This is not yet a proof!!

The set of n-tuples of the field is a vector space and any vector space with a basis with n elements is isomorphic to this space of tuples according to the obvious isomorphism: Σα_iu_i ↔ (α₀, ... α_n−1) but to see this we must show that if Σα_iu_i = Σβ_iu_i then for each i α_i = β_i.
Assume Σα_iu_i = Σβ_iu_i
Σα_iu_i − Σβ_iu_i = 0
Σ(α_iu_i − β_iu_i) = 0
Σ(α_i − β_i)u_i = 0
but by the meaning of ‘independence of the {u_i}’ we have:
for each i α_i − β_i = 0
for each i α_i = β_i

This establishes the one to one nature of the correspondence and thus the isomorphism. It also shows that two vector spaces with equinumerous bases are isomorphic to each other. A vector space with a basis of n elements is called an n-dimensional vector space and is unique up to isomorphism. Indeed this produces all isomorphisms between two such spaces.