Let P := {x € R" : Ax < b} be a nonempty polyhedron, where A E RmXn and b E R™. Note that P is not in standard form. Prove that the following result: P does not contain a vertex if and only if rank(A) < n. (Hint: Think about the implications of the assumption rank(A) < n.)

Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
icon
Related questions
Question
Let P := {x E R" : Ax < b} be a nonempty polyhedron, where A E R™Xn and b E Rm. Note
that P is not in standard form. Prove that the following result:
P does not contain a vertex if and only if rank(A) < n.
(Hint: Think about the implications of the assumption rank(A) < n.)
Transcribed Image Text:Let P := {x E R" : Ax < b} be a nonempty polyhedron, where A E R™Xn and b E Rm. Note that P is not in standard form. Prove that the following result: P does not contain a vertex if and only if rank(A) < n. (Hint: Think about the implications of the assumption rank(A) < n.)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education