Page 399 - Advanced Linear Algebra

Tensor Products 383

~ k !
The map is called the mediating morphism for . If ² ; Á ! ³ is universal for

multilinearity, then is called the tensor product of = ÁÃÁ= and denoted by
;
. The map ! is called the tensor map
=n Ä n = .
V u˜˜˜uV n t V ˜˜˜ V n
1
1
W
f

W
Figure 14.8
As we have seen, the tensor product is unique up to isomorphism.

The basis construction and coordinate-free construction given earlier for the
tensor product of two vector spaces carry over to the multilinear case.

=
~¸ “ 1 ¹ be a basis for for ~ Á à Á . For each
In particular, let 8 Á
ordered -tuple ² Á Ã Á Á Á ³ , construct a new formal symbol
;
n Á Ä n Á and define to be the vector space with basis
“  1 ¹
: ~¸ Á n Ä n Á
The tensor map !¢ = d Ä d = ¦ ; is defined by setting

!² Á Á Ã Á Á ³ ~ Á nÄn Á
and extending by multilinearity. This uniquely defines a multilinear map that is
!
.
universal for multilinear functions from =d Ä d =
Indeed, if ¢ = d Ä d = ¦ > is multilinear, the condition ~ k ! is

equivalent to
³
² Á nÄn Á ³ ~ ² Á Á Ã Á Á
which uniquely defines a linear map ¢; ¦ > . Hence, ²;Á!³ has the universal
property for multilinearity.

Alternatively, we may take the coordinate-free quotient space approach as
follows.

be vector spaces over and let be the vector space
Definition Let =Á Ã Á = - <
. Let be the subspace of generated by all vectors of
<
:
with basis =d Ä d =
the form