iff criterion for n-dimensional vector bundle being trivial

Question

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.

Answer 1

Defn: Two bundles $p_1:E_1\to B$ and $p_2:E_2\to B$ are isomorphic iff we have some map $f:E_1\to E_2$ such that $p_1=p_2\circ f$ and for each $b\in B$, the map $f|_{p_1^{-1}(b)}:p_1^{-1}(b)\to p_2^{-1}(b)$ is an isomorphism of vector spaces.

If the bundle is trivial, by definition we have some isomorphism $f:E\to B\times \mathbb R^n$ such that $f|_{p^{-1}(b)}:p^{-1}(b)\to \mathbb R^n$ is an isomorphism of vector spaces, as desired.

If we have such a $\pi$, define $f:E\to B\times \mathbb R^n$ by $f(x)=(p(x),\pi(x))$. This is an isomorphism since if $p_0:B\times \mathbb R^n\to B$ is the standard projection, we have $p = p_0\circ f$ and for each $b\in B$, $f|_{p^{-1}(b)}=\pi(x)$ is an isomorphism.

Answer 2

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.