Page 109 - Advanced Linear Algebra
P. 109
The Isomorphism Theorems 93
)
1 It is possible that
(l ) ~ * l +
°
with ( * but ) + . Hence, ( * does not imply that a complement
of is isomorphic to a complement of .
(
*
)
2 It is possible that = > and
=~ : l ) and > ~ : l +
(
°
but ) + . Hence, = > does not imply that = °: > °: . However,
according to the previous theorem, if equals > then ) + .)
=
Corollary 3.7 Let be a subspace of a vector space . Then
=
:
dim²= ³ ~ dim²:³ b dim²= °:³
Definition If is a subspace of , then dim ² = ° : ³ is called the codimension of
:
=
: = in and is denoted by codim ² : ³ or codim ² : = ³ .
Thus, the codimension of in is the dimension of any complement of in =
:
=
:
and when is finite-dimensional , we have
=
codim = ²:³ ~ dim ²= ³ c dim ²:³
(This makes no sense, in general, if is not finite-dimensional, since infinite
=
cardinal numbers cannot be subtracted.)
Additional Isomorphism Theorems
There are other isomorphism theorems that are direct consequences of the first
isomorphism theorem. As we have seen, if =~ : l ; then = °; : . This can
be written
:l ; :
; : q ;
This applies to nondirect sums as well.
Theorem 3.7 The second isomorphism theorem ) Let be a vector space
(
=
and let and be subspaces of . Then
;
:
=
:b ; :
; : q ;
Proof. Let ¢ ²:b ;³ ¦ :°²:q ;³ be defined by
² b !³ ~ b ²: q ;³
We leave it to the reader to show that is a well-defined surjective linear
transformation, with kernel . An application of the first isomorphism theorem
;
then completes the proof.
