Page 139 - Advanced Linear Algebra

Modules I: Basic Properties 123

A left-invertible homomorphism must be injective, since

~ ¬ 3 k ~ 3 k ¬ ~
Also, a right-invertible homomorphism ¢( ¦ ) must be surjective, since if
 ), then
² ³µ  im

~ ´ 9 ² ³
For set functions, the converses of these statements hold: is left-invertible if

and only if it is injective and is right-invertible if and only if it is surjective.

However, this is not the case for -maps.
9
be an injective -map. Referring to Figure 4.1,
Let ¢4 ¦ 4 9

H

V_ im(V)
im(V)
)
V_ im(V) -1
M M 1
Figure 4.1
the map im²³ ¢ 4 š im ²³ obtained from by restricting its range to im ²³ is

O
an isomorphism and the left inverses of are precisely the extensions of
3
³ ¢ im

²O im²³ c ²³ š 4 to 4 . Hence, Theorem 4.14 says that the

correspondence

³ with kernel
/ª extension of ² O im²³ c /

is a bijection from the complements of im / ² ³ onto the left inverses of .

be a surjective -map. Referring to Figure 4.2,
9
Now let ¢4 ¦ 4
ker(V)
H V| H
V =(V| ) -1
M R H M 1
Figure 4.2
if ker²³ is complemented, that is, if

4~ ker ² ³ l /