Worksheet 20

1. Recall that $\operatorname{Poly}_n(\mathbb{R})$ is the vector space of real polynomials with degrees $\leq n$. Can you find matrix representation for \(\frac{d}{dx} : \operatorname{Poly}_n(\mathbb{R}) \to \operatorname{Poly}_n(\mathbb{R})?\)

2. Is differentiation a non-singular linear transformation? More precisely, is \(\frac{d}{dx} : \operatorname{Poly}_n(\mathbb{R}) \to \operatorname{Poly}_n(\mathbb{R})\) singular or non-singular?

3. Is integration a non-singular linear transformation? Recall that $\operatorname{Poly}(\mathbb{R})$ be the vector space of all real polynomials. Is \(\int_0^y (-) \ dx : \operatorname{Poly}(\mathbb{R}) \to \operatorname{Poly}(\mathbb{R})\) singular or non-singular?