Any linear set of a projective space **P** is a (possibly degenerate) projective space. The induced space *P*() is a ** subspace** of

*Examples*: , singletons, the set of points on a line, the set of all points of the space are linear sets.

Since the intersection of an arbitrary number of linear sets is a linear set, we can define, for any subset of points,

**Def**: A set of points is called ** collinear** if all points of are incident with a common line. The set is called

The span of a set of three noncollinear points is called a

**Theorem 1.3.1**: (*Greedy Algorithm*) Let be a nonempty linear set of **P**, and let P be a point of **P**. Then

**Theorem 1.3.2**: (*Exchange Property*) Let be a linear set of **P**, and let P be a point of **P** that does not lie in . Then,

**Def**: A set of points of **P** is called ** independent** if for any s/set ' and any point P in \' we have P <>. (When P <\{P}> we say that is

*Examples*: Any singleton, any pair of points, 3 noncollinear points, or any subset of an independent set are independent sets.

**Def**: An independent set of **P** that spans **P** is called a ** basis** of

**Theorem 1.3.3**: A set of **P** is a basis of **P** iff is a minimal spanning set.

**Def**: A projective space **P** is called ** finitely generated** if there exists a finite set of points which span

In the following we shall assume that **P** is finitely generated.

**Theorem 1.3.4**: Let be a finite spanning set of **P**. Then there exists a basis of **P** such that . In particular, **P** has a finite basis.

**Lemma 1.3.5**: Let be an independent set of points of **P**, _{1} and _{2} subsets of . If is finite, then

**Lemma 1.3.6**: (*Exchange Lemma*) Let be a finite basis of **P**, and let P be a point of **P**. Then there is a point Q in with the property that the set

**Theorem 1.3.7**: (*Steinitz exchange theorem for projective spaces*) Let be a finite basis of **P**, and let r = ||. If is an independent set having s points then we have:

- sr
- There is a subset * of with |*| = r - s such that * is a basis of
**P**.

**Corollary 1.3.8**: (*Basis Extension Theorem*) Let **P** be a finitely generated projective space. Any two bases of **P** have the same number of elements. Moreover, any independent set (in particular any basis of a subspace) can be extended to a basis of **P**.

**Def**: If the number of elements in a basis of **P** (and hence all bases) is d + 1, then d is called the ** dimension** of

**Lemma 1.3.9**: Let be a subspace of the finitely generated projective space **P**.

- dim() dim(
**P**), - dim() = dim(
**P**) iff =**P**.

**Def**: Let **P** be a projective space of dimension d. Subspaces of dimension 2 are called ** planes** and subspaces of dimension d-1 are called

We denote the set of all subspaces of

**Def**: The empty set and the whole space are called ** trivial** subspaces. The set of all nontrivial subspaces is denoted by *(P).

Note that (*(**P**), ) is a geometry of rank d.

**Lemma 1.3.10**: Let **P** be a d-dimensional projective space and let **U** be a t-dimensional subspace of **P** (-1td). Then there exist d - t hyperplanes of **P** such that **U** is the intersection of these hyperplanes.

**Theorem 1.3.11**: (*Dimension Formula*) Let **U** and **W** be subspaces of **P**. Then

**Corollary 1.3.12**: Let **P** be a projective space and let **H** be a hyperplane of **P**. Then for any subspace **U** of **P** either

**U****H**, or- dim(
**U****H**) = dim(**U**) - 1.

Note that we have defined projective planes in two ways, you should show that these two definitions coincide. (Homework problem #9)