# Linear 2D transform in the sense of geometric figures?

Consider tranformation which turns one aligned rectangle to another:

This tranformation can be written in matrix form in the following way

where

$a_i=a_{ii}=\frac{x'_{i1}-x'_{i2}}{x_{i1}-x_{i2}}$

$b_i==\frac{x_{i1}x'_{i2}-x_{i2}x'_{i1}}{x_{i1}-x_{i2}}$

and

$a_{12}=a_{21}=0$

i.e. transform matrix is diagonal.

My question is: what about arbitrary matrix? What 2 geometric figures defines it? Is this a pair of two arbitrary quadrangles?

If "yes", then what is the formula, relating angle coordinates to matrix coefficients?

How is it extrapolated to multidimensional?

UPDATE

I realized, that the number of equations in each transformation is $N$, the dimension of space, and the number of unknowns is $N^2+N=N(N+1)$ where $N$ is the number of translation column elements and $N^2$ is the number of linear matrix elements. Hence to make the system complete, we need $N(N+1)/N = (N+1)$ sample pairs of points.

So in the case of 2D we need to provide 2 arbitrary triangles to define transform.

Where to find a formula?

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