First of all why was there a need to invent projective space? One place it appears naturally is when we think about perspective. Imagine railroad tracks stretching out in front of you, if you look out to infinity, at the horizon, the two lines seem to meet. We know that parallel lines don't meet in ordinary Euclidean space but by going to projective space we can make them meet in a formal mathematical sense.

This is how it's done. Start with a field k (think of k=R, the real numbers or k=C, the complex numbers). Now for a nonnegative integer n we are going to define projective n-space. The construction starts with kn+1\{0}, that is, n+1-tuples of elements of k with the origin 0=(0,0,...,0) excluded. We define an equivalence relation on the nonzero points of kn+1 by defining

(a0,a1,....,an+1) equiv (b.a0,b.a1,...,b.an+1)

for any non-zero scalar b in k. Projective n-space denoted by Pn(k) is the set of equivalence classes of the nonzero points kn+1 under this relation.

It's easier to deal with this once we have some good notation. So we write [a0,....,an+1] for the equivalence class of (a0,....,an+1). Thus we have that

[a0,....,an+1]=[b.a0,b.a1,...,b.an+1]

Let's do some examples.

  • n=0 Here P0(k) consists of the points [a], for a nonzero. But we know that [a]=[b.a], for any nonzero b. So P0(k) just consists of one point [1]
  • n=1 This time P1(k), the projective line, consists of the points [a,b], for a,b not both zero. If a=0 then [0,b]=[0,1], so there is only one point with the first coordinate zero. Suppose on the other hand that a is nonzero. Then [a,b]=[1,b/a] and so we see that P1(k) consists of the the points [1,a] with a in k (i.e a copy of the usual 1-space) together with an extra point at infinity, namely [0,1].

  • n=2 Finally P2(k), the projective plane, consists of the points [a,b,c], for a,b,c not all zero. This time if a=0 we get a family of points, namely all [0,b,c] with b,c not both zero. This is just a copy of the projective line that we have just seen. If a is nonzero then we get all points of the form [1,b,c]. This is just a copy of ordinary 2-space k2. So the projective plane is 2-space together with a projective line at infinity.
In general, projective n-space is ordinary n-space together with projective n-1-space at infinity.

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