The field $\displaystyle W=\left\{ \left[ \begin{matrix} a & -b \\ b & a \end{matrix} \right] : a,b\in R \right\}$ is also a vector space over $\displaystyle R$. Do not prove this fact. Prove that $\displaystyle W$ is isomorphic to the complex numbers $\displaystyle C$ as a real vector space.

So far, this is what I have. It's not much, but it's a start.

Let $\displaystyle \varphi :W\rightarrow C$

$\displaystyle \varphi \left( \left[ \begin{matrix} a & -b \\ b & a \end{matrix} \right] \right) =a+bi\in C$

I've been told that I'm to prove that this is one-to-one, onto, and linear.

That's about as much as I know. My brain's a little muddled at the moment due to a short bout of sickness, so I could use a hand.