Sorry for my ignorance, but I want to know how the mechanism of finding the least square solutions or the closest points in Euclidean space works.
For example:
Find the closest point or points to $b =(−1,2)^T$ that lie on the line $x + y = 0$.
I know the answer is
$\frac{\left( \begin{matrix} -1 \\ 2 \end{matrix} \right) \cdot \left( \begin{matrix} 1 \\ -1 \end{matrix} \right)}{\left( \begin{matrix} 1 \\ -1 \end{matrix} \right) \cdot \left( \begin{matrix} 1 \\ -1 \end{matrix} \right)} \cdot \left( \begin{matrix} 1 \\ -1 \end{matrix} \right) = \left( \begin{matrix} -\frac{3}{2} \\ \frac{3}{2} \end{matrix} \right)$
But what does the dot product between $\left( \begin{matrix} -1 \\ 2\end{matrix} \right) \cdot \left( \begin{matrix} 1 \\ -1 \end{matrix} \right)$ tell you? If I commute it the answer is $-3$, but what exactly is $-3$? Also, the denominator is $2$, again, what exactly is $2$ telling us here?