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Math Help - Linear algebra proofs

  1. #1
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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
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  2. #2
    MHF Contributor

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    What do you mean by vector being larger than another? Vector spaces do not, in general, have an order defined on them. Also, what does "yTA" mean?
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  3. #3
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    that would mean every value in the vector is larger than the corresponding value in the other vector. in other words, (2,3) > (1,2)
    yTA means (y transposed) * A
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