What transformations can be set by projecting a straight line onto a straight line (without adding an infinitely distant point)? I said that the homothety with coefficient $k \neq 1$ and the reflection. But I was told that this is not true. Where am i wrong?
The set of transformations of the (affine) plane that preserve lines is the set of affine transformations.
This is equivalent to the set of projective transformations of the protectively completed affine plane.
The category of transformations you indicated was much narrower: in fact they were all isomorphisms of the plane.
Even if you meant that the transformations have to be nonsingular, there are still more homographies than just the homothetic transformations.