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If a map $f$ from the set of polynomials of degree 3 to the real numbers is given by $f(u) = u'(-2)$, how do I find the matrix that represents $f$ with respect to the bases $[1, t, t^2, t^3]$ and $[1]$?

I have worked out $f(1) = 0$, $f(t) = 1$, $f(t^2) = -4$ and $f(t^3) = 12$ and now I'm at a loss.
Any help appreciated.

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Actually you are done already: $$f \equiv \pmatrix{0&1&-4&12}$$

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