of a projective space P is a (possibly degenerate) projective space. The induced space P() is a subspace of P. of the point set 
is linear if for every pair of distinct points P and Q in , each point of the line PQ is in . I.e., (PQ)
, where if g is a line (g) denotes the set of points incident with g.
Any linear set of a projective space P is a (possibly degenerate) projective space. The induced space P() is a subspace of P.
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,
) = <> := 
{ | , a linear set}.
> is the smallest linear set containing .
Def: A set 
of points is called collinear if all points of are incident with a common line. The set is called noncollinear if there is no line incident with all the points of .
The span of a set of three noncollinear points is called a plane.
Theorem 1.3.1: (Greedy Algorithm) Let be a nonempty linear set of P, and let P be a point of P. Then
,P> = 
{(PQ) | Q 
}.
,P> intersects .
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,
<,P>\ then P <,Q>, hence <,P> = <,Q>.
that are spanned by and a point outside of intersect only in the points of .)
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 dependent.
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 P.
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 P.
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
12> = <1> <2>.
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
\{Q}){P}
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:
r* 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 P and denoted by dim(P).
Lemma 1.3.9: Let be a subspace of the finitely generated projective space P.
) dim(P),) = 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 hyperplanes.
We denote the set of all subspaces of P by (P). We call (P) together with the subset relation the projective geometry belonging to the projective space P.
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
W).
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
H, orH) = dim(U) - 1.Note that we have defined projective planes in two ways, you should show that these two definitions coincide. (Homework problem #9)