Any two basis of finite-dimensional vector space have the same length. constituents A finite-dimensional vector space V Basis B_1, B_2 be bases in V requirements Given B_1, B_2 are basis in V, we know that they are both linearly independent and spans V. We have that the length of linearly-independent list \leq length of spanning list. Let’s take first B_1 as linearly independent and B_2 as spanning: We have then len(B_1) \leq len(B_2) Swapping roles: We have then len(B_2) \leq len(B_1) As both of this conditions are true, we have that len(B_1)=len(B_{2}). \blacksquare