If one views { e x ′ = T ( 1 , 0 , 0 ) , e y ′ = T ( 0 , 1 , 0 ) , e z ′ = T ( 0 , 0 , 1 ) } as a new basis, then the coordinates of the new vector v ′ = v x e x ′ + v y e y ′ + v z e z ′ in the new basis are the same as those of v = v x e x + v y e y + v z e z in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.