This observation is made more natural by viewing bases B as a choice of isomorphism Φ Φ B : V → → K n . The space of bases is equivalently the space of such isomorphisms, denoted Iso ( V , K n ) . Active transformations, identified with GL ( V ) , act on Iso ( V , K n ) from the left by composition, while passive transformations, identified with GL ( n , K ) acts on Iso ( V , K n ) from the right by pre-composition.