Page 59 - Advanced Linear Algebra
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Vector Spaces 43
same subspace (say ) or from different subspaces—one from and one from
:
:
; #. When we say that a vector cannot be written as a sum of vectors from the
;
:
distinct subspaces and , we mean that cannot be written as a sum b % &
#
where and can be interpreted as coming from different subspaces, even if
%
&
they can also be interpreted as coming from the same subspace. Thus, if
%Á & : q ; , then # ~ % b & does express # as a sum of vectors from distinct
subspaces.
Theorem 1.5 Let < ~¸: 0¹ be a family of distinct subspaces of . The
=
following are equivalent:
1 )(Independence of the family ) For each 0 ,
p s
: q : ~ ¸ ¹
q £ t
2 )(Uniqueness of expression for ) The zero vector cannot be written as a
sum of nonzero vectors from distinct subspaces of .
<
3 )(Uniqueness of expression ) Every nonzero #= has a unique, except for
order of terms, expression as a sum
# ~ bÄb
of nonzero vectors from distinct subspaces in .
<
Hence, a sum
=~ :
0
)–
is direct if and only if any one of 1 3 holds.
)
Proof. Suppose that 2) fails, that is,
~ b Ä b
where the nonzero 's are from distinct subspaces : . Then and so
c ~ bÄb
which violates 1). Hence, 1) implies 2). If 2) holds and
# ~ b Äb # ~ ! bÄb! and
where the terms are nonzero and the 's belong to distinct subspaces in and
<
similarily for the 's, then
!
~ b Äb c! cÄc!
By collecting terms from the same subspaces, we may write
~ ² c! ³bÄb² c ! ³b b bÄb c! b cÄc!
