I'm self-studying the book by Bott & Tu "Differential forms in algebraic topology" and I'm having problems with exercise 6.2.
It says "Show that two vector bundles on $M$ are isomorphic iff their cocycles relative to some open cover are equivalent".
I have no problems in proving that if the cocycles are equivalent wrt some open cover, then the two bundles are isomorphic, and I can also prove the other implication assuming that the induced map on M is the identity. But I don't know how to prove it for a general isomorphism $f: E \rightarrow E'$.
Any hints will be welcome. Thanks in advance.
If $\{(U_\alpha,\phi_\alpha)\}$ is a trivialization of $E$ and $f:E\rightarrow E'$ is a vector bundle isomorphism, show that $\{(U_\alpha,\psi_\alpha)\}$ is a trivialization of $E'$, where $\psi_\alpha = \phi_\alpha\circ f^{-1}$. Now show that the cocycles have to be the same.