Page 108 - Advanced Linear Algebra
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92 Advanced Linear Algebra
ker²³ ~ ¸# b : Z Z ²# b :³ ~ ¹
~¸# b : #~ ¹
~¸# b : # ker² ³¹
~ ker ³ ² ° :
Z
The uniqueness of is evident.
Theorem 3.4 has a very important corollary, which is often called the first
isomorphism theorem and is obtained by taking :~ ker ² . ³
(
Theorem 3.5 The first isomorphism theorem ) Let ¢= ¦ > be a linear
transformation. Then the linear transformation ¢= °ker ² ³ ¦ > defined by
Z
²# b ker
Z ² ³³ ~ #
is injective and
= ² im
ker ²³ ³
According to Theorem 3.5, the image of any linear transformation on = is
=
isomorphic to a quotient space of . Conversely, any quotient space ° = : of =
=
is the image of a linear transformation on : the canonical projection : . Thus,
up to isomorphism, quotient spaces are equivalent to homomorphic images.
Quotient Spaces, Complements and Codimension
The first isomorphism theorem gives some insight into the relationship between
complements and quotient spaces. Let be a subspace of and let be a
:
;
=
complement of , that is,
:
=~ : l ;
Applying the first isomorphism theorem to the projection operator ;Á: ¢= ¦ ;
gives
; = °:
Theorem 3.6 Let be a subspace of = . All complements of in = : are
:
isomorphic to =°: and hence to each other.
The previous theorem can be rephrased by writing
(l ) ~ (l * ¬ ) *
On the other hand, quotients and complements do not behave as nicely with
respect to isomorphisms as one might casually think. We leave it to the reader to
show the following:
