Question

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 \(D_{e}=S D_{e} S^{-1}\).