Let V be a vector space over Z_p with dim V = n. How many elements are in V?
Do you see why that's true. Every vector in the space can be written as a linear combination of vectors in the basis: $\displaystyle a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n$. There are p possible values for each $\displaystyle a_i$ so by the "fundamental counting principle", there are $\displaystyle (p)(p)\cdot\cdot\cdot(p)$ (n times) so there are $\displaystyle p^n$ possible vectors.
Hi Raoh - Sorry I didn't follow your question completely. Let me try to explain what I was trying to emphasize in my post.
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.