$\displaystyle U = \{(z,w) \in C^2 | 2z = 3\bar{w}\}$

Is U a subspace of $\displaystyle C^2$

1) over C

2) over R

I'm not sure why it wouldn't be a subspace over R. It has the zero vector. I'm always confused by these questions because of the things like $\displaystyle \bar{w}$. I don't know if that gives us important information or it's there to sidetrack us.

Can someone please give me some direction?

Thanks