Page 447 - Advanced Linear Algebra

Affine Geometry 431

' • ²& c&³ b!²& c&³

~ 4 Á b ! Á c ² b !³ Á 5 %
2
~

~ 4 Á b ! Á c² b!c ³ Á 5 % c &
2

~
which is in (c & ~ : , since the last sum is an affine sum. Hence, is a
:
subspace of . We leave the rest of the proof to the reader.
=
The Lattice of Flats
The intersection of subspaces is a subspace, although it may be trivial. For flats,
if the intersection is not empty, then it is also a flat. However, since the
intersection of flats may be empty, the set 7²= ³ does not form a lattice under
intersection. However, we can easily fix this.

Theorem 16.5 Let be a vector space. The set
=
7 7 ²= ³ ~ ²= ³ r ¸J¹
of all flats in , together with the empty set, is a complete lattice in which meet
=
is intersection. In particular:
1) 7 ²= ³ is closed under arbitrary intersection. In fact, if
< ~¸% b : “ 2¹ has nonempty intersection, then

< ~ ² b : % ³ % ~ : b
2 2 2
for some % < . In other words, the base of the intersection is the
intersection of the bases.
2 The join < ) of the family < ~¸% b : “ 2¹ is the intersection of all

flats containing the members of . Also,
<
4 affhull < < ~ 5
)
3 If ?~ % b : and @ ~ & b ; are flats in , then
=
? v @ ~ % b ²º% c &» b : b ;³
If ?q @ £ J , then
?v @ ~ % b ²: b ;³
Proof. For part 1), if

% ²% b : ³
2

then %b : ~ % b : for all  2 and so