I am working through a Linear Algebra text and I am struggling to understand how to find the unique minimum-norm solution to a compatible linear system, similar to the following:
\begin{align} 6x - 3y + 9z = 12 \\ 2x - y + 3z = 4 \end{align}
I know how to verify compatibility by finding if the right hand side is orthogonal to the cokernel. Which in this case is,
\begin{align} coker(A) = (-1,3)^T \end{align}
Which can easily be seen to be orthogonal to the right hand side, which lies in the image. I have also found the general solution,
\begin{align} x = 2 + \frac{1}{2}y - \frac{3}{2}z \end{align}
The solution given in the text is when, \begin{align} x = \frac{4}{7}, y = \frac{-2}{7},z = \frac{6}{7} \end{align}
I am lost as to how they find that solution. I have tried taking the Euclidean dot product of the general solution with the kernel of A, but I end up getting,
\begin{align} x = \frac{32}{59}, y = \frac{-28}{59},z = \frac{48}{59} \end{align}
I am more so following steps than following an intuition, which I am not proud of. I am pretty sure I can project the general solution onto the coimage of A, but I am not confident as to how. I have also not reached the section in the text regarding minimization, in case that has anything to do with this type of problem.