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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 $\mathcal F$?

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