Page 418 - Advanced Linear Algebra

402 Advanced Linear Algebra

!² Á Ã Á ³ ~ vÄv

is universal for symmetric -linear maps with domain = d ; that is, for any

<

symmetric -linear map ¢ = d ¦ < where is a vector space, there is a
unique linear map ¢- ´% ÁÃÁ% µ ¦ < for which

² vÄv ³ ~ ² Á Ã Á ³

)
c
c
2 The pair ²- ´% ÁÃÁ% µÁ !³ , where !¢ = d ¦ - ´% Á ÃÁ% µ is the

multilinear map defined by
!² Á Ã Á ³ ~ wÄw

is universal for antisymmetric -linear maps with domain = d ; that is, for

<

any antisymmetric -linear map ¢ = d ¦ < where is a vector space,
c
there is a unique linear map ¢- ´% ÁÃÁ% µ ¦ < for which

² wÄw ³ ~ ² Á Ã Á ³

Proof. For part 1), the property
² vÄv ³ ~ ² Á Ã Á ³

does indeed uniquely define a linear transformation , provided that it is well-

defined. However,

v Äv ~ vÄv
if and only if the multisets ¸ ÁÃÁ ¹ and ¸ Á ÃÁ ¹ are the same, which

implies that ² Á ÃÁ ³ ~ ² ÁÃÁ ³ , since is symmetric.

For part 2), since is antisymmetric, it is completely determined by the fact that

.

it is alternate and by its values on the basis of ascending words w Ä w
Accordingly, the condition
² wÄw ³ ~ ² Á Ã Á ³

uniquely defines a linear transformation .

The Symmetrization Map

When char²-³ ~ , we can define a linear map :¢ ;²= ³ ¦ :;²= ³ , called
the symmetrization map , by

:! ~ !
[
:
Since ~ , we have