Assuming the relative motion 'v' is along the x dimension then x > x_{0} + v_{x}t
or if we have components of velocity in all dimensions the transform will be:

= 


where:
 v_{x},v_{y},v_{z}= relative velocity of the two reference frames in x,y and z directions.
 x,y,z= position in original frame.
 x',y',z'= position in transformed frame.
in other words point:

is transformed to: 

The nature of this transform is a shear (also known as skew) transform:
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When doing this we choose to make time 'absolute' in that the time lines are left horizontal wheras the position lines are skewed althogh I guess that this is just a covention and we could have skewed the time and made the distance absolute.
Shear (skew) Transform Matrix
The shear transform has the following characteristics:
 determinant = 1
 trace = dimension
for example the two dimensional matrix
a  b 
c_{}  d 
we have:
a+d=2
adbc=1