Page 245 - Advanced Linear Algebra

Structure Theory for Normal Operators 229

then

º Á ² ³O ²!³» ~ º Á i : i !» ~ º Á !» ~ º O ² ³Á !»

:
i

Hence, by definition of adjoint, ²³O ~ ² O ³ i .

:
:
Now let us relate the kernel and image of a linear transformation to those of its
adjoint.
B
=
Theorem 10.3 Let ²= Á > ³ , where and > are finite-dimensional inner

product spaces.
1)
ker²³ ~ im ² ³ ž im ²³ ~ ker² ³ ž
i
and
i

and so
¯ surjective i injective
¯ injective i surjective
2)
ker ² i ³ ~ ker ³ and ker ² i ³ ~ ker i ³
²
²
3)
im ² i ³ ~ im i ³ and im ² i ³ ~ im ³
²
²
4)
² ³ i ~ :Á; ;Á: ž
ž
Proof. For part 1),
" ker ² ³ ¯ i i " ~
i

¯ º "Á = » ~ ¸ ¹
¯ º"Á = » ~ ¸ ¹

¯" im ² ³ ž

and so ker²³ ~ im ² ³ ž . The second equation in part 1) follows by replacing
i

i
by and taking complements.

For part 2), it is clear that ker²³ ‹ ker² i ³ . For the reverse inclusion, we have

i
i
" ~ ¬ º "Á "» ~ ¬ º "Á "» ~ ¬ " ~
and so ker ² i ³ ‹ ker ³ . The second equation follows from the first by
²
i
replacing with . We leave the rest of the proof for the reader.