Projective Geometry - Lecture Notes Chapter 2

2.3 Coordinates

Theorem 2.3.1: Let P1, P2, ..., Pt be points of P(V) with homogeneous coordinates
P1 = (a10:a11: ... :a1d),
... ,
Pt = (at0:at1: ... :atd).
Then the points P1, P2, ..., Pt are independent if and only if the matrix
a10a11...a1d
a20a21...a2d
::::
at0at1...atd
has rank t. In particular, d+1 points form a basis if and only if the matrix whose rows are the homogeneous coordinates of the points is nonsingular.

Theorem 2.3.2: Let V be a vector space of dimension d+1 over the division ring F, and let P(V) be the corresponding projective space. Let H be a hyperplane of P(V). Then the homogeneous coordinates of the points of H are the solutions of a homogeneous equation with coefficients in F. Conversely, any homogeneous equation that is different from the "zero equation" describes a hyperplane of P(V).

Corollary 2.3.3: Any t-dimensional subspace U of a projective space of dimension d given by homogeneous coordinates can be described by a homogeneous system of d-t linear equations. More precisely, there exists a (d-t)×(d+1) matrix H such that a point P = (a0:a1: ...:ad) is a point of U if and only if

(a0:a1: ...:ad)HT = 0.

Theorem 2.3.4: Let P = P(V) be a projective space whose points are given by homogeneous coordinates (a0:a1: ...:ad) with ai in F, (a0,a1, ...,ad) (0,0, ..., 0). Then any hyperplane of P(V) can be represented by [b0:b1: ...:bd] with bj in F, [b0,b1, ...,bd] [0,0, ...,0]. Conversely, to any such (d+1)-tuple [b0:b1: ...:bd] there belongs a hyperplane. Moreover, we have

(a0:a1:...:ad) I [b0:b1: ...:bd] iff a0b0 + a1b1 + ... + adbd = 0.

Corollary 2.3.5: Let P = P(V) be a coordinatized projective space. Then P P. In particular, P is coordinatized as well. Therefore the principle of duality holds for the class of all coordinatized projective spaces of fixed dimension d.

Theorem 2.3.6: Let be the hyperplane of P(V) with equation x0 = 0. Then the affine space A = P\ can be described as follows: