Page 121 - Advanced Linear Algebra
P. 121
The Isomorphism Theorems 105
as being in B²= Á > ³ . Using these identifications, we do have equality in the
finite-dimensional case.
Theorem 3.18 Let and > be finite-dimensional and let ² = Á >B ³ . If we
=
identify = ii with = and > ii with > using the natural maps, then dd is
identified with .
%
Proof. For any %= let the corresponding element of = ii be denoted by and
similarly for > . Then before making any identifications, we have for # = ,
dd ²#³² ³ ~ d ² #´ ³µ ~ #² ³ ~ ² #³ ~ #² ³
for all > * and so
dd ²#³ ~ # > ii
Therefore, using the canonical identifications for both = ii and > ii we have
dd ²#³ ~ #
for all #= .
The next result describes the kernel and image of the operator adjoint.
Theorem 3.19 Let ²= Á > ³ . Then
B
1) ker ² d ³ ~ im ³ ²
2)im ² d ³ ~ ² ker ³
)
Proof. For part 1 ,
ker ² d ³ ~ ¸ > i d ² ³ ~ ¹
i
~ ¸ > ² = ³ ~ ¸ ¹¹
i
~ ¸ > ²im ² ³³ ~ ¸ ¹¹
~ ² im ³
)
For part 2 , if ~ ~ d im² d ³ , then ker ² ³ ker ² ³ and so
ker ² ³ .
For the reverse inclusion, let ker ² ³ = i . We wish to show that
i
~ d ~ for some > . On 2 ~ ker ² ³, there is no problem since
and d ~ agree on 2 for any > i . Let be a complement of ker ² . ³
:
8
:
Then maps a basis ~¸ 0¹ for to a linearly independent set
8 ~¸ 0¹
in > and so we can define > i on 8 by setting
² ³ ~
and extending to all of > . Then ~ ~ d on and therefore on . Thus,
:
8
~ d im ² d ³.
