Exercise 5

Verify that each map is a homomorphism.

• $h\colon\mathcal{P}_{2}\to\mathbb{R}^{2}$ given by

  $$ax^{2}+bx+c\mapsto\begin{pmatrix}a+b\\ a+c\end{pmatrix}$$

• $f\colon\mathbb{R}^{2}\to\mathbb{R}^{3}$ given by

  $$\begin{pmatrix}x\\ y\end{pmatrix}\mapsto\begin{pmatrix}0\\ x-y\\ 3y\end{pmatrix}$$