Deriving the Lorentz Transformation from the Homogeneity of Spacetime

1. The problem statement, all variables and given/known data
Show that the isotropy and homogeneity of space-time and equivalence of different inertial frames (first postulate of relativity) require that the most general transformation between the space-time coordinates (x, y, z, t) and (x', y', z', t') is the linear transformation,
x'=f(v²)x-vf(v²)t; t'=g(v²)t-vh(v²)x; y'=y; z'=z
and its inverse,
x=f(v²)x'+vf(v²)t'; t=g(v²)t'+vh(v²)x'; y=y'; z'=z'

2. Relevant equations


3. The attempt at a solution
Now, I know that homogeneity implies that the transformation must be linear in x and t and that the isotropy of space implies that the coefficients can only be functions of the magnitude of the velocity (not the direction) at most. Therefore, I am stuck at the following:
x'=f(v²)x-vf₀(v²)t; t'=g(v²)t-vh(v²)x; y'=y; z'=z
However, I am having trouble proving that f = f₀. The solution I am looking at says that this follows from the homogeneity of space-time but I am having trouble using that fact to prove it.

 

http://faculty.uml.edu/cbaird/all_homework_solutions/Jackson_11_1_Homework_Solution.pdf