Of the water gets lost in evaporation while being

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of the water gets lost in evaporation while being channeled to the farmer. (a) Formulate this as a linear program. 2
(b) Suppose farmer 2 cannot be served by reservoir 3 because the farmer is at a higher altitude than the the reservoir. How can you take this into account in your model without increasing the number of constraints or variables? Optional linear algebra review This problem will not be graded but you will receive feedback if sub- mitted
(d) Prove that if V and W are 3-dimensional subspaces of < 5 , then V and W must have a nonzero vector in common. HINT: Start with bases for the two subspaces, making six vectors in all. Consider notions of linear independence. (e) True or False: (i) If the columns of a matrix A are linearly independent, then Ax = b has exactly one solution for every b . (ii) A 5 by 7 matrix never has linearly independent columns. 3

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