Page 58 - Advanced Linear Algebra

42 Advanced Linear Algebra

)
(
1 )(Join of the family ) is the sum join of the family :
=
<
=~ :
0
2 )(Independence of the family ) For each 0 ,
p s
: q : ~ ¸ ¹

q £ t
In this case, each is called a direct summand of . If < ~ ¸ : Á Ã Á : ¹ is a
:
=
finite family, the direct sum is often written
=~ : l Ä l :
=
:
;
Finally, if =~ : l ; , then is called a complement of in .
Note that the condition in part 2) of the previous definition is stronger than
saying simply that the members of are pairwise disjoint:
<
:q : ~ J

for all £ 0 .
A word of caution is in order here: If and are subspaces of , then we may
;
=
:
always say that the sum :b ; exists. However, to say that the direct sum of :
and exists or to write : l ; is to imply that : q ; ~ ¸ ¹ . Thus, while the
;
sum of two subspaces always exists, the direct sum of two subspaces does not
always exist. Similar statements apply to families of subspaces of .
=
The reader will be asked in a later chapter to show that the concepts of internal
(
)
and external direct sum are essentially equivalent isomorphic . For this reason,
the term “direct sum” is often used without qualification.
Once we have discussed the concept of a basis, the following theorem can be
easily proved.

Theorem 1.4 Any subspace of a vector space has a complement, that is, if is a
:
subspace of , then there exists a subspace for which = ~ : l . ;
;
=
It should be emphasized that a subspace generally has many complements
(although they are isomorphic ). The reader can easily find examples of this in
s .

We can characterize the uniqueness part of the definition of direct sum in other
:
;
useful ways. First a remark. If and are distinct subspaces of = and if
%Á &  : q ; , then the sum % b & can be thought of as a sum of vectors from the