If I have a map T, and a two bases B and A where x are coordinates in B and y are coordinates in A. Then does the matrix M_AB(T) take coordinates in B and transform them to A, or A to B?
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Is the map T from A to B or vice-versa? The way you have written it, $\displaystyle M_{AB}(T)$, it would be interpreted as from A to B but it is the linear transformation that determines whether that is correct, not the bases or the matrix.
it depends on whether T is a function of x, or of y.
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