Complexification as $\mathbb{C}{\otimes }_{\mathbb{R}}V$ with its canonical real-linear embedding

Let $V$ be a real vector space (Vector space over a field) and regard $\mathbb{C}$ as a real vector space through the constant-class map of The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$. The complexification of $V$ is the real tensor product

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

It becomes a complex vector space through the scalar action

$z\cdot (w\otimes v):=(zw)\otimes v(z,w\in \mathbb{C},v\in V),$

and it carries the canonical real-linear embedding

$\iota :V⟶V_{\mathbb{C}},\iota v:=1\otimes v\mathrm{.}$

The scalar action is well defined. For fixed $z\in \mathbb{C}$ the map $(w,v)\mapsto (zw)\otimes v$ from $\mathbb{C}\times V$ to $V_{\mathbb{C}}$ is additive in each variable and $\mathbb{R}$-balanced, so by Universal property of the tensor product for balanced maps into abelian groups it induces a unique $\mathbb{R}$-linear map ${\mu }_z:V_{\mathbb{C}}\to V_{\mathbb{C}}$ with ${\mu }_z(w\otimes v)=(zw)\otimes v$. The identities ${\mu }_{z+z^{\mathrm{\prime }}}={\mu }_z+{\mu }_{z^{\mathrm{\prime }}}$, ${\mu }_{z{z}^{\mathrm{\prime }}}={\mu }_z\circ {\mu }_{z^{\mathrm{\prime }}}$ and ${\mu }_1=\mathrm{id}$ hold on every elementary tensor and hence everywhere, so the action makes $V_{\mathbb{C}}$ a complex vector space.

The tensor product is over $\mathbb{R}$, and the construction is basis-independent. Every element of $V_{\mathbb{C}}$ is a finite sum ${\sum }_j{z}_j\otimes v_j$ with $z_j\in \mathbb{C}$ and $v_j\in V$; after writing $z_j=a_j+i{b}_j$ each summand is $a_j(1\otimes v_j)+b_j(i\otimes v_j)$.