Page 443 - Advanced Linear Algebra
P. 443
Chapter 16
Affine Geometry
In this chapter, we will study the geometry of a finite-dimensional vector space
= , along with its structure-preserving maps. Throughout this chapter, all vector
spaces are assumed to be finite-dimensional.
Affine Geometry
The cosets of a quotient space have a special geometric name.
Definition Let be a subspace of a vector space . The coset
:
=
# b : ~¸# b :¹
is called a flat in with base and flat representative . We also refer to
:
=
#
#b: as a translate of :. The set 7 ²= ³ of all flats in is called the affine
=
geometry of . The dimension dim ² ²= ³³ of ²= ³ is defined to be dim ²= . ³
=
7
7
While a flat may have many flat representatives, it only has one base since
%b: ~ & b ; implies that % & b; and so %b: ~ & b ; ~ %b; ,
whence :~ ; .
Definition The dimension of a flat %b: is dim ²:³ . A flat of dimension is
-flat
called a . A -flat is a point , a -flat is a line and a -flat is a plane . A flat
of dimension dim²²= ³³ c is called a hyperplane .
7
Definition Two flats ?~ % b : and @ ~ & b ; are said to be parallel if
: ; or ; :. This is denoted by ? @ .
We will denote subspaces of = by the letters :Á;ÁÃ and flats in = by
?Á @ Á Ã .
Here are some of the basic intersection properties of flats.
