Prove the following Farkas' type results. (i) Prove that there exists a 0 satisfying Ax 0...

 

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Prove the following Farkas' type results. (i) Prove that there exists a 0 satisfying Ax 0 satisfying Ax = 0 if and only if for each y Rm, yTA > 0yTA = 0. (iii) Prove that there exists x 0 satisfying x > 0 and Ax = 0 if and only if there is no vector y = Rm satisfying yTA > 0. Prove the following Farkas' type results. (i) Prove that there exists a 0 satisfying Ax 0 satisfying Ax = 0 if and only if for each y Rm, yTA > 0yTA = 0. (iii) Prove that there exists x 0 satisfying x > 0 and Ax = 0 if and only if there is no vector y = Rm satisfying yTA > 0.