Base change/Transformation matrix/R^4/Standard basis and permutated standard basis/Exercise/Solution

The columns of $M_{𝔲}^{𝔳}$ are the coordinates of the vectors $v_j$ with respect to the standard basis $u_i$, hence

$M_{𝔲}^{𝔳}=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right).$

On the other hand, due to $u_1=v_2$, $u_2=v_4$, $u_3=v_1$, $u_4=v_3$ we have

$M_{𝔳}^{𝔲}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\end{array}\right).$

To the exercise