Exercise 2.2.16

To find the best horizontal line to fit $b$, we have $A=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$, $A^T=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$, and $b=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 8\hfill \\ \hfill 8\hfill \\ \hfill 20\hfill \end{array}\right]$

Solve for $A^{T}A\widehat{x}=A^{T}b$, we have $A^{T}A=\left[\begin{array}{c}\hfill 4\hfill \end{array}\right]$, and $A^{T}b=\left[\begin{array}{c}\hfill 36\hfill \end{array}\right]$, we find $\widehat{x}=\left[\begin{array}{c}\hfill 9\hfill \end{array}\right]$, so the best horizontal line fit is $b=C=9$.

The errors are $e=b-p=-9,-1,-1,11$, total errors $E=204$