Let $V$ be a vector space on the field $K$ and let $V^*$ be the dual space of $V$. For every bilinear form $<cdot,cdot>$ on $V$ we define a linear map

begin{equation}

L_{<cdot,cdot>}: V rightarrow V^* : v mapsto <cdot,v>

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^*) : <cdot,cdot> mapsto L_{<cdot,cdot>}

end{equation}

begin{equation}

psi: Hom_K(V,V^*) rightarrow B(V) : f mapsto <cdot,cdot>_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 $<cdot,cdot>$, resp. $f$, if we assume that dim$V = n < infty$.”

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

Thanks in advance!