Suppose $(X,\mathcal O_X)$ is a locally ringed space. Let $\mathcal F$ be a locally free $\mathcal O_X$-module of some fixed rank $n \in \mathbb N$.
Then the dual $\mathcal F^{\lor} := \mathcal{Hom}_{\mathcal O_X}(\mathcal F, \mathcal O_X)$ is locally free of rank $n$ as well.
In the case $n=1$ one has an isomorphism $\mathcal F \otimes \mathcal F^{\lor} \cong \mathcal O_X$, which is used to show that the Picard group $\mathrm{Pic}(X)$ of $X$ really has the structure of a group.
Is it also more generally true that $\mathcal F \otimes \mathcal F^{\lor} \cong \mathcal O_X^{\oplus n^2}$ for arbitrary $n \in \mathbb N$?
Even when $F$ is a sum of line bundles, say $F = \oplus L_i$, one has $$ F \otimes F^\vee \cong \oplus (L_i \otimes L_j^\vee), $$ and there non-trivial summands in the right side unless all the summands in $F$ are isomorphic.