TITLE: How to find a transformation matrix having several original points and their respective transformed results?
QUESTION [5 upvotes]: I have three original points $pt_1, pt_2, pt_3$ which if transformed by an unknown matrix $M$ turn into points $gd_1, gd_2, gd_3$ respectively. How can I find the matrix $M$ (all points are in 3-dimensional space)?
I understand that for original points holds $M\cdot pt_i = gd_i$, so combining all $pt_i$ into matrix $PT$ and all $gd_i$ into $GD$ I'd get a matrix equation $M\cdot PT=GD$ with unknown $M$.
However, many math packages solve matrix equations in form of $A\cdot x=B$, where $x$ is unknown.
Is my idea of combining points into matrices correct and if so how can I solve my matrix equation?

REPLY [2 votes]: I don't believe the accepted answer works for translations. If we have a three-dimensional point and a 3x3 transformation matrix, the zero point will always map to the zero point. 
A general affine transformation requires a four-dimensional matrix.
It may be possible to use a four-dimensional point with the fourth component of the point equal to 1.