Vector space
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Vector space
Let \(\displaystyle{A}\) be a non-empty set and \(\displaystyle{\mathcal{F}}\) be a non-empty collection
of \(\displaystyle{1-1}\) 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.
of \(\displaystyle{1-1}\) 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.
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