Page 455 - Advanced Linear Algebra

Affine Geometry 439

7
I
Theorem 16.12 The linear span function 7¢ ²/³ ¦ ²= ³ from the affine
geometry 7 to the projective geometry I²/³ ²= ³ defined by 7? ~ º?»
satisfies the following properties:
)
1 The linear span function is injective, with inverse given by
7 c < ~ < q /
for all subspaces not contained in the base subspace of .
2
<
/
)
2 The image of the span function is the set of all subspaces of that are not
=
contained in the base subspace of .
2
/
)
3 ?‹ @ if and only if º?» ‹ º@ »
)
4 If ? are flats in with nonempty intersection, then
/
span4 5 ? span ? ² ~ ³
2 2
)
5 For any collection of flats in ,
/
span ? 9 ~ span² ? ³
8
2 2
6 The linear span function preserves dimension, in the sense that
)
²
pdim span ? ² ³ ³ ~ ² ?dim ³
)
7 ?” @ if and only if one of º?» q 2 and º@ » q 2 is contained in the
other.
)
Proof. To prove part 1 , let %b: be a flat in / . Then %  / and so
/ ~ % b 2, which implies that : ‹ 2. Note also that º% b :» ~ º%» b : and
'  º% b:»q/ ~ ²º%»b:³q²%b2³ ¬ ' ~ %b ~ %b
,
for some : 2 and  - . This implies that ² c ³%2 , which
implies that either %2 or ~ . But %/ implies % ¤2 and so ~ ,
which implies that ' ~ %b  %b: . In other words,
º% b :» q / ‹ % b :

Since the reverse inclusion is clear, we have
º% b :» q / ~ % b :
This establishes 1 .
)
To prove 2 , let be a subspace of that is not contained in 2 . We wish to
)
=
<
show that is in the image of the linear span function. Note first that since
<
<‹“2 and dim ²2³ ~ dim ²= ³ c , we have < b 2 ~ = and so
dim²< q 2³ ~ dim²<³ b dim²2³ c dim²< b 2³ ~ dim²<³ c