Page 400 - Advanced Linear Algebra

384 Advanced Linear Algebra

Z

²# Á ÃÁ# c Á"Á# b ÁÃÁ# ³ b ²# ÁÃÁ# c Á" Á# b ÁÃÁ# ³

Z
c²# Á Ã Á # c Á " b " Á # b Á Ã Á # ³

, Z for £ . The quotient space °: is the
<
for Á  - "Á "  = and #  =
and the tensor map is the map
tensor product of =Á Ã Á =
!²#Á Ã Á # ³ ~ ²#Á Ã Á # ³ b :

As before, we denote the coset ²# ÁÃÁ# ³ b : by # n Ä n # and so any

is a sum of decomposable tensors, that is,
element of =n Ä n =

#n Ä n #
where the vector space operations are linear in each variable.
Here are some of the basic properties of multiple tensor products. Proof is left to
the reader.
Theorem 14.12 The tensor product has the following properties. Note that all
vector spaces are over the same field .
-
1 )(Associativity ) There exists an isomorphism
¢ ²= n Än= ³n²> nÄn> ³ ¦ = nÄn= n> nÄn>

for which

´²# n Än# ³n²$ nÄn$ ³µ ~ # nÄn# n$ nÄn$

In particular,
²< n= ³n> š < n²= n> ³ š < n= n>
2 )(Commutativity ) Let be any permutation of the indices ¸ Á Ã Á ¹ . Then

there is an isomorphism
¢ = nÄn= ¦ = ² ³ nÄn= ² ³

for which

²# nÄn# ³ ~ # ² ³ n Än# ² ³
)
3 There is an isomorphism ¢- n = ¦ = for which
² n #³ ~ #
and similarly, there is an isomorphism ¢= n - ¦ = for which
²# n ³ ~ #
Hence, -n = š = š = n - .
The analog of Theorem 14.4 is the following.