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Math Help - complex numbers and vector spaces

  1. #1
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    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
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  2. #2
    A Plied Mathematician
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    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.
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  3. #3
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    p,q - constants in C.

    1) is p*z + q*w back in C^2?
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    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\}
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