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# hw_add_ProblemSet - example the product of(2 1 and-1 1 T...

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The points were split 3-3-2-2 for this problem. I’ll return to part (a) later. For part (b), 1 point was given for each feasibility, boundedness and v 6 = 0 parts. For (c), half points was given for each objective, 2 constraints and variable being free. For part (d), one point was given for using the strong duality on correct problem, and one point for a correct argument why this guarantees the existence of y that satisfies A T y 0 , b T y < 0. Now back to part (a). Please, think a while about why if the product of 2 vectors is negative, it does not imply that one vector is negative and the other one positive. This works for real numbers, but for vectors, there are other possibilities (some vectors are neither positive nor negative), for
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Unformatted text preview: example the product of (2 , 1) and (-1 , 1) T is -1. The same way, if the problem Ax = b,x ≥ 0 is infeasible for some b ≥ 0, it does not mean A < 0. There are lots of other cases when the matrix is positive, or neither positive nor negative, and the problem is infeasible. Also, some of you used v > 0 without explaining if they mean that EVERY coordinate of v is positive, or that AT LEAST ONE coordinate is. I did not penalize this, but next time be more careful.Working with vectors is sometimes more complicated than working with numbers, but if you’re not sure if you can generalize some statement, think about a 2-dimensional case ﬁrst. 1...
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