The Proof of The Fundamental Theorem of Projective Geometry

 The Fundamental Theorem of Projective Geometry If L =(A,B,C,D) and L' = (A',B',C',D') are lists of points of RP2, each with no three collinear, then there is a unique element of P(2) mapping L to L'. Much as in affine geometry, this follows a special case with the standard points X = [1,0,0], Y = [0,1,0], Z = [0,0,1], U = [1,1,1] in one of the lists. The (X,Y,Z,U) Theorem If L =(P,Q,R,S) is a list of points of RP2, with no three collinear, then there is a unique element of P(2) mapping (X,Y,Z,U) to L. Proof of the (X,Y,Z,U) Theorem Let P, Q, R, S be the p-points [p], [q], [r], [s] respectively, and let x, y, z, u be the vectors (1,0,0), (0,1,0), (0,0,1), (1,1,1). Observe that, as no three of P,Q,R,S are collinear, no three of p,q,r,s are linearly dependent. Suppose that t is defined by the matrix A. Then t(U) = S if and only if [Au] = [s], i.e. Au = λs with λ ≠ 0. If we replace A by (1/λ)A, we get the same t, so we may assume that Au = s Now, Ax, Ay and Az are just the columns of A (in order), so t maps X, Y, Z to P, Q, R if and only if the columns of A are of the form αp, βq, γr, with α, β, γ ≠ 0. Also Au is the sum of the columns of A so that t has the required images if and only if αp + βq + γr = s. Since p, q, r are linearly independent, this has a unique solution α, β, γ. Since s is not dependent on any two of p, q, r, α, β, γ are non-zero. Thus the matrix A with columns αp, βq, γr is invertible, so defines an element of P(2). From the above argument, A is unique up to scaling, so t is unique. Proof of the Fundamental Theorem By the (X,Y,Z,U) Theorem, there exist elements r, s of P(2) such that r maps (X,Y,Z,U) to L and s maps (X,Y,Z,U) to L'. Then t = sor-1 maps L to L'. Suppose that u also maps L to L'. Then uor maps (X,Y,Z,U) to L'. By the uniqueness clause of the Theorem, only s maps (X,Y,Z,U) to L'. Thus uor = s, so u = sor-1 = t, i.e. t is unique.