Matrix of the derivate on polynomials

Let the space of polynomials of degree at most 4 over $$\mathbb R$$ be equipped with basis $$A=(x^4+x^3,\ x^3+x^2,\ x^2+x,\ x+1,\ x^4+1)$$. Determine the matrix $$[D_x]_{AK}$$ for the mapping $$D_x$$ that assigns $$f(x)$$ its derivative $$f'(x)$$.

(Consider $$K=(x^0,…,x^4)$$ as the canonical basis.)

• Resolution

It holds that $$[D_x]_{AK}=[D_x]_{KK}[id]_{AK}$$, (i.e. we first change the basis and then we derive), where

$$[id]_{AK}= \begin{pmatrix} 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ \end{pmatrix}$$ and $$[D_x]_{KK}= \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix}$$.

• Result

The desired matrix is $$[D_x]_{AK}= \begin{pmatrix} 0 & 0 & 1 & 1 & 0 \\ 0 & 2 & 2 & 0 & 0 \\ 3 & 3 & 0 & 0 & 0 \\ 4 & 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix}$$.