lorentz[x_,t_,v_] = {(x- v t)/Sqrt[1-v^2],(t - v x)/Sqrt[1-v^2]}

galilean[x_,t_,v_] = {x- v t,t}

Manipulate[
Column[{v,Show[
ListPlot[Flatten[Table[galilean[x,t,v],{x,-5,5},{t,-5,5}],1], PlotRange->{{-10,10},{-10,10}},
AspectRatio->1,ImageSize->Large,PlotStyle->PointSize[Medium], AxesLabel->{x',ct'},
PlotLabel->Galilean transformation],
ListPlot[{lorentz[-100,-100,0],lorentz[100,100,0],lorentz[100,-100,0],lorentz[-100,100,0]},Joined->True,PlotStyle->{Dashed,Black}],
ListPlot[{galilean[5,1,v],galilean[3,2,v]},PlotStyle->{PointSize[Large],Green}]
]},Center]
, {{v,0},-.999,.999}]

Manipulate[
Column[{v,Show[
ListPlot[Flatten[Table[lorentz[x,t,v],{x,-5,5},{t,-5,5}],1], PlotRange->{{-10,10},{-10,10}},AspectRatio->1,ImageSize->Large,PlotStyle->PointSize[Medium], AxesLabel->{x',ct'},
PlotLabel -> Lorentz transformation],
ListPlot[{lorentz[-100,-100,0],lorentz[100,100,0],lorentz[100,-100,0],lorentz[-100,100,0]},Joined->True,PlotStyle->{Dashed,Black}],
ListPlot[{lorentz[0,0,v],lorentz[3,2,v]},PlotStyle->{PointSize[Large],Green}]
]},Center]
, {{v,0},-.999,.999}]

lorentz[5,4,0.8]