Tensor Product of Locally Free Sheaves

[Tsakanikas Nickos]

Let \( \left( X, \mathcal{O}_{X} \right) \) be a ringed space. If \( \mathscr{F} \) and \( \mathscr{G} \) are locally free sheaves on \( X \), then show that \( \mathscr{F} \otimes_{\mathcal{O}_{X}} \mathscr{G} \) is also a locally free sheaf on \( X \).