Page 107 - Advanced Linear Algebra

The Isomorphism Theorems 91

W
V W

S s
W'

V/S
Figure 3.2: The universal property

The universal property states that if ker²³ Œ : , then there is a unique

Z
¢= °: ¦ > for which
Z k : ~

Another way to say this is that any such ²= Á > ³ can be factored through
B
.
the canonical projection :
Theorem 3.4 Let : be a subspace of = and let  B ² = Á > ³ satisfy
:‹ ker ² ³. Then, as pictured in Figure 3.2, there is a unique linear

Z
transformation ¢= °: ¦ > with the property that
Z k : ~
Z
Z
Moreover, ker²³ ~ ker² ³°: and im ²³ ~ im ² . ³

Proof. We have no other choice but to define by the condition Z k : ~ ,
Z

that is,
Z ²# b :³ ~ #
This function is well-defined if and only if
# b :~ " b :¬ Z ²# b :³ ~ Z ²" b :³
which is equivalent to each of the following statements:

# b :~ " b :¬ # ~ "
#c"  : ¬ ²#c"³ ~

%: ¬ %~

:‹ ker ² ³

Z
Thus, ¢= °: ¦ > is well-defined. Also,
Z
Z
im² ³ ~¸ ²# b :³“#= ¹~¸ #“#= ¹~ im² ³

and