Page 385 - Advanced Linear Algebra

Tensor Products 369

but this is not necessary for our discussion and adds a complication, since the
bases may be infinite.

Suppose that M and N ~¸$ “ 0¹ ~¸% “ 1¹ are also bases for and
<

= , respectively, and that

'~ ²$ n % ³
Á

~ ~
'
where :~ ² ³ is a coordinate matrix of with respect to these bases. We
Á
claim that the coordinate matrices and have the same rank, which can then
:
9
be defined as the rank of the tensor ' < n = .
is a finite linear combination of basis vectors in , perhaps
Each $Á Ã Á $ 8
and perhaps involving other vectors in . We can
8
involving some of "Á Ã Á "
further reindex 8 so that each $ is a linear combination of the vectors
8 ~²" Á à Á " ³, where  and set
Z

< ~ span ²" ÁÃÁ" ³

Next, extend ²$ ÁÃÁ$ ³ to a basis M Z ~ ²$ ÁÃÁ$ Á$ b ÁÃÁ$ ³ for < .

(Since we no longer need the rest of the basis M , we have commandeered the
symbols $ Á b à Á $ , for simplicity. Hence
)

$~ " Á for ~ Á Ã Á

~
where (~ ² ³ is invertible of size d .
Á
Now repeat this process on the second coordinate. Reindex the basis so that
9
the subspace =~ span²# ÁÃÁ# ³ contains % ÁÃÁ% and extend to a basis

Z
N ~ ²% Á ÃÁ% Á% b ÁÃÁ% ³ for = . Then

%~ # Á for ~ Á Ã Á

~
where )~ ² ³ is invertible of size d .
Á
Next, write

'~ ²" n # ³

Á
~ ~
~ for € or € . Thus, the d matrix 9 ² comes
by setting Á ~ ³ Á
9

from by adding c rows of 's to the bottom and then c columns of
9
9's. In particular, and have the same rank.