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Let $\mathcal{P}_{3}$ be the space of polynomials of degree at most 3 anD let $D: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}$ be the derivative operator.
a) Using the basis $e_{1}=1, e_{2}=x, e_{3}=x^{2}, \epsilon_{4}=x^{3}$ find the matrix $D_{e}$ representing D.
b) Using the basis $\epsilon_{1}=x^{3}, \epsilon_{2}=x^{2}, \epsilon_{3}=x, \epsilon_{4}=1$ find the matrix $D_{e}$ representing D.
c) Show that the matrices $D_{e}$ and $D_{e}$ are similar by finding an invertible map $S$ : $\mathcal{P}_{3} \rightarrow \mathcal{P}_{3}$ with the property that $D_{e}=S D_{e} S^{-1}$.
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