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Linear Transformations using polynomials

  1. May 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Let P3 be the space of all polynomials (with real coefficients) of degree at most 3. Let
    D : P3 -> P3 be the linear transformation given by taking the derivative of a polynomial.
    That is
    D(a + bx + cx2 + dx3) = b + 2cx + 3dx2:
    Let B be the standard basis {1; x; x2; x3} of P3.
    (a) Find the matrix MD of D with respect to the standard basis.
    (b) Explain, without doing any matrix calculations, why (MD)4 = 0.

    2. Relevant equations



    3. The attempt at a solution
    i know it may be a simple question but i dont even know where to begin
    but in a attempt is
    MD = [1 1 1 1;
    0 2 3 0]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 5, 2009 #2

    tiny-tim

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    Welcome to PF!

    Hi mbud! Welcome to PF! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    To find M, you only need the effect of M on the four basis elements. Sooo …

    Hint: what is x2 as a vector? what is Dx2 as a vector? :wink:
     
  4. May 5, 2009 #3

    HallsofIvy

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    More specifically, since the standard basis is 1, x, x2, and x3, take the derivative of each and write it in terms of those. The coefficients give each column of the matrix.
     
  5. May 5, 2009 #4
    Thanks, for the help, but i still dont know how to explain (MD)^4 = 0
    I can show it using matrices, but when it comes to words, im flabbergasted.
     
  6. May 5, 2009 #5

    tiny-tim

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    Hint: if V is the vector form of a polynomial ax3 + bx2 + cx + d, what is (MD)4V the vector of? :wink:
     
  7. May 5, 2009 #6

    matt grime

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    Differentiation does what to the degree of a polynomial? This is nothing to do with vectors, or vector spaces, or matrices - sometimes you just have to look at things and think for a little bit.
     
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