Page 448 - Advanced Linear Algebra
P. 448
432 Advanced Linear Algebra
²% b : ³ ~ ²% b : ³ ~ % b :
2 2 2
We leave proof of part 2) to the reader.
For part 3), since %Á & ? v @ , it follows that
?v @ ~ % b < ~ & b <
<
for some subspace of . Thus, %c& < . Also, % b: % b< implies that
=
: < and similarly ; < , whence : b ; < and so if > ~
º% c&»b: b; , then > < . Hence, % b> % b< ~ ? v@ . On the
other hand,
? ~ %b: %b>
and
@ ~ & b ;~ % c ²% c &³ b ; % b >
and so ?v @ % b > . Thus, ?v @ ~ % b > .
@
If ?q @ £ J , then we may take the flat representatives for and to be any
?
element ' ? q @ , in which case part 1) gives
?v @ ~ 'b ²º'c '» b : b ;³ ~ 'b : b ;
and since % ? v @ , we also have ? v@ ~ % b: b; .
We can now describe the dimension of the join of two flats.
Theorem 16.6 Let ?~ % b : and @ ~ & b ; be flats in .
=
)
1 If ?q @ £ J , then
dim²? v @ ³ ~ dim²: b ;³ ~ dim²?³ b dim²@ ³ c dim²? q @ ³
)
2 If ?q @ ~ J , then
dim²? v @ ³ ~ dim²: b ;³ b
Proof. We have seen that if ?q @ £ J , then
?v @ ~ % b : b ;
and so
dim²? v @ ³ ~ dim²: b ;³
On the other hand, if ?q @ ~ J , then
? v @ ~ %b²º%c&» b: b;³
and since dim²º% c &»³ ~ , we get
