Joined: Mon Nov 09, 2015 1:52 pm Posts: 426

Let \(\displaystyle{A}\) be a nonempty set and \(\displaystyle{\mathcal{F}}\) be a nonempty collection
of \(\displaystyle{11}\) and onto functions \(\displaystyle{f:A\to \mathbb{R}^n}\) such that : if
\(\displaystyle{f\,,g\in\mathcal{F}}\) then \(\displaystyle{f\circ g^{1}:\mathbb{R}^n\to \mathbb{R}^n}\)
is \(\displaystyle{\mathbb{R}}\)  linear isomorphism.
Prove that the set \(\displaystyle{A}\) is a vector space (uniquely defined) and each \(\displaystyle{f\in\mathcal{F}}\)
is \(\displaystyle{\mathbb{R}}\)  linear isomorphism.

