Page 456 - Advanced Linear Algebra
P. 456
440 Advanced Linear Algebra
Now let £%< c 2 . Then
% ¤ 2¬ º%» b 2~ =
¬ % b / for some £ -Á 2
¬ %/
Thus, % < q/ for some £ - . Hence, the flat %b²< q2³ lies in /
and
dim² % b ²< q 2³³ ~ dim²< q 2³ ~ dim²<³ c
<
which implies that span² % b ²< q 2³³ ~ º %» b ²< q 2³ lies in and has
the same dimension as . In other words,
<
span² % b ²< q 2³³ ~ º %» b ²< q 2³ ~ <
We leave proof of the remaining parts of the theorem as exercises.
Exercises
'
'
1. Show that if % Á Ã Á % = , then the set : ~¸ % ~ ¹ is a
subspace of .
=
2. Prove that if ? = is nonempty then
affhull²?³ ~ % b º? c %»
3. Prove that the set ? ~ ¸² Á ³Á ² Á ³Á ² Á ³¹ in ²{ ³ is closed under the
formation of lines, but not affine hulls.
4. Prove that a flat contains the origin if and only if it is a subspace.
5. Prove that a flat is a subspace if and only if for some % ? we have
?
% ? for some £ -.
6. Show that the join of a collection 9 ~¸% b : 2¹ of flats in is the
=
intersection of all flats that contain all flats in .
9
7. Is the collection of all flats in a lattice under set inclusion? If not, how
=
can you “fix” this?
8. Suppose that ?~ % b : and @ ~ & b ; . Prove that if dim ²?³ ~ dim ²@ ³
and ? @ , then : ~ ; .
9. Suppose that ?~ % b : and @ ~ & b ; are disjoint hyperplanes in = .
Show that :~ ; .
10. (The parallel postulate) Let be a flat in and ¤ = # ? . Show that there is
?
exactly one flat containing , parallel to and having the same dimension
?
#
as .
?
)
11. a Find an example to show that the join ?v @ of two flats may not be
the set of all lines connecting all points in the union of these flats.
)
b Show that if ? and are flats with ? q @ £ J , then ? v @ is the
@
union of all lines %& where % ? and & @ .
12. Show that if ? @ and ? q @ ~ J , then
dim²? v @ ³ ~ max dim²?³Á dim²@ ³¹ b
¸
