Let $M$ be a smooth manifold and let $r \in \mathbb{N}$. Let $\mathrm{Vect}_r(M)$ be the category whose objects are smooth real vector bundles $\pi_E:E \to M$ of rank $r$ and whose morphisms are smooth vector bundle isomorphisms over $\operatorname{id}_M$. Let $\mathrm{Prin}_{GL_r(\mathbb{R})}(M)$ be the category whose objects are smooth principal right $GL_r(\mathbb{R})$-bundles $\pi_P:P \to M$ and whose morphisms are smooth principal bundle isomorphisms over $\operatorname{id}_M$.

For a rank-$r$ vector bundle $E \to M$, let $\operatorname{Fr}(E) \to M$ denote its linear frame bundle, whose fiber over $p \in M$ is

\begin{align*} \operatorname{Fr}(E)_p=\{\nu:\mathbb{R}^r \to E_p \mid \nu \text{ is a linear isomorphism}\}, \end{align*}

with right action $\nu \cdot g := \nu \circ g$ for $g \in GL_r(\mathbb{R})$. Then the assignment

\begin{align*} E \mapsto \operatorname{Fr}(E) \end{align*}

extends to an equivalence of categories

\begin{align*} \operatorname{Fr}: \mathrm{Vect}_r(M) \to \mathrm{Prin}_{GL_r(\mathbb{R})}(M). \end{align*}

A quasi-inverse sends a principal right $GL_r(\mathbb{R})$-bundle $P \to M$ to the associated vector bundle

\begin{align*} P \times_{GL_r(\mathbb{R})} \mathbb{R}^r \to M, \end{align*}

formed using the standard left action of $GL_r(\mathbb{R})$ on $\mathbb{R}^r$.