Transition functions of trivial vector bundle

A bundle is trivial if and only if the cocycle of its transition functions becomes cohomologous to the trivial cocycle $1$ after suitable refinement of the covering on which the transition functions are defined. I don't think one gets any additional information from knowing that the trivial bundle $F$ in question is a subbundle of a larger (possibly nontrivial) bundle $E$. Also, it is not true in general that a basis of sections for a subbundle can be extended globally to a basis for a larger bundle.

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Differential Geometry

Vector Bundles

Algebraic Geometry

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