Page 454 - Advanced Linear Algebra

438 Advanced Linear Algebra

planes º3 » and º3 » have the property that their intersection is a line through

the origin, even if the lines are parallel . We are now ready to define projective
geometries.
Let be a vector space of any dimension and let be a hyperplane in not
=
=
/
containing the origin. To each flat in , we associate the subspace ? º » of =
?
/
?
generated by . Thus, the linear span function ¢ 7 7 ² / ³ ¦ I ² = ³ maps affine
=
subspaces of to subspaces ? º » of . The span function is not surjective:
?
/
Its image is the set of all subspaces that are not contained in the base subspace
2 / of the flat .
The linear span function is one-to-one and its inverse is intersection with ,
/
7 c < ~ < q /
for any subspace not contained in .
2
<
The affine geometry 7²/³ is, as we have remarked, somewhat incomplete. In
the case dim²/³ ~ , every pair of points determines a line but not every pair
of lines determines a point.
7
Now, since the linear span function is injective, we can identify ² 7 / ³ with
7
=
its image 7² ²/³³ , which is the set of all subspaces of not contained in the
base subspace 2 . This view of 7 ² / ³ allows us to “complete” 7 ² / ³ by
including the base subspace . In the three-dimensional case of Figure 16.1, the
2
base plane, in effect, adds a projective line at infinity. With this inclusion, every
pair of lines intersects, parallel lines intersecting at a point on the line at infinity.
This two-dimensional projective geometry is called the projective plane .
Definition Let = be a vector space. The set ² I = ³ of all subspaces of is
=
called the projective geometry of . The projective dimension pdim ² : ³ of

=
: ²= ³ is defined as
I
pdim²:³ ~ dim ²:³ c
The projective dimension of F²=³ is defined to be pdim ²=³ ~ dim ²=³ c . A
subspace of projective dimension is called a projective point and a subspace

of projective dimension is called a projective line .

Thus, referring to Figure 16.1, a projective point is a line through the origin and,
provided that it is not contained in the base plane 2 , it meets / in an affine
point. Similarly, a projective line is a plane through the origin and, provided that
it is not , it will meet in an affine line. In short,
2
/
span ² ³affine point ~ line through the origin ~ projective point
span ² ³affine line ~ plane through the origin ~ projective line

The linear span function has the following properties.