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We know a vector bundle as a manifold $E$ over a manifold $M$ is defined using transition maps over a collection of open sets each giving us a trivialization. Since $M$ is a manifold, it has an atlas. Can we intersect the two collections of open sets and obtain a chart for $E$ which also gives us the trivialization?

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Yep! Given an open set $U$ such that $E$ is trivial over $U$ and an open set $V$ that is diffeomorphic to an open subset of $\mathbb{R}^n$, $U\cap V$ is an open set which is both diffeomorphic to an open subset of $\mathbb{R}^n$ and on which $E$ is trivial. Letting $U$ and $V$ range over open covers of $M$, you get an open cover of $M$ by sets of this form. So you can always find an atlas for $M$ such that $E$ is trivial on each set in the atlas.

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