Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them. If dim V = dim W, a 1-to-1 correspondence between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W. It is an isomorphism, by its very definition. Therefore, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space over a given field is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n -dimensional F -vector space V is isomorphic to F. However, there is no "canonical" or preferred isomorphism; an isomorphism φ : F → V is equivalent to the choice of a basis of V, by mapping the standard basis of F to V, via φ.