# Whitehead's axioms of projective geometry and a vector space over a field

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are:

G1: Every line contains at least 3 points

G2: Every two points, A and B, lie on a unique line, AB.

G3: If lines AB and CD intersect, then so do lines AC and BD
(where it is assumed that A and D are distinct from B and C).

Now let $V$ be a vector space over a field $K$.
We denote by $P(V)$ the set of one-dimensional subspaces of $V$.
If $V$ is finite dimensional, this is the usual definition of a projective space over $K$.
We say a two-dimensional subspace of $V$ a line of $P(V)$.
Then it is natural to expect that points and lines of $P(V)$ satisfy the above axioms.
I think it is easy to prove G1 and G2.
But how can we prove G3?

#### Solutions Collecting From Web of "Whitehead's axioms of projective geometry and a vector space over a field"

Claim: $AB$ intersects $CD$ if and only if the underlying vectors of $A,B,C,D$ are linearly dependent.

(in fact, if $AB \neq CD$, they span a space of dimension $3$)