Before revision: Note the equivalence between the two kinds of transformations: the coordinates of the new point in the active transformation and the new coordinates of the point in the passive transformation are the same, namely
After revision: In abstract vector spacesThe distinction between active and passive transformations can be seen mathematically by considering abstract vector spaces.Note the equivalence between the two kinds of transformations: the coordinates of the new point in the active transformation and the new coordinates of the point in the passive transformation are the same, namely