Exercise 3

Find the range space and rank of each map.

• $h\colon\mathbb{R}^{2}\to\mathcal{P}_{3}$ given by

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

• $h\colon\mathcal{M}_{2\!\times\!2}\to\mathbb{R}$ given by

  $$\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mapsto a+d$$

• $h\colon\mathcal{M}_{2\!\times\!2}\to\mathcal{P}_{2}$ given by

  $$\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mapsto a+b+c+dx^{2}$$

• the zero map $Z\colon\mathbb{R}^{3}\to\mathbb{R}^{4}$