Let V be a vector space over Z_p with dim V = n. How many elements are in V?
What I meant was that each of the linear combination (and there are p^n linear combination because of the reason you mentioned) is in one-one and onto maping with all the vectors in the vector space. And the reason that happens is only because v1,v2.....vn form a basis. The phrase 'in a unique way' is important to stress the one-one part of the mapping. 'Every' specifies the onto part.