a2 - x for the problem and a vector d R n such that Ad , d...

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CO350 LINEAR PROGRAMMING - ASSIGNMENT 2 Due Date: Friday May 21 at 2 PM. Please make sure that your name and student number, the course number, and the name of your instructor are clearly written on the front of your assignment, and that all pages are securely stapled. You may discuss the assignment questions with other students, but you must write your solutions up independently, and all submitted work must be your own. Exercise 1. Let { a 1 ,...,a k } ⊂ R m be a set of linearly independent vectors. (a) Prove that if a k +1 R m is not in the subspace spanned by a 1 ,...,a k then the set { a 1 ,...,a k +1 } is linearly independent. (b) Using part (a) prove that the set { a 1 ,...,a k } is contained in a basis of R m . Exercise 2. Consider the linear program max { c T x : Ax b,x 0 } , where A R m × n , c R n and b R m . (a) Prove that if there exists a vector y R m such that y T A 0 , y 0 and b T y < 0 then the problem has no feasible solution. (b) Prove that if there exists a feasible solution
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Unformatted text preview: x for the problem and a vector d R n such that Ad , d and c T d &gt; then the problem is unbounded. Exercise 3. Consider the linear program max { c T x : Ax = 0 ,x } , where A R m n and c R n . Prove that if x = 0 is not an optimal solution then the linear program is unbounded. Exercise 4. Consider the problem max { x : x &lt; 1 } . Explain why the problem is not a linear program and show that it is not infeasible, not unbounded and has no optimal solution. Exercise 5. Convert the following linear program into standard equality form: min 7 x 1-3 x 3 x 1 + x 2 = 2-2 x 1 + x 2-4 x 3 1 2 x 2 + x 3 7 x 2 ,x 3 Exercise 6. Let A R m n , b R m and c R n . Convert the problem max { c T x : Ax b } into standard equality form. 1...
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