Page 394 - Advanced Linear Algebra

378 Advanced Linear Algebra

Theorem 14.8 There is a unique linear transformation
Z
Z
Z
Z
¢
B²<Á < ³ n B²=Á = ³ ¦ B²< n =Á < n = ³
defined by ²n ³ ~ p where

²p ³²" n #³ ~ " n #

Moreover, is an embedding and is an isomorphism if all vector spaces are

finite-dimensional. Thus, the tensor product n of linear transformations is
(via this embedding ) a linear transformation on tensor products.
Let us note a few special cases of the previous theorem.
—
Corollary 14.9 Let us use the symbol ? @ Æ to denote the fact that there is an
embedding of ? into @ that is an isomorphism if ? and @ are finite-
dimensional.
Z
)
1 Taking <~ - gives
i
Z
<n ²= Á = ³ Æ Z — B ²< n = Á = ³
B
where
² n ³²" n #³ ~ ²"³ ²#³

for  < i .
)
Z
Z
2 Taking <~ - and = ~ - gives
—
i
i
<n = Æ ²< n = ³ i
where
² n ³²" n #³ ~ ²"³ ²#³
)
Z
3 Taking =~ - and noting that ²-Á = ³ š = Z and < n - š < gives
B
(letting >~ = Z )
—
Z
B Z Ʋ<Á < ³ n > B ²<Á < n > ³
where

² n $³²"³ ~ " n $
)
Z
(
4 Taking <~ - and = ~ - gives letting > ~ = Z )
—
i
<n > ÆB ²<Á > ³
where
² n $³²"³ ~ ²"³$