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Matrix (Complex Numbers)

  1. Nov 28, 2007 #1
    1. The problem statement, all variables and given/known data

    The set of complex numbers C is a vector space over R. Note that {1, i} is the basis for C as a real vector space. Define:

    T(z) = (3+4i)z

    What is the matrix for T in the basis {1,i}

    2. Relevant equations

    Dimension of the matrix (n,m) = n x m

    3. The attempt at a solution

    I know the dimension of this matrix is 1 x i = i. But I don't know where to go from here. We haven't learned matrices for complex numbers, and I'm very confused by the concept of having something as i-dimensional.
     
  2. jcsd
  3. Nov 28, 2007 #2

    Dick

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    You should be confused about something being 'i dimensional'. The good news is that it is not. It's TWO dimensional. There are TWO 'vectors' in the basis, 1 and i. Split T(1) and T(i) into real and imaginary parts. Their coefficients are the columns of your matrix. Note the matrix of T is REAL.
     
  4. Nov 28, 2007 #3
    so T(1) = 3+4i
    T(i) = 3i+4i^2 = 3i-4

    so Mat T =

    [ 3 4
    3 -4]
     
  5. Nov 28, 2007 #4

    Dick

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    Try it out. 1=(1,0) and i=(0,1) (column vectors). If you do that you should realize that you should put (3,4) and (3,-4) into the columns, not the rows.
     
  6. Nov 28, 2007 #5
    So should the matrix be

    [3 3
    4 -4]
     
  7. Nov 14, 2008 #6
    I think it should be
    [ 3 -4
    4 3 ]
     
  8. Nov 14, 2008 #7

    Dick

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    I agree.
     
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