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Homework Help: Linear transformation from given matrices - bases unknown

  1. Jul 21, 2009 #1
    1. The problem statement, all variables and given/known data

    Find a linear transformation

    [tex]T: P_2 \rightarrow M_{22}[/tex]

    such that




    3. The attempt at a solution

    In all the examples I have access to the linear transformation is already defined, the bases given and the question is to find the matrix (easy enough).

    For this question, I could've reversed this approach had I known what the bases were. However, the bases aren't given and I'm stumped.

    Could someone offer me a hint here please?

  2. jcsd
  3. Jul 21, 2009 #2


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    T(ax^2+bx+c)=aT(x^2)+bT(x)+cT(1). If you knew what T(x^2), T(x) and T(1) were it would be easy to write down a formula for T acting on a general element of P2. Can you find them from the given information?
  4. Jul 22, 2009 #3
    Hi Dick!

    I think I know where you're going, but not sure how to get there. Let's see if I understand what you're saying though and give it a shot:

    From what you've given me

    I thought that we could then say




    However, I think we'll still be acting legit when swapping rows 1 and 2 of the second equation so that




    If we then subtract equation 3 from 1, we get



    which the same as equation 2 except for the minus sign. If all of what I've done so far is still ok and if my reasoning still holds, this means that




    Having done all of this I have to admit that
    1. I don't know if what I've done is mathematically acceptable and
    2. if everything up until this point is ok, what do I do next?

    Thanks for the help!
  5. Jul 22, 2009 #4


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    It's perfectly mathematically acceptable. Now just add things up T(ax^2+bx+c)=aT(x^2)+bT(x)+c=?? What's the resulting matrix?
  6. Jul 29, 2009 #5
    Hi Dick!

    Sorry about the slow response time, but I've been rather busy this past week.

    I believe our final answer is:




    But I'm not so sure about the notation, do we write it as


  7. Jul 29, 2009 #6


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    How about just writing T(ax^2+bx+c)? I think that's a little clearer.
  8. Jul 29, 2009 #7
    Ok cool. Once again thank you for your help! :smile:
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