Real vector space/Complexification/Tensor product/Example

Let ${}V$ be a real vector space. The tensorization with the ${}\mathbb {R}$-algebra ${}\mathbb {C}$, that is,

${}V_{\mathbb {C} }:=\mathbb {C} \otimes _{\mathbb {R} }V\,,$

is called the complexification of ${}V$. If ${}V$ has the dimension ${}n$, then ${}V_{\mathbb {C} }$ has, considered as a complex vector space, also the dimension ${}n$. If we consider ${}V_{\mathbb {C} }$ as a real vector space, then it has the real dimension ${}2n$.