## 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
 a10 a11 ... a1d a20 a21 ... a2d : : : : at0 at1 ... 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