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