Note that projective spaces are not vector spaces. There is no meaningful way of adding vectors nor multiplying by a scalar. Meaning we have to be very careful

Note that projective spaces are not vector spaces. There is no meaningful way of adding vectors nor multiplying by a scalar. Meaning we have to be very careful about treating our projective transformations as linear maps, as they are, in fact, nothing like that. Of course people doing graphics often pick a section of the projective space ("last component is 1"), then map that onto R^3 and off they go, sort of ignoring the "planes at infinity" and praying for the best. This is messy but I do not see a better way.