## Projective Geometry - Lecture Notes Chapter 2

### 2.3 Coordinates

**Theorem 2.3.1**: Let P_{1}, P_{2}, ..., P_{t} be points of **P**(V) with homogeneous coordinates
P_{1} = (a_{10}:a_{11}: ... :a_{1d}),

... ,

P_{t} = (a_{t0}:a_{t1}: ... :a_{td}).
Then the points P_{1}, P_{2}, ..., P_{t} are independent if and only if the matrix
a_{10} | a_{11} | ... | a_{1d} |

a_{20} | a_{21} | ... | a_{2d} |

: | : | : | : |

a_{t0} | a_{t1} | ... | a_{td} |

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 = (a_{0}:a_{1}: ...:a_{d}) is a point of **U** if and only if

(a_{0}:a_{1}: ...:a_{d})H^{T} = 0.
**Theorem 2.3.4**: Let **P = P**(V) be a projective space whose points are given by homogeneous coordinates (a_{0}:a_{1}: ...:a_{d}) with a_{i} in F, (a_{0},a_{1}, ...,a_{d}) (0,0, ..., 0). Then any hyperplane of **P**(V) can be represented by [b_{0}:b_{1}: ...:b_{d}] with b_{j} in F, [b_{0},b_{1}, ...,b_{d}] [0,0, ...,0]. Conversely, to any such (d+1)-tuple [b_{0}:b_{1}: ...:b_{d}] there belongs a hyperplane. Moreover, we have

(a_{0}:a_{1}:...:a_{d}) I [b_{0}:b_{1}: ...:b_{d}] iff a_{0}b_{0} + a_{1}b_{1} + ... + a_{d}b_{d} = 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 x_{0} = 0. Then the affine space **A = P**\ can be described as follows:

- The points of
**A** are the vectors (a_{1},...,a_{d}) of the d-dimensional vector space F^{d};
- the lines of
**A** are the cosets of the 1-dimensional subspaces of F^{d}; that is the set u + <v>, where u,v in F^{d}, v 0;
- incidence is set-theoretical containment.