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

  1. Dec 12, 2012 #1
    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!.
     
  2. jcsd
  3. Dec 12, 2012 #2

    Mark44

    Staff: Mentor

    Re: Linear Algebra question regarding linear operators and matrix rep. relative to ba

    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)?
     
  4. Dec 12, 2012 #3
    Re: Linear Algebra question regarding linear operators and matrix rep. relative to ba

    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.
     
  5. Dec 12, 2012 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Linear Algebra question regarding linear operators and matrix rep. relative to ba

    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.
     
  6. Dec 12, 2012 #5
    Re: Linear Algebra question regarding linear operators and matrix rep. relative to ba

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

    thanks for the help, you made it clear for me.
     
  7. Dec 12, 2012 #6
    Re: Linear Algebra question regarding linear operators and matrix rep. relative to ba

    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
     
  8. Dec 12, 2012 #7

    Dick

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    Science Advisor
    Homework Helper

    Re: Linear Algebra question regarding linear operators and matrix rep. relative to ba

    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.
     
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