Explanation Given: The F be the nonempty feasible set of all solutions x of a linear programming problem A x ≤ b with x ≥ 0 . Result used: A set X ⊆ ℝ n is convex if the convex combination of any two points in X is also in X; that is, for all x 1 , x 2 ∈ X , then the point x = α 1 x 1 + α 2 x 2 ⊆ X such that α 1 + α 2 = 1 , α 1 ≥ 0 , α 2 ≥ 0 and for all x 1 ≥ 0 , x 2 ≥ 0 . Calculation: Noticed that, the set F be the feasible set of all solutions x of a linear programming problem A x ≤ b with x ≥ 0 . Let us assume that the points p = [ x 1 x 2 x 3 ⋮ x n ] and q = [ y 1 y 2 y 3 ⋮ y n ] is in the set ℝ n . That is, the points p and q satisfy the equation A x ≤ b with p ≥ 0 and A p ≤ b , A q ≤ b with p ≥ 0 , q ≥ 0 . Since, the point p ≥ 0 , and q ≥ 0 imply all x i ≥ 0 , y j ≥ 0 for all i = 1 , 2 , 3 , 4 , ⋅ ⋅ ⋅ ⋅ , n and j = 1 , 2 , 3 , 4 , ⋅ ⋅ ⋅ ⋅ , n . To show that the set F is convex set in ℝ n , it is suffice to show that the number r = α 1 p + α 2 q is a solution of the equation A x ≤ b with x ≥ 0 and α 1 + α 2 = 1 that is r ∈ ℝ n

Linear Algebra and Its Applications (5th Edition)

5th Edition

ISBN: 9780321982384

Author: David C. Lay, Steven R. Lay, Judi J. McDonald

Publisher: PEARSON

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Chapter 9.2, Problem 16E

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To show: The set F is a convex set in ℝn .

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Chapter 9.2, Problem 15E

Chapter 9.2, Problem 17E

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Linear Algebra and Its Applications (5th Edition)

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The set F is a convex set in ℝ n .

Solution Summary: The author explains that mathcalF be the feasible set of all solutions x of a linear programming problem.

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