Base change/Transformation matrix/R^3/Standard basis and permuted standard basis/Exercise

Determine the transformation matrices ${}M_{\mathfrak {v}}^{\mathfrak {u}}$ and ${}M_{\mathfrak {u}}^{\mathfrak {v}}$ for the standard basis ${}{\mathfrak {u}}$ and the basis ${}{\mathfrak {v}}$ given by the vectors

$v_{1}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\,v_{2}={\begin{pmatrix}0\\0\\1\end{pmatrix}},\,{\text{ and }}\,v_{3}={\begin{pmatrix}1\\0\\0\end{pmatrix}}$

in ${}\mathbb {R} ^{3}$.

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