You need to check vector addition and scalar multiplication. You're correct in that, for 2), the conjugation makes no difference. For 1), it might make a difference.
Is U a subspace of
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 . I don't know if that gives us important information or it's there to sidetrack us.
Can someone please give me some direction?