The basis in Jennifer’s grid represent a transformation as matrix below. It can convert the vector in Jennifer’s language into the vector in our language.
The inverse switches the direction.
If we apply the inversion of basis matrix to the vector in our language, we can obtain the vector in Jennifer’s language.
The function is inverse change of previous pic.
This transformation can be written in
A^(-1)MA. This represent convert other language into our language, then apply a transformation to it, finally convert our language to origin language.
This transformation means a transformation on the other language.