Exercise 4.2.4

$P_1=\left[\begin{array}{cc}1\hfill & 0\hfill \\ 0\hfill & 0\hfill \end{array}\right],P_2=\frac{a{a}^T}{a^{T}a}=\frac{1}{2}\left[\begin{array}{cc}\hfill 1& \hfill -1\\ \hfill -1& \hfill 1\end{array}\right]\begin{array}{ccc}\hfill & P_1\text{ projects onto }(1,0),P_2\text{ projects onto }(1,-1)\hfill & \hfill \\ \hfill & P_1{P}_2\ne 0\text{ and }{P}_1+P_2\text{ is not a projection matrix. }\hfill \\ \hfill & {\left(P_1+P_2\right)}^2\text{ is different from }{P}_1+P_2.\hfill \end{array}$