In geometry, an affine plane is defined as a system of points which fullfill:
1) Any two distinct points lie on a unique line. 2) Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. 3) There exist three non-collinear points.
How does this definition tell me that an affine plane is an affine space which has dimension 2. Every book says that an affine space has dimension 2, but I cannot proof it.
My definition of an affine space A is that: Given a set A, whose elements are taken as points. Furthermore, let a K-vector space V_A be given and a map C: A x A, which uniquely assigns a connection vector pq to each ordered pair (p, q) with p, q in A; such that the following axioms are satisfied: a) For every vector v in V_a and every point p in A there is exactly one point q in A with v = {pq}. b) {pq} + {qr} = {pr}.