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July 5th, 2010, 10:06 AM   #1
 
Joined: Jul 2010

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Linear algebra proofs

Hi, can someone please help me figure out these proofs?
They're from Combinatorial Optimization by Cook, Cunningham, Pulleyblank,Schrijver.
Thanks

2.15. Prove that there exists a vector x >= 0 such that Ax <= b, if and only if for each y >= 0
satisfying yTA >= 0 one has yT b >= 0.

2.16. Prove that there exists a vector x > 0 such that Ax = 0, if and only if for each y
satisfying yTA >= 0 one has yTA = 0. (Stiemke's theorem (Stiemke [1915]).)

2.17. Prove that there exists a vector x != 0 satisfying x >= 0 and Ax = 0, if and only if
there is no vector y satisfying yTA > 0. (Gordan's theorem (Gordan [1873]).)

2.18. Prove that there exists a vector x satisfying Ax < b, if and only if y = 0 is the only
solution for y >= 0; yTA = 0; yT b <= 0.

2.19. Prove that there exists a vector x satisfying Ax < b and A'x <= b', if and only if for
all vectors y; y' >= 0 one has:
(i) if yTA + y'TA' = 0 then yT b + y'T b' >= 0, and
(ii) if yTA + y'TA' = 0 and y != 0 then yT b + y'T b' > 0.

Thanks guys
nieldv is offline  
 
July 8th, 2010, 10:39 PM   #2
 
Joined: Apr 2008

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Re: Linear algebra proofs

Why don't you show us your work and ideas, and we can take it from there. Most of these seem to be definitional. Though I must add, I don't know what you mean by T. What is T?
jason.spade is offline  
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