Because there is only a single ordered basis formula_3, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing formula_7 or formula_8 therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation, then, because all isomorphisms among zero-dimensional vector spaces preserve the ordered basis, they also preserve the orientation. This is unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms.