Page 392 - Advanced Linear Algebra

376 Advanced Linear Algebra

²"Á #³ ~ " n #

The same argument that we used in the proof of Theorem 14.7 will work here.
Z

"

Namely, the map ²"Á #³ ª " n # from < d = to < n = Z is bilinear in and
Z
Z
# and so there is a unique linear map ² p ³¢ < n= ¦ < n= for which

²p ³²" n #³ ~ " n #

The function
Z
Z
Z
Z
¢
B²<Á < ³ d B²=Á = ³ ¦ B²< n =Á < n = ³
defined by
²Á
³ ~ p
is bilinear, since

²² b ³ p ³²" n #³ ~ ² b ³²"³ n #

~² " b "³ n #

~ ´ " n #µ b ´ " n #µ

~ ² p ³²" n #³ b ² p ³²" n #³

~ ² ² p ³ b ² p ³³²" n #³
and similarly for the second coordinate. Hence, there is a unique linear
transformation
Z
Z
Z
Z
B²<Á < ³ n B²=Á = ³ ¦ B²< n =Á < n = ³
¢
satisfying
²n ³ ~ p
that is,

´ ² n³µ²" n #³ ~ " n#

Z
Z
To see that is injective, if  ²<Á < ³ n ²= Á = ³ is nonzero, then we may
B

B
write

~ n
~
Z
Z
B
B
where the  ²<Á < ³ are nonzero and the set ¸ ¹ ‹ ²= Á = ³ is linearly

independent. If ² ³ ~ , then for all "  < and #  = we have

~ ² ³²" n #³ ~ ² n ³²" n #³ ~ ²"³ n ²#³

~ ~

Since £ , it follows that £ for some and so we may choose a "  <

such that ²"³ £ for some . Moreover, we may assume, by reindexing if