Page 438 - Advanced Linear Algebra

422 Advanced Linear Algebra

To finish the proof of part 2), we must prove that
:q s bb ~ J ¬ : q s ž b £ J

:
Let 8 ~¸) Á à Á ) ¹ be a basis for . Then 5 ~² Á à Á ³ : ž if and

for all . In matrix terms, if
only if 5ž )
4 ~ ² ³~²) “) “Ä“) ³

Á

has rows 9Á à Á 9 , then 5  : ž if and only if 54 ~ , that is,
9 bÄb 9 ~

Now, : ž contains a strictly positive vector 5 ~ ² Á à Á ³ if and only if this
equation holds, where ‚ for all and € for some . Moreover, we may

assume without loss of generality that ' ~ , or equivalently, that is in the

convex hull of the row space of 4 . Hence,
9
:q s ž b £ J ¯  9
Thus, we wish to prove that
:q s bb ~ J ¬  9
or equivalently,
¤ 9 ¬ : q s bb £ J
Now we have something to separate. Since is closed and convex, Theorem
9
15.5 implies that there is a nonzero vector ) ~ ² ÁÃÁ ³  s for which

º)Á » ‚ ))) €
9
Consider the vector
# ~ ) bÄb )  :

The th coordinate of is

#
))

Á bÄb Á ~ º)Á 9 » ‚ ) €

and so is strongly positive. Hence, #  : q s bb , which is therefore
#
nonempty.
Inhomogeneous Systems
We now turn our attention to inhomogeneous systems
(% ~
The following lemma is required.