Exercise 17

Consider the space of polynomials $\mathcal{P}_{n}$.

• Show that for each $i$, the $i$-th derivative operator $d^{i}/dx^{i}$ is a linear transformation of that space.

• Conclude that for any scalars $c_{k},\ldots,c_{0}$ this map is a linear transformation of that space.

  $$f\;\mapsto\;c_{k}\frac{d^{k}}{dx^{k}}f+c_{k-1}\frac{d^{k-1}}{dx^{k-1}}f+\dots+c_{1}\frac{d}{dx}f+c_{0}f$$