Exercise 2.2.20

For the closest parabola $b=C+\mathit{Dt}+E{t}^2$, we have $\mathit{Ax}=b$, i.e. $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 9\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 16\hfill \end{array}\right]\left[\begin{array}{c}\hfill C\hfill \\ \hfill D\hfill \\ \hfill E\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 8\hfill \\ \hfill 8\hfill \\ \hfill 20\hfill \end{array}\right]$, and the normal equations $A^{T}A\widehat{x}=A^{T}b$ is: $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 9\hfill & \hfill 16\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 9\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 16\hfill \end{array}\right]\left[\begin{array}{c}\hfill C\hfill \\ \hfill D\hfill \\ \hfill E\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 9\hfill & \hfill 16\hfill \end{array}\right]\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 8\hfill \\ \hfill 8\hfill \\ \hfill 20\hfill \end{array}\right]$

In Figure II.3a, we are still finding the projection of $b$ onto the column space of $A$, which is now composed by 3 columns.