Complexification of a real vector space ($V_{\mathbb{C}}=V{\otimes }_{\mathbb{R}}\mathbb{C}$)

The complexification of a real vector space $V$ allows complex scalar coefficients while preserving its real linear relations. It can be constructed as the tensor product $V{\otimes }_{\mathbb{R}}\mathbb{C}$, or as formal sums $u+iv$ with $u,v\in V$. A real basis of $V$ becomes a complex basis of $V_{\mathbb{C}}$, so its complex dimension equals the original real dimension. Every real linear map extends uniquely by complex linearity. In particular a real representation extends by viewing its real matrices as complex matrices. Complexifying the realification of a complex representation produces the original complex representation together with its complex conjugate; the coordinate change $(u,v)\mapsto (u+iv,u-iv)$ exhibits the two blocks.