Page 419 - Advanced Linear Algebra

Tensor Products 403

²:!³ ~ ! ~ ! ~ ! ~ :!
[ [ [
: : :
and so :! is symmetric. The reason for the factor ° [ is that if is a symmetric
#
tensor, then #~# and so

:# ~ # ~ # ~ #
[ [
: :
that is, the symmetrization map fixes all symmetric tensors. It follows that for

any tensor !; ²= ³ ,

:! ~ :!
:
Thus, is idempotent and is therefore the projection map of ; ² = ³ onto

im²:³ ~ :; ²= ³.
The Determinant
The universal property for antisymmetric multilinear maps has the following
corollary.

Corollary 14.20 Let be a vector space of dimension over a field . Let
=
-

, ~ ² ÁÃÁ ³ be an ordered basis for = . Then there is at most one

antisymmetric -linear form ¢ = d ¦ - for which

² Á Ã Á ³ ~

Proof. According to the universal property for antisymmetric -linear forms, for

every antisymmetric -linear form ¢ = d ¦ - satisfying ² Á Ã Á ³ ~ ,

there is a unique linear map ¢ = ¦ - for which
² wÄw ³ ~ ² Á Ã Á ³ ~

But has dimension and so there is only one linear map ¢ = = ¦ -

with ² wÄw ³ ~ . Therefore, if and are two such forms, then

~ ~ , from which it follows that
~ k ! ~ k ! ~
We now wish to construct an antisymmetric form ¢ = d ¦ - , which is unique
by the previous theorem. Let be a basis for . For any  # = , write # ´ µ 8Á for
=
8
the th coordinate of the coordinate matrix # ´ µ 8 . Thus,

#~ ´#µ 8 Á

For clarity, and since we will not change the basis, let us write ´#µ for ´#µ Á 8 .