Lines in projective space

I have the following definitions:

Given a vector space V over a field k, we can define the projective space PV=(V{0})/ where identifies all points that lie on the same line through the origin.

A projective subspace PW of PV is of the form π(W{0}), where π is the residue class map and W is a vector subspace of V. Define dim(PV)=dim(V)1. A line in PV is a 1-dimensional projective subspace.


Now I’m finding it difficult to visualise what a line in projective space actually is. I can understand why any two lines in a projective plane intersect. Suppose I’m in P3 and want to write ‘an equation’ for the line that goes through the points p=(1:0:0:0) and q=(a:b:c:d). How could I do that? Does my question even make sense? I’m concerned because PV isn’t actually a vector space, so can I think of points inside it as vectors?

Thanks

Answer

If you have two distinct points A=[a0::an],B=[b0::bn]Pn, they correspond to two vectors a=(a0,,an),b=(b0,,bn)kn+1.
These vectors span a plane Λkn+1 whose vectors are the ua+vb,(u,vk).
The corresponding line ¯AB=P(Λ)Pn has its points of the form [ua0+vb0::uan+vbn](u,vk, not both zero ).

In your particular case the projective line ¯pq joining p=[1:0:0:0] and q=[a:b:c:d] has points with coordinates [u+va:vb:vc:vd]

Attribution
Source : Link , Question Author : Jonathan , Answer Author : Georges Elencwajg

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