For example, I have set of points in 3D. Points lie on straight line. Transformed set of points lies on straight line too. How to check if transformation is affine?
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According to the fundamental theorem of affine geometry, it suffices to check that the transformation is a bijective collineation:
The proof can be found in Berger's Geometry 1 (Springer, 1987, pp. 52-56).
an affine transformation between two vector spaces $$F:X\rightarrow Y$$ (one might define it more general) is defined as $$y = F(x) = Ax + y_0$$
where $A$ is a constant map (might be represented as matrix) and $y_0\in Y$ is a constant element.
So, to check if a transformation is affine you might try to proof that such $A$ and $y_0$ exist.