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Basis - Complex Vector Space and Real Vector Space

  1. Feb 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Let {e1,...,en} be a basis for a complex vector space X. Find a basis for X regarded as a real vector space. What is the dimension of X in either case?


    2. Relevant equations



    3. The attempt at a solution
    I'm really not sure where to begin with this question.
    Are the ej's something like (i, 0, 0,...), (0,i,0,...)?
    If so, could we take {ie1,....,ien} as the basis for X regarded as a real vector space?

    Also, I think that the dimension in either case is n. Does anyone know if that is correct?

    Thank you very much.
     
  2. jcsd
  3. Feb 10, 2009 #2

    Office_Shredder

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    Consider the case when n=1 and try to generalize. Notice that over C:

    a*e1 = (x+iy)e1 = xe1 + yie1

    And this is zero if and only if x and y are, so we immediately get that e1 and ie1 are linearly independent.
     
  4. Feb 10, 2009 #3
    Thank you for your response.

    I definitely see why that is true, but I am confused as to what role that fact plays into this problem. Do you think you could explain it? Thanks so much.
     
  5. Feb 10, 2009 #4

    Dick

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    The point is that over C, e1 and i*e1 are not linearly independent. Over R, they are.
     
  6. Aug 24, 2010 #5
    A blast from the past:
    Sorry, I did a search, and got this post; thought it would be better to followup on
    it instead of doing a new post:


    Is this considered to be the canonical way of turning an n- complex vector space into
    a 2n-real vector space.?. I mean, there are many ways of getting a real basis once
    we are given a complex basis, but this one seems to be nice in that the original basis
    seems to be somehow "embedded" in the real basis {e1,ie1,...,en,ien}.

    Also: is there a construction that allows us to go in the opposite direction, i.e.,
    we are given an n-real vector space V_r , and we want to construct a complex
    vector space in which V_r is " embedded" , in the sense that if we were to forget/drop
    the complex part, we would get V_r back, i.e., if we took the basis {e1,ie1,..,en,ien}
    as above, and we ignored the vectors iej , to get the vector space V_r with basis:
    {e1,e2,...,en}. Is this the complexification of V_r.?

    Thanks.
     
  7. Aug 24, 2010 #6

    Dick

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    You are really just talking about choices of basis here. {e1,(1+i)e1...en,(1+i)en} is also a 2n vector real basis for the complex space. The 'real' part is still embedded in there. It's all pretty arbitrary, as far as I know.
     
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