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

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

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

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

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

$\displaystyle f' = 2ax + b, ~ f'' = 2a$

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

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

which is sorta correct,except: The coords of $\displaystyle f$ 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?