Page 106 - Advanced Linear Algebra
P. 106
90 Advanced Linear Algebra
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Figure 3.1: The correspondence theorem
(
Theorem 3.3 The correspondence theorem) Let be a subspace of . Then
=
:
the function that assigns to each intermediate subspace : ; = the
(
subspace ;°: of = °: is an order-preserving with respect to set inclusion)
one-to-one correspondence between the set of all subspaces of containing :
=
and the set of all subspaces of =°: .
Proof. We prove only that the correspondence is surjective. Let
?~ ¸" b : " <¹
be a subspace of =°: and let be the union of all cosets in :
;
?
;~ ²" b :³
"<
We show that : ; = and that ;°:~ ? . If %Á & ; , then % b : and
&b : are in ? and since ? =°:, we have
% b:Á ²%b&³ b: ?
which implies that %Á % b & ; . Hence, is a subspace of containing .
:
;
=
Moreover, if !b: ;°: , then ! ; and so !b: ? . Conversely, if
"b: ?, then " ; and therefore "b: ;°:. Thus, ? ~ ;°:.
The Universal Property of Quotients and the First
Isomorphism Theorem
:
Let be a subspace of . The pair ² = = ° : Á : ³ has a very special property,
known as the universal property —a term that comes from the world of category
theory.
Figure 3.2 shows a linear transformation B ²= Á > ³ , along with the
from to the quotient space = ° . :
=
canonical projection :
