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Referring to: 2.5 The projective geometry of 1D by Richard Hartley and Andrew Zissermann. Can I compute a projective transformation matrix 2x2 from two sets of 3 1D colinear points?

I want to find the projective transformation of 2 sets of 3 colinear points on a 2D sketch.

pointset x and x'

solve for H using A:

\begin{equation*} \begin{bmatrix} -x_1 & -1 & u_1x_1 & u_1 \\ -x_2 & -1 & u_2x_2 & u_2 \\ x_3 & -1 & u3x_3 & u_3 \\ 0 & 0 & 0 & 1 \end{bmatrix} * \begin{bmatrix} h_{11} \\ h_{12} \\ h_{21} \\ h_{22}\end{bmatrix} =\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \end{equation*}

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