[wgardner]

One more thought on this:

I guess I wrote the equations the way I did because that's always the way that I've seen the linear regression derived in the first place.

Namely, that the way you compute the OPR values is view the problem as
M = A O + N
then form the squared error as
N' N = (M - A O)' (M - A O)
then compute the derivative of N' N with respect to O and solve for O, which gives you Oest = Inv(A' A) A M.

Is there a different way of expressing this derivation without resorting to a vector N of the errors that are being minimized?