Page 404 - Advanced Linear Algebra

388 Advanced Linear Algebra

is the zero map then

is an isomorphism, since if # p Äp# p pÄp

²# ³Ä ²# ³ ²% ³Ä ²% ³# b n Än# n b nÄn ~

for all  = i and %  = , which implies that

# n Än# n nÄn ~

by

As usual, we denote the map # p Äp# p pÄp
# n Än# n nÄn

Theorem 11.15 For   and   ,
i n² c ³
i n

;²= ³ š B²²= ³ n = n Á = n² c ³ n ²= ³ ³

When ~ and ~ , we get
i n
i n

³
;²= ³ š B²²= ³ n = n Á -³ ~ ²²= ³ n = n i

as before.
Let us look at some special cases. For ~ we have
in n² c ³
;²= ³ š B²²= ³ Á = ³

where
²# n Än# ³² nÄn ³ ~ ²# ³Ä ²# ³# b n Än#

When ~ ~ , we get for ~ and ~ ,
;²= ³ š B B²- n= Á = n-³ š ²= ³
where
²# n ³²$³ ~ ²$³#
and for ~ and ~ ,
i
i
;²= ³ š B i i B²= n-Á - n= ³ š ²= Á = ³
where
²# n ³² ³ ~ ²#³
Finally, when ~ ~ , we get a multilinear form
²# n ³² Á $³ ~ ²#³ ²$³
Consider also a tensor n of type ² Á ³ . When ~ ~ we get a
multilinear functional n ¢ ²= d = ³ ¦ - defined by