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

See similar textbooks

Videos

Question

Chapter 9.2, Problem 16E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.2, Problem 15E

Chapter 9.2, Problem 17E

Chapter 9 Solutions

Linear Algebra and Its Applications (5th Edition)

Ch. 9.1 - Find the optimal strategies and the value of the...Ch. 9.1 - Player R has a supply of dimes and quarters....Ch. 9.1 - Prob. 2E Ch. 9.1 - In the traditional Japanese childrens game janken...Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Find all saddle points for the matrix games in...Ch. 9.1 - Find all saddle points for the matrix games in...Ch. 9.1 - Find all saddle points for the matrix games in...Ch. 9.1 - Prob. 9E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Similar questions

23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes 1/2 of the food and 1/3 of the manufactured goods. Manufacturing consumes 1/2 of the food and 2/3 of the manufactured goods. Assuming the economy is closed and in equilibrium, find the relative outputs of the farming and manufacturing industries.

Recommended textbooks for you

Algebra for College Students
Algebra
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning

Algebra for College Students

Algebra

ISBN:9781285195780

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:9781285463247

Author:David Poole

Publisher:Cengage Learning

College Algebra (MindTap Course List)

Algebra

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:Cengage Learning

SEE MORE TEXTBOOKS

Solve ANY Optimization Problem in 5 Steps w/ Examples. What are they and How do you solve them?; Author: Ace Tutors;https://www.youtube.com/watch?v=BfOSKc_sncg;License: Standard YouTube License, CC-BY

Optimization Problems in Calculus; Author: Professor Dave Explains;https://www.youtube.com/watch?v=q1U6AmIa_uQ;License: Standard YouTube License, CC-BY

Introduction to Optimization; Author: Math with Dr. Claire;https://www.youtube.com/watch?v=YLzgYm2tN8E;License: Standard YouTube License, CC-BY

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.

Videos

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

Want to see the full answer?

Chapter 9 Solutions