Page 432 - Advanced Linear Algebra

416 Advanced Linear Algebra

)
)
( ( c !  c ! 
4
and so
)
) ) ² ³ c ²!³ ~ ² c! ³% bÄb² c ! ³% )

 ( ) c ! % ) ( b Ä b ( c ! ) % )
(
)  4 ) c !
~
9
9
Finally, since ²+³ ~ ²:³ , it follows that ²:³ is compact.
Linear and Affine Hyperplanes
We next discuss hyperplanes in s . A linear hyperplane in s is an ² c ³ -
dimensional subspace of s . As such, it is the solution set of a linear equation
of the form
% bÄb % ~

or
º5Á %» ~

where 5 ~ ² ÁÃÁ ³ is nonzero and % ~ ²% Á ÃÁ% ³ . Geometrically

speaking, this is the set of all vectors in s that are perpendicular (normal) to
the vector .
5
An affine hyperplane , or just hyperplane , in s is a linear hyperplane that has
been translated by a vector. Thus, it is the solution set to an equation of the form
²% c ³ bÄb ²% c ³ ~

or equivalently,
º5Á %» ~
. We denote this hyperplane by

where ~ bÄb
> s²5Á ³ ~ ¸%  “ º5Á%» ~ ¹
Note that the hyperplane
> )) ³ ~ ¸%  s²5Á 5 “ º5Á %» ~ 5)) ¹
5
contains the point 5 , which is the point of ² > 5 Á )) ³ closest to the origin,
since Cauchy's inequality gives
%
5
)) ~ º 5 Á % »  ))) )
5

and so ))5 ) ) for all %  > ²5Á )) ³ . Moreover, we leave it as an
%
5