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Math Help - [SOLVED] Matrix of a lin. transform

  1. #1
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    [SOLVED] Matrix of a lin. transform

    Let T : P_2(R) \to P_2(R) be the linear transform defined by T(f)=2f'' +3f'-f.

    Let E and F be to bases: E=\{1, x, x^2\}, ~F=\{1+x,1+x+x^2,1-2x+x^2\}.

    Determine the matrix representation, C, of T relative to E.
    -----

    My approach (don't know if it's correct):

    Let f = ax^2+bx+c \in P_2.

    f' = 2ax + b, ~ f'' = 2a

    \therefore T(f) = (-a)x^2+(6a-b)x+(4a+3b-c)

    \therefore C = \begin{bmatrix}-1&0&0\\6&-1&0\\4&3&-1\end{bmatrix}

    which is sorta correct, except: The coords of f relative to E are (c,b,a) not (a,b,c)! So if I'm given a vector X relative to E, CX won't be correct.

    What should I have done?
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  2. #2
    Senior Member
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    ah, nevermind, I realised C should be:

    C = \begin{bmatrix}-1&3&4\\0&-1&6\\0&0&-1\end{bmatrix}

    since T(c,b,a)=(-c + 3b + 4a, -b + 6a, -a)
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