Exercise 4.2.30

(a) The column space is the line through $a=\left[\begin{array}{c}3\hfill \\ 4\hfill \end{array}\right]$so $P_C=\frac{{\mathrm{aa}}^T}{a^{T}a}=\frac{1}{25}\left[\begin{array}{cc}\hfill 9\hfill & \hfill 12\hfill \\ \hfill 12\hfill & \hfill 16\hfill \end{array}\right]$. The formula $P=A{\left(A^{T}A\right)}^{-1}{A}^T$ needs independent columns-this $A$ has dependent columns. The update formula is correct.

(b) The row space is the line through $v=(1,2,2)$ and $P_R=v{v}^T∕v^{T}v$. Always $P_{C}A=A$ (columns of $A$ project to themselves) and $A{P}_R=A$. Then $P_{C}A{P}_R=A$.