Exercise 3

Show that these are homomorphisms. Are they inverse to each other?

• $d/dx\colon\mathcal{P}_{3}\to\mathcal{P}_{2}$ given by $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}$ maps to $a_{1}+2a_{2}x+3a_{3}x^{2}$

• $\int\colon\mathcal{P}_{2}\to\mathcal{P}_{3}$ given by $b_{0}+b_{1}x+b_{2}x^{2}$ maps to $b_{0}x+(b_{1}/2)x^{2}+(b_{2}/3)x^{3}$