Exercise 3.C.2

Proof. Just differentiate any basis of $P_3(ℝ)$, take whatever is left as basis of $P_2(ℝ)$ (except the $0$ vector) and adjust the order. For example, differentiating the standard basis $1,x,x^2,x^3$ we get $0,1,2x,3x^2$. Therefore, the matrix of $D$ with respect to the basis $x^3,x^2,x,1$ of $P_3(ℝ)$ and the basis $3x^2,2x,1$ of $P_2(ℝ)$ is

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