Checking that $\Bbb C \cong \Bbb R^2$ as real vector spaces.

Question

Prove that $\Bbb C \cong \Bbb R^2$ as real vector spaces and use this to show that $\dim(\Bbb C) = 2$.

This is an isomorphism question and I know that I am supposed to construct a map from $\Bbb C$ to $\Bbb R^2$ and prove that the map is linear but I am not too sure on how to construct the map.

Answer 1

Map each x+iy to (x,y). This is a biyection. Since the (real vector space) operations in both C and R^2 are coordinatewise, you have then the linearity. As a consequence, x+i0 and 0+iy form a base of C as a real vector space.

Answer 2

Follow your nose! $$\Large{T: \Bbb R^2 \to \Bbb C, \qquad T((x,y)) = x+iy}$$