Page 391 - Advanced Linear Algebra
P. 391
Tensor Products 375
is bilinear since, for example,
´² b ³p µ²"n#³ ~ ² b ³²"³h ²#³
~ ²"³ ²#³ b ²"³ ²#³
~ ´ ² p ³ b ² p ³µ²" n #³
which shows that . is linear in its first coordinate. Hence, the universal
property implies that there exists a unique linear map
i
i
¢< n = ¦ ²< n = ³ i
for which
² n ³ ~ p
i
To see that is an injection, if < n = i is nonzero, then we may write in
the form
~ n
~
where the < i are nonzero and ¸ ¹ = i is linearly
independent. If ² ³ ~ , then for any " < and # = , we have
~ ² ³²" n #³ ~ ² n ³²" n #³ ~ ²"³ ²#³
~ ~
Hence, for each nonzero "< , the linear functional
²"³
~
is the zero map and so the linear independence of ¸ ¹ implies that ²"³ ~
"
for all . Since is arbitrary, it follows that ~ for all and so ~ .
Finally, in the finite-dimensional case, the map is a bijection since
i
i
i
dim²< n = ³ ~ dim²²< n = ³ ³ B
Combining the isomorphisms of Theorem 14.4 and Theorem 14.7, we have, for
finite-dimensional vector spaces and ,
=
<
i
i
i
< n = ²< n = ³ hom ²<Á = Â -³
The Tensor Product of Linear Transformations
We wish to generalize Theorem 14.7 to arbitrary linear transformations. Let
Z
Z
B ²<Á < ³ and B ²= Á = ³. While the product ²"³ ²#³ does not make
sense, the tensor product "n # does and is bilinear in and , that is, the
"
#
following function is bilinear:
