The Galilean transformation is used to transform between the coordinate of two reference frames which differ only by constant relative motion within the construct of Newtonian physics. This is a passive transformation point of view. In special relativity the Galilean transformations are replaced by Lorentz transformations.
The notation that can be found below describes the relationship under the Galilean transformation between the coordinates (x, y, z, t) and (x’, y’, x’, t’) of a single arbitrary event as measured in two coordinate systems S and S’, in uniform relative motion in their common x and x’ directions, with their spatial origins coinciding at t=t’=0.
x’ = x – vt
y’ = y
z’ = z
t’ = t
The transformation can also be considered a shear mapping and is described with matrix acting on a vector. When the motion is parallel to the x-axis, the transformation acts on only two components:
(x',t')= (x,t)(1 0)