Let $\cP{n}$ denote the vector space of real polynomials of degree $n$ or less, and let $\D{n}:\cP{n}\rightarrow \cP{n-1}$ denote the ordinary derivative. Linear forms on $\cP{n}$ can be given in terms of evaluations, and so we introduce the following notation. For every scalar $k\in\reals$ , let $\ev{n}_k\in (\cP{n})\dual$ denote the evaluation functional $$\ev{n}_k:p\mapsto p(k),\quad p\in \cP{n}.$$ Note: the degree superscript matters! For example: $$\ev1_2 = 2\, \ev1_1- \ev1_0,$$ whereas $\ev2_0, \ev2_1, \ev2_2$ are linearly independent. Let us consider the dual homomorphism $\D2\dual$ , i.e. the adjoint of $\D2$ . We have the following relations:

By contrast, $\D2$ will be represented by a $2\times 3$ matrix. The dual basis of $\cP1$ is $$-x+1,\quad x$$ and the dual basis of $\cP2$ is $$\frac{1}{2} (x-1)(x-2),\quad x(2-x),\quad \frac{1}{2} x(x-1).$$ Relative to these bases, $\D2$ is represented by the transpose of the matrix for $\D2\dual$ , namely $$ \begin{pmatrix} -\frac{3}{2} & 2 & - \frac{1}{2} \\ - \frac{1}{2} & 0 & \frac{1}{2} \end{pmatrix} $$ This corresponds to the following three relations: