Context:
When deriving the Lorentz transformation of special relativity fundamentally from the invariance between the interval $ds^{2}=(cdt)^{2}-(dx)^{2}-(dy)^{2}-(dz)^{2}$ in a frame $K$ and its primed value $ds^{'2}=(cdt')^{2}-(dx')^{2}-(dy')^{2}-(dz')^{2}$ in the frame $K'$, it is stated in Jackson's book on classical electrodynamics (3rd ed, p. 525) that we must have $ds$ and $ds'$ proportional to each other and the proportionality factor is dependent on the velocity magnitude between the frames in isotropic homogeneous space.
Similarly, Landau and Lifshitz, in their 2nd book on Theoretical Physics (4th ed, p.4) say that since having zero $ds$ will lead to zero $ds'$ and since $ds$ and $ds'$ when not zero are "infinitesimally of the same order" as each other, then they must be linearly related and the factor between them depends on the velocity magnitude.
Question:
Why do they have to be linearly related (using basic principles) following these simple observations? And how was it inferred that the factor is a function of velocity?
