Look at the wikipedia article (paragraph: Spherical wavefronts of light) about the derivation of Lorentz transformation between two reference frames $O(x,y,z,t)$ $O'(x',y',z',t')$ in standard position. At a certain point one reads :
" The relation between $x$ and $x'$ should be in linear form and in such a way that it should reduce to Galilean transformation at $v << c$. Therefore, such a relation can be written of the form: \begin{align} x' &= \gamma \left( x - v t \right)\\ \end{align} where γ, not necessarily a constant, is to be determined "
I don't understand why we are searching a transformation written in this form.
Thanks in advance.
It must be a linear form because it needs to preserve vector addition, which is precisely what linear transformations do. Otherwise, I can interpret vectors $a,b,c$ as force vectors and find that an inertial frame ($a+b+c=0$) transformed into a non-inertial frame. So linearity is a must.
To be honest I'm not too convinced by their second portion either. I go about this by
requiring that the light vectors $[c,1]^T$ be eigenvectors of unspecified eigenvalues
requiring that the transformation for a given boost send some vector $[v,1]^T$ to $x=0$ (with the $t$ value left unspecified)
requiring that the inverse be found by simply making velocity negative. i.e. the same transformation takes $[0,1]^T$ to $[-vt,t]^T$ with some unspecified $t$ value.
After some simple algebra the Lorentz transforms emerge. Even though it's a little more tedious, I like this method because it invokes all of the major physical properties that the Lorentz transform needs to have through the derivation.