eq={x==u+v,y==2v}
T=Solve[eq,{x,y}][[1]]
Tinv=Solve[eq,{u,v}][[1]]
RegionPlot[{u≥0 && u≤1 && v≥0 && v≤1},{u,0,2},{v,0,2}]
RegionPlot[{u≥0 && u≤1 && v≥0 && v≤1}/.Tinv,{x,0,2},{y,0,2}]

Jac=D[{x,y}/.T,{{u,v}}]
MatrixForm[Jac]
Det[Jac]

Jacinv=D[{u,v}/.Tinv,{{x,y}}]
MatrixForm[Jacinv]
Det[Jacinv]