An abstract vector space of dimension n over a field k is the set of all formal expressions a_1 v_1 + a_2 v_2 + ... + a_n v_n, where {v_1, v_2, ..., v_n} is a given set of n objects (called a basis) and (a_1, a_2, ..., a_n) is any n-tuple of elements of k.

free module | quotient vector space | vector space