Help understanding the projective space Pln

Hi,

I'm having trouble understanding the projective space Pkn. Now, I have 2 definitions for it. The first being that it is the set of all 1 dimensional k vector subspaces of k^n+1, and the other being that it is represented by the set of equivalence classes [a0,a1.....an] with (a0,a1....an) in non zero in K^n+1. Now I understand both definitions , but what I don't understand is how Pk0 is a single point, and how Pk1 is a line? For example, Pk0=[a0], how is Pk0 just a single point? I'm struggling to gather the intuition regarding this. And I also have Pk1 to be [a,1] disjoint union [1,0] , with a in K. Somehow this gives Pk^1 to be K and a point at infinity, but I have no idea how.

Thanks