# bilinear form – Show that the rank of \$phi\$ and \$psi\$ is equal to the rank of \$\$, resp. \$f\$

Let $$V$$ be a vector space on the field $$K$$ and let $$V^*$$ be the dual space of $$V$$. For every bilinear form $$$$ on $$V$$ we define a linear map
$$begin{equation} L_{}: V rightarrow V^* : v mapsto end{equation}$$

Let B(V) be the set of all bilinear forms on $$V$$ and consider the functions
$$begin{equation} phi: B(V) rightarrow Hom_K(V,V^*) : mapsto L_{} end{equation}$$
$$begin{equation} psi: Hom_K(V,V^*) rightarrow B(V) : f mapsto _f end{equation}$$

I have already shown that $$phi$$ and $$psi$$ are each other’s inverse en are thus bijections.

I am stuck on the next question: “Show that the rank of $$phi$$ and $$psi$$ is equal to the rank of $$$$, resp. $$f$$, if we assume that dim$$V = n < infty$$.”

I know that the rank of the bilinear form $$$$ is equal to the rank of a Grammian matrix of the bilinear form $$$$, but I couldn’t get any further.