Exercise 2.2.19

To project $b$ onto the line through $a=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 3\hfill \\ \hfill 4\hfill \end{array}\right]$, the projection of $b$ on $a$ is thus $p=\frac{a^{T}b}{a^{T}a}a$, we can prove that $b-p$ is perpendicular to $a$. So we have $\widehat{x}=\frac{56}{13}$, $p=\widehat{x}a=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill \frac{56}{13}\hfill \\ \hfill \frac{168}{13}\hfill \\ \hfill \frac{224}{13}\hfill \end{array}\right]$, and $e=b-p=\left[\begin{array}{c}\hfill -\frac{56}{13}\hfill \\ \hfill \frac{48}{13}\hfill \\ \hfill -\frac{64}{13}\hfill \\ \hfill \frac{36}{13}\hfill \end{array}\right]$, and it’s easyto see that $e^{T}a=0$, the shortest distance from $b$ to the line through $a$ is $\left|e\right|=64.095$.

The best $C$ in problem 16 and the best $D$ in problem 18 do NOT agree with the best $(Ĉ,\widehat{D})$ in problem 11-14. That is because the two columns $(1,1,1,1)$ and $(0,1,3,4)$ are not perpendicular.