Explanation Given: The point c is in ℝ n and if f is defined on ℝ n by f ( x ) = c T x , then f is called a linear functional. Result used: 1. 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 line t x 1 + ( 1 − t ) x 2 ⊆ X for t ∈ [ 0 , 1 ] . 2. A function f is said to be convex if f ( t x 1 + ( 1 − t ) x 2 ) ≤ t ⋅ f ( x 1 ) + ( 1 − t ) ⋅ f ( x 2 ) , for t ∈ [ 0 , 1 ] and x 1 , x 2 ∈ f . Calculation: Note that the point c is in ℝ n and if f is defined on ℝ n by f ( x ) = c T x . It is also clear that for any real number d , the set { x : f ( x ) = d } is a level set of linear functional f . Thus, the set S = { x : f ( x ) = d } implies that for any p , q ∈ S . To show the level set S is convex, then it is suffice to show that x = t p + ( 1 − t ) q ∈ S for any p , q ∈ S and t ∈ [ 0 , 1 ]

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 19E

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To show: The level set {x:f(x)=d} for any real number d is convex.

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Chapter 9.2, Problem 18E

Chapter 9.3, Problem 1PP

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Linear Algebra and Its Applications (5th Edition)

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The level set { x : f ( x ) = d } for any real number d is convex.

Solution Summary: The author explains that the level set leftx:f (x)=d right for any real number is convex.

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