Page 380 - Advanced Linear Algebra

364 Advanced Linear Algebra

equal to . We will consider the latter issue in some detail a bit later in the

chapter.
:
The fact that is a basis for <n = gives the following.
Theorem 14.2 For finite-dimensional vector spaces and ,
=
<
dim²< n = ³ ~ dim²<³ h dim²= ³

Construction II: Coordinate Free
The previous construction of the tensor product is reasonably intuitive, but has
the disadvantage of not being coordinate free. The following approach does not
require the choice of a basis.

Let - be the vector space over with basis < d = . Let be the subspace
:
- <d=
generated by all vectors of the form
of - <d=
²"Á$³ b ²#Á$³ c ² " b #Á$³ (14.1 )
and
²"Á #³ b ²"Á $³ c ²"Á # b $³ (14.2 )

$
where Á  - and "Á # and are in the appropriate spaces. Note that these
vectors are precisely what we must “identify” as the zero vector in order to
enforce bilinearity. Put another way, these vectors are if the ordered pairs are

replaced by tensors according to our previous construction.
Accordingly, the quotient space

- <d=
<n = •
:
is also sometimes taken as the definition of the tensor product of and .
=
<
(Strictly speaking, we should not be using the symbol <n = until we have
shown that this is the tensor product.) The elements of <n = have the form
4 5 ²"Á #³ b : ~ !²" Á # ³ b :

However, since ²"Á#³ c ² "Á#³  : and ²"Á#³ c ²"Á #³  : , we can absorb
the scalar in either coordinate, that is,

´²"Á#³ b :µ ~ ² "Á#³ b : ~ ²"Á #³ b :
and so the elements of <n = can be written simply as

´²" Á # ³ b :µ

It is customary to denote the coset ²"Á #³ b : by " n # , and so any element of