Choice of trivializations in the definition of a vector bundle

Question

In the definition of vector bundle $E$ it is required that at every point $p$ there exists a local diffeomorphism $\phi_p$ from an open neighbourhood of p $U_p$ to $U_p \times V$. If we now take any linear invertible function $A :V \rightarrow V$, we can define $\phi'_p = A \circ\phi_p$ (with $A$ just acting on the fiber I hope the notation is clear) which should still be a diffeomorphism. Is $\phi'_p$ still an admissible diffeomorphism? Do we change something important considering in the definition $\phi'_p$ instead of $\phi_p$? Of course the structure functions would change but is the vector bundle itself changing? Told another way, does the choice of the local diffeomorphisms have an impact on the properties of the vector bundle?

Answer 1

Basically you don't change the isomorphic class of your vector bundle. If $p:P\rightarrow M$ is a vector bundle, $(U_i,\phi_i)$ is a trivialization, $\phi:p^{-1}(U_i)\rightarrow U_i\times V$, $\phi_i\circ {\phi_j}^{-1}(x,y)=(x,g_{ij}(y)$ is a $1$-cocycle: $g_{ij}g_{jk}=g_{ik}$. The isomorphic class of the vector bundle is defined by this cocycle. if you define $\psi_i=(Id,A_i)\circ\phi_i$ you modify the 1-cocycle by a boundary and the resulting bundle is isomorphic to the first one. This is explained in every introductory book on the subject.