Real Versus Complex

Abstract

Let W be a vector space of dimension n over the complex number field $\mathbb{C}$. Then W can be considered a vector space over the real subfield $\mathbb{R} \subset \mathbb{C}$ as well. The resulting vector space over $\mathbb{R}$ is denoted by $W_{\mathbb{R}}$ and called the realification of the complex vector space W. For every basis e ₁, e ₂, …, e _n of W over $\mathbb{C}$, the vectors e ₁, e ₂, …, e _n , ie ₁, ie ₂, …, ie _n form a basis of $W_{\mathbb{R}}$ over $\mathbb{R}$, because for every w ∈ W, the uniqueness of the expansion

$$\displaystyle{w =\sum (x_{\nu } + i\,y_{\nu }) \cdot e_{\nu }\, ,\;\text{where}\;(x_{\nu } + i\,y_{\nu }) \in \mathbb{C}\, ,}$$

is equivalent to the uniqueness of the expansion

$$\displaystyle{w =\sum x_{\nu } \cdot e_{\nu } +\sum y_{\nu } \cdot ie_{\nu }\, ,\;\text{where}\;x_{\nu },\,y_{\nu } \in \mathbb{R}\,.}$$

Therefore, $\dim _{\mathbb{R}}W_{\mathbb{R}} = 2\dim _{\mathbb{C}}W$. Note that the realification of a complex vector space is always even-dimensional.