Page 383 - Advanced Linear Algebra

Tensor Products 367

Bilinearity on <d = Equals Linearity on <n =
The universal property for bilinearity says that to each bilinear function
¢ <d = ¦ > , there corresponds a unique linear function ¢ <n = ¦ > ,
called the mediating morphism for . Thus, we can define the mediating

morphism map
B¢ hom
²<Á =Â > ³ ¦ ²< n =Á > ³
by setting ~ . In other words, is the unique linear map for which

² ³²" n #³ ~ ²"Á #³

Observe that is itself linear, since if Á  hom ²<Á= Â> ³ , then

´ ² ³ b ² ³µ²" n #³ ~ ²"Á #³ b ²"Á #³ ~ ² b ³²"Á #³

and so ² ³ b ² ³ is the mediating morphism for b , that is,

² ³ b ² ³ ~ ² b ³

Also, is surjective, since if ¢< n = ¦ > is any linear map, then
~ k !¢ < d = ¦ > is bilinear and has mediating morphism , that is,

~ . Finally, is injective, for if ~ , then ~ k ! ~ . We have

established the following result.
-
Theorem 14.4 Let , and > be vector spaces over . Then the mediating
<=
morphism map ²<Á =Â > ³ ¦ B¢ hom , where ²< n =Á > ³ is the unique
linear map satisfying ~ k ! , is an isomorphism and so

B¢ hom
²<Á = > ³ š ²< n =Á > ³
When Is a Tensor Product Zero?
Armed with the universal property of bilinearity, we can now discuss some of
the basic properties of tensor products. Let us first consider the question of
is zero.
when a tensor "n #
The bilinearity of the tensor product gives
n# ~ ² b ³ n# ~ n#b n#

and so n# ~ . Similarly, " n ~ . Now suppose that
"n # ~

where we may assume that none of the vectors " and # are . Let

¢ <d = ¦ > be a bilinear map and let ¢ <n = ¦ > be its mediating
morphism, that is, k! ~ . Then