The columns of M u v {\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}} are the coordinates of the vectors v j {\displaystyle {}v_{j}} with respect to the standard basis u i {\displaystyle {}u_{i}} , hence
On the other hand, due to u 1 = v 2 {\displaystyle {}u_{1}=v_{2}} , u 2 = v 4 {\displaystyle {}u_{2}=v_{4}} , u 3 = v 1 {\displaystyle {}u_{3}=v_{1}} , u 4 = v 3 {\displaystyle {}u_{4}=v_{3}} we have