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$T:P_{3}\rightarrow P_{3}$ defined by $T(p(t))=tp'(t)+p(0)$ is a linear transformation.

Determine whether $T$ is invertible.

If yes, find $T^{-1}(q(t))$, where $q(t)$ is a polynomial of degree at most three.

One last thing... I have got that if $p(t)=at^3+bt^2+c^t+d$, then $T(p(t))=3at^3+2bt^2+c^t+d$. I understand that I can write the inverse of T as a matrix $[\frac{1}{3}, \frac{1}{2}, 1, 1]$, but how to put that into a nice form like the one $p(t)$ is in?

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If you're asking a follow-up, please link back and give credit to the person who already answered the original question. –  user61527 Oct 17 '13 at 6:42

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So if you let $q = \frac 13 at^3 + \frac 12 b t^2 + ct + d$ (the coefficients are just the inverses), what can you say about $T(q)$?

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Well $T(q)$ is just $p(t)$. The questions asks me to find $T^{-1}(q(t))$ though. –  666sys Oct 17 '13 at 6:51
    
But $T(q) = p \iff q = T^{-1}(p)$. So just rename $p$ as $q$ :) –  martini Oct 17 '13 at 6:52

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