Page 393 - Advanced Linear Algebra

Tensor Products 377

necessary, that the set ¸ ²"³Á à Á ²"³¹ is a maximal linearly independent

subset of ¸ ²"³Á à Á ²"³¹ . Hence, for each € , we have

²"³ ~ Á ²"³

~
and so

~ ²"³ n ²#³

~

~ ²"³ n ²#³ b @ Á ²"³ n ²#³
A

~ ~ b ~

~ ²"³n ²#³b Á ²"³n ²#³ !

~ ~ b ~

~ ² " ³ n ² # ³b ²"³n @ Á ²#³ A

~ ~ ~ b

~ ² " ³ n @ ² # ³ b ² # Á ³ A
~ ~ b
Thus, the linear independence of ¸ ²"³Á à Á ²"³¹ implies that for each

 ,

²#³ b Á ²#³ ~

~ b
for all #= and so

b Á ~

~ b
But this contradicts the fact that the set ¸ ¹ is linearly independent. Hence, it

cannot happen that for £ and so is injective.
² ³ ~
Z
Z
The embedding of B Z B²<Á < ³ n ²= Z into BÁ = ³ ²< n = Á < n = ³ means that
each n can be thought of as the linear transformation p from < n = to
Z
<n = , defined by
Z

²p ³²" n #³ ~ " n #

In fact, the notation n is often used to denote both the tensor product of
)
(
vectors linear transformations and the linear map p , and we will do this as

well. In summary, we can say that the tensor product n of linear
transformations is (up to isomorphism) a linear transformation on tensor
products.