[Physics] Homogeneity and isotropy and derivation of the Lorentz transformations

coordinate systemsinertial-frameslorentz-symmetrymetric-tensorspecial-relativity

In deriving the Lorentz transformations I have found (from reading a few different sets lecture notes) that it is argued that they must be linear and thus there general form must be $$x'=Ax+Bt,\quad t'=Dx+Et$$ (assuming relative motion between two inertial frames $S$ and $S'$ along one axis).

My question is, can the linearity of Lorentz transformations be argued purely from Einstein's two postulates, or does one have to assume homogeneity of space and time, and isotropy of space?

I can kind of see that they must be linear purely from the fact that one wishes to map between two inertial frames and hence, in particular, straight lines should be mapped to straight lines (otherwise a particle observed to be unaccelerated in one inertial frame will appear to be accelerating in another). Also, the inverse of a linear transformation is also linear, which is required otherwise such transformations would single it privileged inertial reference frames, violating the principle of relativity.

However, doesn't the mere existence of global inertial frames require spatial homogeneity and isotropy, as otherwise any measurements made by an observer in a given inertial frame would depend on the location of the observer within the inertial frame, and in which direction they make the measurement?!

If one starts off with the assumption of homogeneity and isotropy, then I can definitely see why the transformations should be linear, since homogeneity requires that the form of transformation should not depend on the location of the two inertial frames in space, this the derivative of the transformation should be independent of location, i.e. it should be a constant. Isotropy of space also implies that the transformation should not depend on the relative velocity between the two frames, but at most, the relative speed between them.

I would really appreciate it if someone could enlighten me on this subject?