# Transition functions of the dual sheaf

Let $$(X, \mathcal{O})$$ be a ringed space and $$\mathcal{F}$$ an $$\mathcal O$$-module on $$X$$, which furthermore is assumed to be locally free of some finite rank $$n \in \mathbb N$$.

Then the dual sheaf $$\mathcal{F}^{\vee} := \mathcal{Hom}_{\mathcal O}(\mathcal F, \mathcal O)$$ is locally free of rank $$n$$ again.

Given two open subsets $$U_i, U_j \subseteq X$$ with intersection $$V$$ trivializing $$\mathcal F$$, one obtains a transition function $$T_{ij} \in \mathrm{Aut}_{\Gamma(V, \mathcal O)}(\Gamma(V,\mathcal F))$$.

It is easy to check that $$\mathcal F^\vee$$ is trivial on $$U_i, U_j$$ and hence $$V$$ as well.

What is the relation between the transition functions $$T_{ij}^\vee$$ of $$\mathcal F^\vee$$ and $$T_{ij}$$ of $$\mathcal F$$?