Page 105 - Advanced Linear Algebra
P. 105
The Isomorphism Theorems 89
Theorem 3.1 Let be a subspace of . The binary relation
=
:
"# ¯ " c #:
is an equivalence relation on , whose equivalence classes are the cosets
=
# b : ~¸# b :¹
=
:
=
:
of in . The set ° = : of all cosets of in , called the quotient space of =
modulo , is a vector space under the well-defined operations
:
²"b:³ ~ "b:
²" b:³b²#b:³ ~ ²"b#³b:
The zero vector in =°: is the coset b : ~ : .
The Natural Projection and the Correspondence Theorem
:
=
If is a subspace of , then we can define a map : ¢ = ¦ = ° : by sending
each vector to the coset containing it:
: ²#³ ~ # b :
This map is called the canonical projection or natural projection of onto
=
=°:, or simply projection modulo : . (Not to be confused with the projection
(
operators :Á; .) It is easily seen to be linear, for we have writing for : )
² " b #³ ~ ² " b #³ b : ~ ²" b :³ b ²# b :³ ~ ²"³ b ²#³
The canonical projection is clearly surjective. To determine the kernel of , note
that
# ker ² ³ ¯ ²#³ ~ ¯# b : ~: ¯#:
and so
ker²³ ~ :
Theorem 3.2 The canonical projection : ¢= ¦ = °: defined by
: ²#³ ~ # b :
is a surjective linear transformation with ker ² ³ : ~ . :
=
If is a subspace of , then the subspaces of the quotient space ° = : have the
:
form ;°: for some intermediate subspace satisfying : ; = . In fact, as
;
provides a one-to-one
shown in Figure 3.1, the projection map :
correspondence between intermediate subspaces : ; = and subspaces of
the quotient space =°: . The proof of the following theorem is left as an
exercise.
