Page 382 - Advanced Linear Algebra

366 Advanced Linear Algebra

there is a unique linear transformation ¢- <d= ¦ > for which
k ~

Note that sends the vectors (14.1) and (14.2) that generate to the zero vector

:
and so :‹ ker ² ³ . For example,

´ ²"Á$³ b ²#Á$³ c ² " b #Á$³µ
~ ´ ²"Á$³ b ²#Á$³ c ² " b #Á$³µ

~ ²"Á $³ b ²#Á $³ c ² " b #Á $³
~ ²"Á $³ b ²#Á $³ c ² " b #Á $³
~
(
and similarly for the second coordinate. Hence, Theorem 3.4 the universal
property described in Example 14.2) implies that there exists a unique linear
transformation ¢< n = ¦ > for which
k ~
Hence,

k! ~ k k ~ k ~
Z
As to uniqueness, if k! ~ , then
Z ´²"Á#³ b :µ ~ ²"Á#³ ~ ´²"Á#³ b :µ

and since the cosets ²"Á #³ b : generate - <d= °: , we conclude that Z ~ .

Thus, is the mediating morphism and ²< n = Á !³ is universal for bilinearity.
Let us take a moment to compare the two previous constructions. Let
Z
Z
¸ “  0¹ and ¸ “  1¹ be bases for and , respectively. Let ²; Á ! ³ be
=
<

the tensor product as constructed using these two bases and let
°:Á!³ be the tensor product construction using quotient spaces.
²;Á!³ ~ ²- <d=
Since both of these pairs are universal for bilinearity, Theorem 14.1 implies that
the mediating morphism for with respect to , that is, the map ! ! Z ¢ ; Z ¦ ;
defined by
² n ³ ~ ² Á ³ b :

is a vector space isomorphism. Therefore, the basis ¸² n ³¹ of ; Z is sent to

;
the set ¸² Á ³ b :¹ , which is therefore a basis for .

=
In other words, given any two bases ¸ “  0¹ and ¸ “  1¹ for and ,
<

respectively, the tensors n form a basis for < n = , regardless of which
construction of the tensor product we use. Therefore, we are free to think of
either as a formal symbol belonging to a basis for < n = or as the coset
n
² Á ³ b : belonging to a basis for < n = .