Page 421 - Advanced Linear Algebra
P. 421
Tensor Products 405
Finally,
² Á Ã Á ³ ~ ²c ³ ´ µ Ä´ µ
:
~ ² c ³ Ä Á Á
:
~
Thus, the map is the unique antisymmetric -linear form on = d for which
² Á Ã Á ³ ~ .
Under the ordered basis ; ~² Á Ã Á ³ , we can view as the space - of
=
coordinate vectors and view = d as the space 4 ² - ³ of d matrices, via the
isomorphism
v ´# µ Ä ´# µ y
²# ÁÃÁ# ³ ª Å Å
w z
´# µ Ä ´# µ
where all coordinate matrices are with respect to .
;
With this viewpoint, becomes an antisymmetric -form on the columns of a
matrix (~ ² ³ given by
Á
²(³ ~ ²c ³ Á Ä Á
:
This is called the determinant of the matrix .
(
Properties of the Determinant
Let us explore some of the properties of the determinant function.
Theorem 14.21 If ( 4 ²-³ , then
!
²(³ ~ ²( ³
Proof. We have
²(³ ~ ²c ³ Á Ä Á
:
~ ² c ³ c c Ä Á c Á
c
:
~ ² c ³ Ä Á Á
:
!
~ ²( ³
as desired.
