Page 420 - Advanced Linear Algebra

404 Advanced Linear Algebra

Consider the map ¢ = d ¦ - defined by

²#Á Ã Á # ³ ~ ²c ³ ´#µ Ä´# µ

:
Then is multilinear since

² # b " Á Ã Á # ³ ~ ²c ³ ´ # b " µ Ä´# µ

:
~ ² c ³ ² ´ # µ b ´ " µ ³ Ä ´ # µ ()

:

~ ²c ³ ´# µ Ä´# µ

:

b ²c ³ ´" µ Ä´# µ

:
~ ²# Á Ã Á # ³ b ²" Á # Á Ã Á # ³

and similarly for any coordinate position.
To see that is alternating, and therefore antisymmetric since char ² - ³ £ ,

suppose for instance that #~ # . For any permutation  : , let
Z ~² ³

Z

Then %~ % for % £ Á and

Z ~ and Z ~
Hence, Z £ . Also, since ² Z Z ~ ³ , if the sets Á ¸ Z ¹ and Á ¸ Z ¹ intersect,
then they are identical. Thus, the distinct sets ¸Á Z ¹ form a partition of : . It
follows that

²#Á #Á #Á Ã Á # ³ ~ ²c ³ ´#µ ´#µ Ä´# µ

:
Z

~ > ²c ³ ´# µ ´# µ Ä´# µ b ²c ³ ´# µ ´# µ Ä´# µ Z ?

Z

Z

pairs ¸Á ¹
Z

But
´#µ ´#µ ~ ´#µ ´#µ Z Z

and since ²c ³ ~ c²c ³ Z , the sum of the two terms involving the pair ¸ Á Z ¹

is . Hence, ²# Á# ÁÃÁ# ³ ~ . A similar argument holds for any coordinate

pair.