# Projective space

The collection of all subspaces of an incidence system $\pi = \{ {\mathcal P} , {\mathcal L} , \textrm{ I } \}$, where the elements of the set ${\mathcal P}$ are called points, the elements of the set ${\mathcal L}$ are called lines and I is the incidence relation. A subspace of $\pi$ is defined to be a subset $S$ of ${\mathcal P}$ for which the following condition holds: If $p , q \in S$ and $p \neq q$, then the set of points of the line passing through $p$ and $q$ also belongs to $S$. The incidence system $\pi$ satisfies the following requirements:

1) for any two different points $p$ and $q$ there exists a unique line $L$ such that $p \textrm{ I } L$ and $q \textrm{ I } L$;

2) every line is incident to at least three points;

3) if two different lines $L$ and $M$ intersect at a point $p$ and if the following four relations hold: $q \textrm{ I } L$, $r \textrm{ I } L$, $s \textrm{ I } M$, $l \textrm{ I } M$, then the straight lines passing through the pairs of points $r , l$ and $s , q$ intersect.

A subspace $S$ is generated by a set $s$ of points in ${\mathcal P}$( written $S = \langle s \rangle$) if $S$ is the intersection of all subspaces containing $s$. A set $s$ of points is said to be independent if for any $x \in s$ one has $x \notin \langle s \setminus \{ x \} \rangle$. An ordered maximal and independent set of points of a subspace $S$ is called a basis of $S$, and the number $d ( S)$ of its elements is called the dimension of the subspace $S$. A subspace of dimension $0$ is a point, a subspace of dimension $1$ is a projective straight line, a subspace of dimension $2$ is called a projective plane.

In a projective space the operations of addition and intersection of spaces are defined. The sum $P _ {m} + P _ {k}$ of two subspaces $P _ {m}$ and $P _ {k}$ is defined to be the smallest of the subspaces containing both $P _ {m}$ and $P _ {k}$. The intersection $P _ {m} \cap P _ {k}$ of two subspaces $P _ {m}$ and $P _ {k}$ is defined to be the largest of the subspaces contained in both $P _ {m}$ and $P _ {k}$. The dimensions of the subspaces $P _ {m}$, $P _ {k}$, of their sum, and of their intersection are connected by the relation

$$m + k = d ( P _ {m} \cap P _ {k} ) + d ( P _ {m} + P _ {k} ) .$$

For any $P _ {m}$ there is a $P _ {n-} m- 1$ such that $P _ {m} \cap P _ {n-} m- 1 = P _ {-} 1 = \emptyset$ and $P _ {m} + P _ {n-} m- 1 = P _ {n}$( $P _ {n-} m- 1$ is a complement of $P _ {m}$ in $P _ {n}$), and if $P _ {m} \subset P _ {r}$, then

$$( P _ {m} + P _ {k} ) \cap P _ {r} = P _ {m} + P _ {k} \cap P _ {r}$$

for any $P _ {k}$( Dedekind's rule), that is, with respect to the operation just introduced the projective space is a complemented modular lattice.

A projective space of dimension exceeding two is Desarguesian (see Desargues assumption) and hence is isomorphic to a projective space (left or right) over a suitable skew-field $k$. The (for example) left projective space $P _ {n} ^ {l} ( k)$ of dimension $n$ over a skew-field $k$ is the collection of linear subspaces of an $( n+ 1)$- dimensional left linear space $A _ {n+} 1 ^ {l} ( k)$ over $k$; the points of $P _ {n} ^ {l} ( k)$ are the lines of $A _ {n+} 1 ^ {l} ( k)$, i.e. the left equivalence classes of rows $( x _ {0} \dots x _ {n} )$ consisting of elements of $k$ which are not simultaneously equal to zero (two rows $( x _ {0} \dots x _ {n} )$ and $( y _ {0} \dots y _ {n} )$ are left equivalent if there is a $\lambda \in k$ such that $x _ {i} = \lambda y _ {i}$, $i = 0 \dots n$); the subspaces $P _ {m} ^ {l} ( k)$, $m = 1 \dots n$, are the $( m+ 1)$- dimensional subspaces $A _ {m+} 1 ^ {l} ( k)$. It is possible to establish a correspondence between a left $P _ {n} ^ {l} ( k)$ and a right $P _ {n} ^ {r} ( k)$ projective space under which to a subspace $P _ {s} ^ {l} ( k)$ corresponds $P _ {n-} s- 1 ^ {r} ( k)$( the subspaces $P _ {s} ^ {l} ( k)$ and $P _ {n-} s- 1 ^ {r} ( k)$ are called dual to one another), to an intersection of subspaces corresponds a sum, and to a sum corresponds an intersection. If an assertion based only on properties of linear subspaces, their intersections and sums is true for $P _ {n} ^ {l} ( k)$, then the corresponding assertion is true for $P _ {n} ^ {r} ( k)$. This correspondence between the properties of the spaces $P _ {n} ^ {r} ( k)$ and $P _ {n} ^ {l} ( k)$ is called the duality principle for projective spaces (see ).

A finite skew-field is necessarily commutative; consequently, a finite projective space of dimension exceeding two and of order $q$ is isomorphic to the projective space $\mathop{\rm PG} ( n , q )$ over the Galois field. The finite projective space $\mathop{\rm PG} ( n , q )$ contains $( q ^ {n+} 1 - 1 ) / ( q - 1 )$ points and $\prod _ {i=} 0 ^ {r} ( q ^ {n+} 1- i - 1 ) / ( q ^ {r+} 1- i - 1 )$ subspaces of dimension $r$( see ).

A collineation of a projective space is a permutation of its points that maps lines to lines so that subspaces are mapped to subspaces. A non-trivial collineation of the projective space has at most one centre and at most one axis. The group of collineations of a finite projective space $\mathop{\rm PG} ( n , p ^ {h} )$ has order

$$hp ^ {hn(} n+ 1)/2 \prod _ { i= } 1 ^ { n+ } 1 ( p ^ {hi} - 1 ) .$$

Every projective space $\mathop{\rm PG} ( n , q )$ admits a cyclic transitive group of collineations (see ).

A correlation $\delta$ of a projective space is a permutation of subspaces that reverses inclusions, that is, if $S \subset T$, then $S ^ \delta \supset T ^ \delta$. A projective space admits a correlation only if it is finite-dimensional. An important role in projective geometry is played by the correlations of order two, also called polarities (Polarity).

How to Cite This Entry:
Projective space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_space&oldid=48327
This article was adapted from an original article by V.V. Afanas'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article