Let $E \to M$ be a smooth rank $r$ real vector bundle, and let $\operatorname{Fr}(E)$ be the principal $GL_r(\mathbb R)$-bundle of ordered frames of $E$. There is a natural vector bundle isomorphism
paragraph
admin
\begin{align*}
\operatorname{Fr}(E) \times_{GL_r(\mathbb R)} \mathbb R^r \cong E,
\end{align*}
latex_env
admin
where $GL_r(\mathbb R)$ acts on $\mathbb R^r$ by the standard representation.