manifold structure on on a finite dimensional real vector space

I am reading Warner's Differentiable Manifolds I do not get one example which is

Let $$V$$ be a finite dimensional real vector space. Then $$V$$ has a natural manifold structure. If $$\{e_i\}$$ is a basis then the elements of the dual basis $$\{r_i\}$$ are the coordinate functions of a global coordinate system on $$V$$.

I don't understand how "the elements of the dual basis $$\{r_i\}$$ are the coordinate functions of a global coordinate system on $$V$$." Could any one explain me about that? Then how such a global coordinate system uniquely determines a differentiable structure on $$V$$? And why this structure is indipendent of choice of basis?

First of all for a manifold structure I need each point must have an open neighborhood $$U$$ homeomorphic to some open subset of $$\mathbb{R}^n$$. Here am I getting such notions?

The space $\mathbb{R}^n$ has coordinate functions $x_j:\mathbb{R}^n\to\mathbb{R}$, projection onto the $j^{th}$ axis. If $(\phi,U)$ is a coordinate system on a manifold $M$, then we get coordinate functions on $U$ by composing $\phi$ with the $x_j$.
Warner is just saying that by choosing a basis on a real vector space $V$, you induce a bjiective linear map (hence homeomorphism) $A$ between $V$ and $\mathbb{R}^n$, and that homeomorphism is a global coordinate system with coordinate functions $x_j\circ A = r_j$. The open neighborhood about each point is the entire space $V$.
• Thank you Neal, But could you tell me how a nbd of a point is the whole vector space $V$? – Marso Jun 19 '12 at 13:04