Page 423 - Advanced Linear Algebra

Tensor Products 407

4. Let 8 be a basis for and let 9 ~¸" “ 0¹ < ~ ¸# “ 1¹ be a basis

for . Show that the set
=
: ~¸" n # “ 0Á 1¹

is a basis for <n = by showing that it is linearly independent and spans.
5. Prove that the following property of a pair ²> Á ¢< d = ¦ > ³ with
bilinear characterizes the tensor product ²< n= Á !¢ < d= ¦ < n= ³ up
to isomorphism, and thus could have been used as the definition of tensor

product: For a pair ²> Á ¢< d = ¦ > ³ with bilinear if ¸" ¹ is a basis

=
<
for and # ¸ ¹ is a basis for , then ¸ ² " Á # ³ ¹ is a basis for > .
6. Prove that <n = š = n < .
7. Let ? and @ be nonempty sets. Use the universal property of tensor
products to prove that < < ?d@ š ? n < @ .
Z
Z
8. Let "Á "  < and #Á #  = . Assuming that " n # £ , show that
Z
Z
Z
c

Z
" n #~" n # if and only if " ~ " and # ~ #, for £ .
9. Let ~¸ ¹ be a basis for and ~¸ ¹ be a basis for . Show that any
9
=
<
8

function ¢ 8 d 9 ¦ > can be extended to a linear function
¢ < n = ¦ > . Deduce that the function can be extended in a unique

V
way to a bilinear map ¢ < d = ¦ > . Show that all bilinear maps are
obtained in this way.
be subspaces of . Show that
<
10. Let :Á :
²: n = ³ q ²: n = ³ š ²: q : ³ n =

11. Let :‹ < and ; ‹ = be subspaces of vector spaces < and = ,
respectively. Show that
²: n= ³q²< n ;³ š : n;
12. Let : Á : ‹ < and ; Á ; ‹ = be subspaces of < and = , respectively.
Show that
²: n ; ³ q ²: n ; ³ š ²: q : ³ n ²; n ; ³

13. Find an example of two vector spaces < and = and a nonzero vector
(
%< n = that has at least two distinct not including order of the terms)
representations of the form

%~ " n #
~
where the 's are linearly independent and so are the 's.
"

#
14. Let ? denote the identity operator on a vector space ? . Prove that
= > p = ~ n > .