Exercise 10

Represent each linear map with respect to each pair of bases.

• $d/dx\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$ with respect to $B,B$ where $B=\langle 1,x,\dots,x^{n}\rangle$, given by

  $$a_{0}+a_{1}x+a_{2}x^{2}+\dots+a_{n}x^{n}\mapsto a_{1}+2a_{2}x+\dots+na_{n}x^{n-1}$$

• $\int\colon\mathcal{P}_{n}\to\mathcal{P}_{n+1}$ with respect to $B_{n},B_{n+1}$ where $B_{i}=\langle 1,x,\dots,x^{i}\rangle$, given by

  $$a_{0}+a_{1}x+a_{2}x^{2}+\dots+a_{n}x^{n}\mapsto a_{0}x+\frac{a_{1}}{2}x^{2}+\dots+\frac{a_{n}}{n+1}x^{n+1}$$

• $\int^{1}_{0}\colon\mathcal{P}_{n}\to\mathbb{R}$ with respect to $B,{\cal E}_{1}$ where $B=\langle 1,x,\dots,x^{n}\rangle$ and ${\cal E}_{1}=\langle 1\rangle$, given by

  $$a_{0}+a_{1}x+a_{2}x^{2}+\dots+a_{n}x^{n}\mapsto a_{0}+\frac{a_{1}}{2}+\dots+\frac{a_{n}}{n+1}$$

• $\text{eval}_{3}\colon\mathcal{P}_{n}\to\mathbb{R}$ with respect to $B,{\cal E}_{1}$ where $B=\langle 1,x,\dots,x^{n}\rangle$ and ${\cal E}_{1}=\langle 1\rangle$, given by

  $$a_{0}+a_{1}x+a_{2}x^{2}+\dots+a_{n}x^{n}\mapsto a_{0}+a_{1}\cdot 3+a_{2}\cdot 3^{2}+\dots+a_{n}\cdot 3^{n}$$

• $\text{slide}_{-1}\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$ with respect to $B,B$ where $B=\langle 1,x,\ldots,x^{n}\rangle$, given by

  $$a_{0}+a_{1}x+a_{2}x^{2}+\dots+a_{n}x^{n}\mapsto a_{0}+a_{1}\cdot(x+1)+\dots+a_{n}\cdot(x+1)^{n}$$