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358 Advanced Linear Algebra

k

of mediating morphisms implies that ~ . Similarly k ~ and so and
are inverses of one another, making the desired bijection.
f g f
A S A T A S

W V VW L
g f f

T S S
Figure 14.3

Examples of Universality

Now let us look at some examples of the universal property. Let Vect²-³ denote
the family of all vector spaces over the base field . We use the term family
(
-
informally to represent what in set theory is formally referred to as a class. A
class is a “collection” that is too large to be considered a set. For example,
Vect²-³ is a class. )
Example 14.1 (Bases ) Let be a nonempty set and let
8
1) I ~ ² Vect - ³
2) < set functions from to members of <
8 ~
3) > ~ linear transformations

If = is a vector space with basis , then the pair = 8 8 ² 8 Á ¢ 8 ¦ = 8 ³ , where is
the inclusion map # ~ # , is universal for ² Á<> ³ . To see this, note that the
condition that  < can be factored through ,

~ k

is equivalent to the statement that for each basis vector # 8#~ # . But this
uniquely defines a linear transformation .

In fact, the universality of the pair ²= Á ³ is precisely the statement that a linear
8
transformation is uniquely determined by assigning its values arbitrarily on a

basis , the function doing the arbitrary assignment in this context. Note also
8

that Theorem 14.1 implies that if ²> Á ¢ 8 ¦ > ³ is also universal for ² Á > < , ³
= to > , that is, > =
then there is a bijective mediating morphism from 8 and 8
are isomorphic.
Example 14.2 (Quotient spaces and canonical projections ) Let be a vector
=
space and let be a subspace of . Let
2
=
1) I ~ ² Vect - ³