Page 484 - Advanced Linear Algebra
P. 484
468 Advanced Linear Algebra
#~ " b
for Á s , then
#~ " b " b
and since "¤ s , it follows that ~ and so ~ or ~ . But £ since
#¤ s and £ since ¸"Á #¹ are linearly independent.
Thus, + Z is a subspace of and
+
+~ s l + Z
Z
We now look at + Z , which is a real vector space. If + ~ ¸ ¹ , then + ~ s and
we are done, so assume otherwise. If + Z is nonzero, then ~c where
Z
s. Hence, ~ c + satisfies ~ c . If
Z
+ ~ ~¸ ¹
s
s
s
s
then +~ s l ~ d and we are done. If not, then is a proper subspace of
+ .
Z
In the quaternion field, there is an element for which b ~ . So we seek a
Z
+ ± with this property. To this end, define a bilinear form on + by
Z
s
º"Á #» ~ c²"# b #"³
Then it is easy to see that this form is a real inner product on + (
positive
Z
)
definite, symmetric and bilinear . Hence, if s is a proper subspace of + Z , then
Z
+~ p :
s
where p denotes the orthogonal direct sum. If " : is nonzero, then
c
"~ c for and so if s ~ " , then
~ c and b ~
Now, s is a subspace of and so
:
Z
s
+~ p p ;
s
Setting ~ , we have
cº Á » ~ b ~ b ~
and
cº Á » ~ b ~ b ~
and so ; and we can write
Z
s
s
+~ p p p <
s
