Linear Algebra question regarding linear operators and matrix rep. relative to basis

gothloli

1. The problem statement, all variables and given/known data
Let w = a + bi be a complex number and let T : C -> C be defined by T(z) = w · z.
Considering C as a vector space over R, find the matrix B representing T relative to
the basis {1, i} of C.


2. Relevant equations



3. The attempt at a solution
I think you use eigenvalues and eigenvectors, if T is diagonalizable, but my main problem is finding standard basis of T since z is not defined, hence can you guide me there thanks, please help I have an exam tomorrow!.

Mark44

I don't see that eigenvalues or eigenvectors enter into this at all.
"standard basis of T" makes no sense to me. A basis is associated with a vector space, not a transformation.

You are given a basis for C; namely {1, i}. What is T(1)? What is T(i)?

gothloli

sorry I meant to say standard matrix of T not basis. Then can you tell me how to solve the question please, I have an exam tomorrow, I'm so confused, I just need help.

Dick

w=a+bi. As Mark44 suggested, if you find T(1) and T(i) then those will be the column vectors of the 2x2 matrix for T in the basis {1,i}. What are they? Express them in terms of the basis.

gothloli

okay so I get (a -b)
(b a)

thanks for the help, you made it clear for me.

gothloli

I get the matrix (a -b)
(b a)
I don't have time to find the matrix input on this thing.

Thanks for the help, you made it clear

Dick

I'm clear you've got it. That's what's important. Don't worry about the notation. I fudge it a lot myself. I'd express that as [[a,-b],[b,a]] and just hope people get it.