# How to find a transformation matrix having several original points and their respective transformed results?

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?

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Form matrices whose rows are $pt_k$ and $gd_k$. Then $$\begin{bmatrix}pt_1\\pt_2\\pt_3\end{bmatrix}M=\begin{bmatrix}gd_1\\gd_2\\gd_3\end{bmatrix}$$ Then we have $$M=\begin{bmatrix}pt_1\\pt_2\\pt_3\end{bmatrix}^{-1}\begin{bmatrix}gd_1\\gd_2\\gd_3\end{bmatrix}$$ So your idea is correct.
Wait, why rows? In your mind, if $A$ is a matrix, $X$ and $B$ are vectors, then which notation is correct: $A\cdot X=B$ or $X\cdot A=B$? –  mbaitoff Feb 24 '13 at 8:17