Homotopy of base spaces for vector bunles over $\mathbb S ^1$

Question

If i construct vector bundles over $\mathbb{S}^1$ in the following fashion:

Write $\mathbb{S}^1$ as the union of its upper and lower hemispheres $D_+,D_-$, with $D_+\cap D_-=\mathbb{S}^0$. Given a map $f:\mathbb{S}^0\mapsto GL_n(\mathbb{R})$, i can construct a vectorbundle $$E_f:= (D_+\times\mathbb R^n) \sqcup (D_-\times\mathbb R^n) / \sim$$ by identifying $(x,v) \in \partial D_-\times\mathbb R^n$ with $(x,f(x)(v)) \in \partial D_+\times\mathbb R^n$, and choosing the natural projection $E_f\mapsto \mathbb S^1$ asn the vectorbundle.

I want to prove, that if two maps $f,g:\mathbb{S}^0\mapsto GL_n(\mathbb{R})$ are homotopic, then $E_f \simeq E_g$.

Answer 1

Your question is answered in page 23 of Hatcher's book draft on vector bundles and K-theory, freely (and legally) available on the internet here.