Suppose $W$ is a complex vector space. Let $\overline{W}$ denote the complex vector space $W$, but with scalar multiplication replaced by $(z,w)\mapsto \bar{z}\cdot w$.
I want to show that $(W_\mathbb{R})^\mathbb{C}$ is isomorphic (as complex vector spaces) to the external direct product $W\boxplus\overline{W}$ without resorting to dimension arguments. My text isn't clear, but I think $(W_\mathbb{R})^\mathbb{C}$ denotes the complexification of $W$ when viewed as a real vector space.
I tried to define a map $f\colon (W_\mathbb{R})^\Bbb{C}\to W\boxplus\overline{W}$ by $u+vi\mapsto (u,v)$. So $f$ is a bijection, and an additive homomorphism. I tried showing it respects scalar multiplication like this.
I get $ f((a+bi)(u+vi))=f(au-bv+(bu+av)i)=(au-bv,bu+av) $ and $ (a+bi)f(u+vi)=(a+bi)(u,v)=((a+bi)u,(a-bi)v) $ so I think my map is wrong. Is there an explicit isomorphism?
By the way, this is problem 2.33(b) of Roman's Advanced Linear Algebra.