Page 118 - Advanced Linear Algebra
P. 118
102 Advanced Linear Algebra
Definition Let 4 be a nonempty subset of a vector space . The annihilator
=
4 4 of is
i
4 ~ ¸ = ²4³ ~ ¸ ¹¹
The term annihilator is quite descriptive, since 4 consists of all linear
(
)
functionals that annihilate send to every vector in 4 . It is not hard to see
that 4 is a subspace of = i , even when 4 is not a subspace of .
=
The basic properties of annihilators are contained in the following theorem.
Theorem 3.14
1 )(Order-reversing ) If 4 5 and are nonempty subsets of , then
=
4 5 ¬ 5 4
)
2 If dim²= ³ B , then for any nonempty subset 4 of the natural map
=
¢ span²4³ 4
is an isomorphism from span²4³ onto 4 . In particular, if is a
:
subspace of , then : . :
=
3 If and are subspaces of , then
)
;
=
:
²: q;³ ~ : b ; and ²: b ;³ ~ : q;
Proof. We leave proof of part 1 for the reader. For part 2 , since
)
)
4 ~ ² ² span 4 ³ ³
it is sufficient to prove that ¢: : is an isomorphism, where : is a
subspace of . Now, we know that is a monomorphism, so it remains to prove
=
that . If : , then ~ has the property that for all : ,
: ~ :
² ³ ~ ~
and so ~ : , which implies that : : . Moreover, if # : , then
for all : we have
²#³ ~ #² ³ ~
:
and so every linear functional that annihilates also annihilates . But if ¤ # # , :
then there is a linear functional = i for which ²:³ ~ ¸ ¹ and ²#³ £ .
(We leave proof of this as an exercise. ) Hence, #: and so #~ # : and
so : : .
)
For part 3 , it is clear that annihilates b : ; if and only if annihilates both
: ; and . Hence, ²: b;³ ~ : q ; . Also, if ~ b : b; where
: and ; , then Á ²: q ;³ and so ²: q ;³ . Thus,
