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Math Help - Change of Basis Help

  1. #1
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    Change of Basis Help

    Consider the map T : P2 --> P3 such that T(f(x)) = (x^2)f'(x+1)
    Write the matrix representing this transformation with respect to the standard bases of P2 (1, x, x^2) and P3 (1, x, x^2, x^3).

    I really don't understand how to do this because when you plug in x^2 and x^2, you get an answer that's of a degree higher than 3.

    If you're feeling generous, could you help with these that I also don't get concerning this problem:
    a) Find a basis for the kernel of T and the image of T
    b) Is T injective? Surjective?
    c) Does T have a left inverse? Does it have a right inverse? Find either if it exists or prove that it does not exist.
    d) Find the matrix representing the same transformation with respect to the basis (x^2 - 1, x - 1, 1) of P2 and the standard basis of P3.
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  2. #2
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    Quote Originally Posted by letitbemww View Post
    Consider the map T : P2 --> P3 such that T(f(x)) = (x^2)f'(x+1)
    Write the matrix representing this transformation with respect to the standard bases of P2 (1, x, x^2) and P3 (1, x, x^2, x^3).

    I really don't understand how to do this because when you plug in x^2 and x^2, you get an answer that's of a degree higher than 3.


    No, you don't: T(x^2):=(x^2)(x^2+1)'=x^2(2x)=2x^3\in P_3

    For the rest of your questions give us some self work and say where did you get stuck

    Tonio



    If you're feeling generous, could you help with these that I also don't get concerning this problem:
    a) Find a basis for the kernel of T and the image of T
    b) Is T injective? Surjective?
    c) Does T have a left inverse? Does it have a right inverse? Find either if it exists or prove that it does not exist.
    d) Find the matrix representing the same transformation with respect to the basis (x^2 - 1, x - 1, 1) of P2 and the standard basis of P3.

    .
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  3. #3
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    We help those who help themselves. Now that you know that this really is a valid linear transformation, what are T(1), T(x), and T(x^2)? How would you write those in terms of the basis {1, x, x^2, x^3}?

    If you honestly don't know how to do any of (a) thorugh (d), you have more problems than we can help you with! Talk to your teacher about it.
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