Use the least square approximation to find the closest line (the line of "Best Fit") to the points:
$$(-6,-1), \quad (-2,2), \quad (1,1), \quad (7,6)$$
I'm attempting to use the least squares approximation formulation that is as follows:
$$A^TAx = A^Tb$$
However, I'm confused because I'm given four vectors. Does that mean I can use the first three vectors $(-6,-1),(-2,2),(1,1)$ to create my matrix A, and then use the last vector $(7,6)$ for the "$b$" value? I understand the process is very straight forward once proper substitution has been done, but I tried the method I described above and got the wrong answer.Thank you guys.
You are looking for an equation $y=mx+c$. Ideally it would pass through all of the given points: that is, you would have $-6m+c = -1$, and similarly for other points. This is a set of four equations with two unknowns $m,c$. Its matrix representation is $$ \begin{pmatrix} -6 & 1\\ -2 & 1 \\ 1 & 1 \\ 7 & 1 \end{pmatrix} \begin{pmatrix} m \\ c \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \\ 1 \\6 \end{pmatrix} $$ These are $A$ and $b$ you are looking for. Typically, the system $Ax=b$ has no solutions (there is no line through all the points); this is why we solve $A^TAx=A^Tb$ instead, obtaining $x$ that minimizes the residual $\|Ax-b\|^2$.