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

The columns of ${}M_{\mathfrak {u}}^{\mathfrak {v}}$ are the coordinates of the vectors ${}v_{j}$ with respect to the standard basis ${}u_{i}$, hence

${}M_{\mathfrak {u}}^{\mathfrak {v}}={\begin{pmatrix}0&1&0&0\\0&0&0&1\\1&0&0&0\\0&0&1&0\end{pmatrix}}\,.$

On the other hand, due to ${}u_{1}=v_{2}$, ${}u_{2}=v_{4}$, ${}u_{3}=v_{1}$, ${}u_{4}=v_{3}$ we have

${}M_{\mathfrak {v}}^{\mathfrak {u}}={\begin{pmatrix}0&0&1&0\\1&0&0&0\\0&0&0&1\\0&1&0&0\end{pmatrix}}\,.$

To the exercise