dual spaces to its matrix representation.
Anyway, through the composition of isomorphisms you have an isomorphism between $L(V,W)$ and $L(W^*,V^*)$ as required.