Chapter 3.8, Problem 15E

Consider the following table of data:

$\begin{array}{ccccc}t& t_1& t_2& \cdot \cdot \cdot & t_m\\ y& y_1& y_2& \cdot \cdot \cdot & y_m\end{array}$

For given functions $g_1$ and $g_2$, consider a function f defined by $f\left(t\right)=a_1{g}_1\left(t\right)+a_2{g}_2\left(t\right)$. Show that

${\displaystyle \underset{i=1}{\overset{m}{\sum }}{\left[f\left(t_i\right)-y_i\right]}^2={\Vert Ax-b\Vert }^2}$.

where

$A=\left[\begin{array}{cc}{g}_1\left(t_1\right)& g_2\left(t_1\right)\\ g_1\left(t_2\right)& g_2\left(t_2\right)\\ & \\ g_1\left(t_m\right)& g_m\left(t_m\right)\end{array}\right]$, $x=\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]$, and $b=\left[\begin{array}{c}{y}_1\\ y_2\\ \\ y_m\end{array}\right]$