Prove that if a smooth manifold $M$ is contractible then every vector bundle over $M$ is trivial

Question

I've seen that this can be proved by using that if two functions are homotopic then the pullbacks of such functions are isomorphic, but the only "easy" proof of this I found is in Hatcher's vector bundles book and I don't find this proof very clear. This was left to me as a homework exercise and all we've seen of vector bundles are the basic definitions, constrictions by cocycles and that every vector bundle has a riemmanian metric so I don't know how to proceed with only this, any help would be appreciated

Answer 1

You might also try Hussemoller's book "Fiber Bundles", Chapter 2, Corollary 4.8.

Usually one proves, in the same breath, that if $f,g : X \to Y$ are two homotopic maps and if $B$ is a vector bundle over $Y$ then $f^*(B)$, $g^*(B)$ are isomorphic vector bundles over $X$. In Hussemoller's book this is is Chapter 2, Theorem 4.7.