# Thread: complex numbers and vector spaces

1. ## complex numbers and vector spaces

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

Is U a subspace of $C^2$
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 $\bar{w}$. 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

2. 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.

3. p,q - constants in C.

1) is p*z + q*w back in $C^2$?

4. Originally Posted by jayshizwiz
$U = \{(z,w) \in C^2 | 2z = 3\bar{w}\}$

Is U a subspace of $C^2$
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 $\bar{w}$. 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

First, simplify!

$\{(z,w) \in C^2 | 2z = 3\bar{w}\}=\{(z,w) \in C^2 | 2(x+iy) = 3\bar{a+ib}\}=\{(z,w) \in C^2 | 2x+2iy = 3a-3ib\}=\{(x,y,a,b) \in R^2 | 2x=3a \& 2y=-3b\}$