Find Matrix B for Complex Number T(z)

[gothloli]

Homework Statement

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.

Homework Equations

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]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I think you use eigenvalues and eigenvectors, if T is diagonalizable

I don't see that eigenvalues or eigenvectors enter into this at all.

, but my main problem is finding standard basis of T

"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)?

since z is not defined, hence can you guide me there thanks, please help I have an exam tomorrow!.

 

[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]

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.

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]

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.

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

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

 

[gothloli]

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.

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 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

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.