Exercise 2

For each map, give the chain of range spaces and the chain of null spaces, and the generalized range space and the generalized null space.

• $t_{0}\colon\mathcal{P}_{2}\to\mathcal{P}_{2}$, $a+bx+cx^{2}\mapsto b+cx^{2}$

• $t_{1}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$,

  $$\begin{pmatrix}a\\ b\end{pmatrix}\mapsto\begin{pmatrix}0\\ a\end{pmatrix}$$

• $t_{2}\colon\mathcal{P}_{2}\to\mathcal{P}_{2}$, $a+bx+cx^{2}\mapsto b+cx+ax^{2}$

• $t_{3}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$,

  $$\begin{pmatrix}a\\ b\\ c\end{pmatrix}\mapsto\begin{pmatrix}a\\ a\\ b\end{pmatrix}$$