Exercise 4.3.1

$A=\left[\begin{array}{cc}1\hfill & 0\hfill \\ 1\hfill & 1\hfill \\ 1\hfill & 3\hfill \\ 1\hfill & 4\hfill \end{array}\right]$and $b=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 8\hfill \\ \hfill 8\hfill \\ \hfill 20\hfill \end{array}\right]$give $A^{T}A=\left[\begin{array}{cc}\hfill 4\hfill & \hfill 8\hfill \\ \hfill 8\hfill & \hfill 26\hfill \end{array}\right]$and $A^{T}b=\left[\begin{array}{c}\hfill 36\hfill \\ \hfill 112\hfill \end{array}\right]$.

$A^{T}A\hat{x}=A^{T}b$ gives $\hat{x}=\left[\begin{array}{c}1\hfill \\ 4\hfill \end{array}\right]$ and $p=A\hat{x}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 5\hfill \\ \hfill 13\hfill \\ \hfill 17\hfill \end{array}\right]$ and $e=b-p=\left[\begin{array}{c}\hfill -1\\ \hfill 3\\ \hfill -5\\ \hfill \end{array}\right]$