Page 386 - Advanced Linear Algebra

370 Advanced Linear Algebra

'
The expression for in terms of the basis vectors $ ÁÃÁ$ and % Á ÃÁ% can
also be extended using coefficients to

'~ ²$ n % ³

Á

~ ~
where the d matrix : ~ ² ³ has the same rank as .
:

Á
Now at last, we can compute. First, bilinearity gives

$n % ~ Á Á " n #

~ ~
and so

'~ ²$ n % ³ ~ Á 8 Á Á Á " n # 9

~ ~ ~ ~ ~ ~

~ 8 ² ³ Á 9 " n Á Á #
~ ~ ~ ~

!
Á Á
~ ²( : ³ 9 " n #
8
~ ~ ~

~ ( ! ) Á : 2 n 3 # "
~ ~
Thus

!
²" n # ³ ~ ' ~ ²( : )³ Á ²" n # ³

Á
~ ~ ~ ~
!
(
)
and so 9~ ( : ) . Since and are invertible, we deduce that

rk²9³~ rk²9 ³~ rk²: ³~ rk²:³

as desired. Moreover, in block matrix terms, we can write
9 :
9~ > ? and : ~ > ?

block block
and if we write
( ! i )
! Á i Á
(~ > ? and ) ~ > ?
i i block i i block
!
then 9~ ( : ) implies that