Let V be vector space with dimension of $\displaystyle n $ above C (complex plane). Now we look at him (V) like vector space above R. What his dimension above R?
Yes, since we can "construct" every element in $\displaystyle \mathbb{C}$ using 2 elements of $\displaystyle \mathbb{R}$, we may conclude that the dimension of a vector space spanned over $\displaystyle \mathbb{R}$ is twice than of the vector space spanned over $\displaystyle \mathbb{C}$.