Different basis over $\Bbb{R}$ and $\Bbb{C}$ V is a finite dimensional vector space over $\Bbb{C}$ and 
{v$_1$,...,v$_n$}
be a basis of V. 
Show {v$_1$,iv$_1$,...,v$_n$,iv$_n$} is a basis of V over $\Bbb{R}$ and conclude:
dim$_{\Bbb{R}}$V=2dim$_{\Bbb{C}}$V.
I have proved this is true for the case V = $\Bbb{C}^2$ using e1, ie1, e2 and ie2. How can I extend this to a general V?

Let $\alpha_1,\beta_1,\alpha_2,\beta_2,\ldots,\alpha_n,\beta_n\in\Bbb R$ such that
$$\alpha_1 v_1+\beta_1 i v_1+\cdots+\alpha_n+\beta_n i v_n=0$$
so with $z_i=\alpha_i+i\beta_i$ we have
$$z_1v_1+\cdots+z_n v_n=0\Rightarrow z_i=0 \;\forall i$$
since $(v_1,\ldots,v_n)$ are linearly independant, moreover, it's clear that $(v_1,iv_1,\ldots,v_n,iv_n)$ spans the linear space $V$ hence we have the desired result.
