Feb_8_notes.pdf - MATB61 Feb 8th notes 1 Show one of the...

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MATB61 Feb. 8th notes 1. Show one of the following linear systems has a solution if and only if the other one has no solution.
2. We want to prove above by using the following facts. One of the following linear systems has a solution if and only if the other one has no solution. A~x = ~ b, ~x 0 (3) and A T ~ y 0 , ~ y T ~ b < 0 (4) Proof. We notice that there is no sign constraint in system (1). Thus, we should rewrite the (1) to an equivalent system which dose have such a constraint. Let ~x = ~u - ~v and ~u,~v 0 . Rewrite (1) in terms ~u,~v as the following. A~x = ~ b A ( ~u - ~v ) = ~ b
A~u - A~v = ~ b h A - A i " ~u ~v # = ~ b, (5) where you can view M = h A - A i is a bigger matrix and ~ c = " ~u ~v # 0 . Now you may compare system (5) with (3) and you should see they are equivalent. ( ie, (5) gives you M~ c = ~ b,~ c 0.) Also note that (5) is same as (1). Apply the fact in our case, you should get the following.

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