Exercise 4.2.12

$P_1=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill \end{array}\right]=$projection matrix onto the column space of $A$ (the $\mathit{xy}$ plane) $P_2=\left[\begin{array}{ccc}0.5\hfill & 0.5\hfill & 0\hfill \\ 0.5\hfill & 0.5\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]=\begin{array}{ccc}\hfill & \text{ Projection matrix }A{\left(A^{T}A\right)}^{-1}{A}^T\text{ onto the second column space. }\hfill & \hfill \\ \hfill & \text{ Certainly }{\left(P_2\right)}^2=P_2.\text{ A true projection matrix. }\hfill \end{array}$