dual homomorphism of the derivative

Let ${𝒫}_n$ denote the vector space of realpolynomials of degree $n$ or less, and let ${\mathrm{D}}_n:{𝒫}_n\to {𝒫}_{n-1}$ denote the ordinary derivative. Linear forms on ${𝒫}_n$can be given in terms of evaluations, and so we introduce thefollowing notation. For every scalar $k\in ℝ$, let ${\mathrm{Ev}}_k^{(n)}\in {({𝒫}_n)}^{*}$ denote the evaluation functional

$${\mathrm{Ev}}_k^{(n)}:p\mapsto p(k),p\in {𝒫}_n.$$

Note: the degree superscript matters! For example:

$${\mathrm{Ev}}_2^{(1)}=2{\mathrm{Ev}}_1^{(1)}-{\mathrm{Ev}}_0^{(1)},$$

whereas ${\mathrm{Ev}}_0^{(2)},{\mathrm{Ev}}_1^{(2)},{\mathrm{Ev}}_2^{(2)}$ arelinearly independent. Let us consider the dual homomorphism${\mathrm{D}}_2^{*}$, i.e. the adjoint of ${\mathrm{D}}_2$. We have the followingrelations:

In other words, taking ${\mathrm{Ev}}_0^{(1)},{\mathrm{Ev}}_1^{(1)}$ as the basis of${({𝒫}_1)}^{*}$ and ${\mathrm{Ev}}_0^{(2)},{\mathrm{Ev}}_1^{(2)},{\mathrm{Ev}}_2^{(2)}$ as the basis of${({𝒫}_2)}^{*}$, the matrix that represents ${\mathrm{D}}_2^{*}$ is just

Note the contravariant relationship between ${\mathrm{D}}_2$ and ${\mathrm{D}}_2^{*}$. Theformer turns second degree polynomials into first degree polynomials,where as the latter turns first degree evaluations into second degreeevaluations. The matrix of ${\mathrm{D}}_2^{*}$ has 2 columns and 3 rowsprecisely because ${\mathrm{D}}_2^{*}$ is a homomorphism from a 2-dimensionalvector space to a 3-dimensional vector space.

By contrast, ${\mathrm{D}}_2$ will be represented by a $2\times 3$ matrix. Thedual basis of ${𝒫}_1$ is

$$-x+1,x$$

and the dual basis of ${𝒫}_2$ is

$$\frac{1}{2}(x-1)(x-2),x(2-x),\frac{1}{2}x(x-1).$$

Relative to these bases, ${\mathrm{D}}_2$ is represented by the transpose of thematrix for ${\mathrm{D}}_2^{*}$, namely

This corresponds to the following three relations: