Transition functions of trivial vector bundle

Question

I have a question on trivial vector bundles. The question is as follows:

Can we characterize the transition functions of a trivial vector bundle in some way?

To be very concrete: suppose we have a vector bundle $E$, say of rank $3$, on an algebraic variety $X$ (or manifold if you prefer), and we look at trivial sub-bundles $0\to F\to E$ of rank $2$. What can we say about the transition functions of $F$ if we know those of $E$?

In this particular circumstance, we do have $3\times 3$ matrices at hand, and we want to produce $2\times 2$ matrices. The question is which ones are good.

At the beginning, I thought I could extend an argument of pure linear algebra: if we have an exact sequence of vector spaces $0\to V\to W\to W/V\to 0$, we can complete a basis $(e_1,e_2)$ of $V$ to a basis $(e_1,e_2,e_3)$ of $W$, so that $W\to W/V$ is represented by the row vector $e_3$ and $V\to W$ by the $3\times 2$ matrix containing $e_1,e_2$ written as columns. This didn't bring me anywhere, even because I can't figure how to use the triviality hypothesis.

Answer 1

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.

Answer 2

Vector bundle is trivial iff the transition functions have the form $g_{ij}=g_{i}g_{j}^{-1}$ where $g_i \in {\rm{GL}}(n,\mathcal{O}_{U_i})$.

Composite the trivilization with $g_i^{-1}$ we get another trivilization and the new transition functions are $g_{ij}\equiv{1}$.