Matrix of the derivate on polynomials

Task number: 2568

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 \([D_x]_{AK}= \begin{pmatrix} 0 & 0 & 1 & 1 & 0 \\ 0 & 2 & 2 & 0 & 0 \\ 3 & 3 & 0 & 0 & 0 \\ 4 & 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} \).

Difficulty level: Easy task (using definitions and simple reasoning)
Solution require uncommon idea
Cs translation
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