iff criterion for n-dimensional vector bundle being trivial Prove that an $n$ dimensional vector bundle $p: E \to B$ is trivial iff there exists a continuous map $\pi: E \to \mathbb{R}^n$ whose restriction to each fiber $p^{-1}(b), b \in B,$ is a vector space isomorphism.
This is for my high school math independent study and my teacher is unable to help.

Morphisms $\pi:E\to \mathbb R^n$ of the type you describe  and trivializations $j:E\stackrel {\cong}{\to }  B\times \mathbb R^n$ of $E$ correspond to each other under the bijective correspondence $$   \pi :E\to \mathbb R^n\iff    j:E\stackrel {\cong}{\to }  B\times \mathbb R^n:e\mapsto (p(e),\pi(e))          $$  
Edit
The map $\pi$ is interesting because it is a particular case of a Gauss map, a map $E\to \mathbb R^N$ supposed  only to be a linear monomorphism on the fibres of $E$.
These Gauss maps permit one to classify vector bundles on a paracompact space $B$ by homotopy classes of continuous maps from $B$ into an infinite Grassmannian.
This is explained in Chapter 3 of Husemoller's Fibre Bundles.  
