Page 55 - Advanced Linear Algebra
P. 55
Vector Spaces 39
Theorem 1.2 A nontrivial vector space over an infinite field is not the
-
=
union of a finite number of proper subspaces.
, where we may assume that
Proof. Suppose that =~ : r Ä r :
\
: : rÄr:
Let $ : ± ²: rÄr: ³ and let # ¤ : . Consider the infinite set
(~ ¸ $ b # -¹
#
$
which is the “line” through , parallel to . We want to show that each :
contains at most one vector from the infinite set , which is contrary to the fact
(
. This will prove the theorem.
that =~ : r Ä r :
If $ b # : for £ , then $ : implies #: , contrary to assumption.
Next, suppose that $ b # : and $ b # : , for , where £ .
Then
: ² $b#³c² $b#³ ~ ² c ³$
and so $: , which is also contrary to assumption.
To determine the smallest subspace of containing the subspaces and , we
=
:
;
make the following definition.
Definition Let and be subspaces of . The sum b : ; is defined by
:
;
=
: b ; ~ ¸" b # ":Á # ;¹
More generally, the sum of any collection ¸: 2¹ of subspaces is the set
:
of all finite sums of vectors from the union :
H b Ä c b :~ : I
2 2
It is not hard to show that the sum of any collection of subspaces of is a
=
subspace of and that the sum is the least upper bound under set inclusion:
=
:b ; ~ lub ¸:Á ;¹
More generally,
:~ lub ¸: 2¹
2
If a partially ordered set has the property that every pair of elements has a
7
least upper bound and greatest lower bound, then is called a lattice . If has
7
7
a smallest element and a largest element and has the property that every
collection of elements has a least upper bound and greatest lower bound, then 7
