The following derivations are similar to *here*.

Lorentz transformations for space with time

Let unprimed *x* and *t* be from inertial frame K and primed *x′* and *t′* be from inertial frame K′. Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant coefficients, A, *B*, *C*, and *D*:

*x′ = Ax + Bt*

*t′ = Ct + Dx*

A body at rest in the K′ frame at position *x*′ = 0 moves with constant velocity *v* in the K frame. Hence the transformation must yield *x*′ = 0 if *x* = *vt*. Therefore, *B* = −*Av* and the first equation becomes

*x′ = A* (*x – vt*).

*Using the principle of relativity*

According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame K′ to frame K should have the same form as the original but with the velocity in the opposite direction, i.e., replacing *v* with *−v*:

*x = A* (*x′* − (−*vt′*))* = A* (*x′ + vt′*).