I've got strange notation of change of basis matrix in my book and I'd like to have it explained a little bit. It says, if:
$M _{\mathcal A}^{\mathcal B}(id) \cdot \vec{v} _{\mathcal B} = \vec{v}_{\mathcal A}$
then $M _{A}^{B}(id)$ is change of basis matrix from the basis $A$ to the basis $B$. It completely breaks my intuition, because in the result, we have a vector represented in the basis $A$!
On same page in this book, they say: $\phi(\vec{v})=\vec{w} \iff M _{\mathcal B}^{\mathcal A}(\phi) \cdot \vec{v} _{\mathcal A} = \vec{w}_{\mathcal B} $ It brings even more confusion.
Any help would be appreciated.