What is the matrix for T in a complex number basis?

[mrroboto]

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.

 

[Dick]

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.

 

[mrroboto]

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

so Mat T =

[ 3 4
3 -4]

 

[Dick]

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.

 

[mrroboto]

So should the matrix be

[3 3
4 -4]

 

[matrixwarrior]

I think it should be
[ 3 -4
4 3 ]

 

[Dick]

I think it should be
[ 3 -4
4 3 ]

I agree.