Matrix (Complex Numbers)

  • Thread starter mrroboto
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  • #1
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Homework Statement



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}

Homework Equations



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

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.
 

Answers and Replies

  • #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.
 
  • #3
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so T(1) = 3+4i
T(i) = 3i+4i^2 = 3i-4

so Mat T =

[ 3 4
3 -4]
 
  • #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.
 
  • #5
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So should the matrix be

[3 3
4 -4]
 
  • #6
I think it should be
[ 3 -4
4 3 ]
 
  • #7
Dick
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