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3. For a nonempty set S C R" and a positive real number r E R (and r > 0), detine the set rS as follows: rS = {z € R" | z = rx, x E S}. Here rx is the multiplication of the vector x E R" by the scalarr e R; in your more familiar notation, rã. The set rS is the set of all points that are obtained by multiplying r with the vectors x E S. For a given nomempty set SC R" and a given positive number r > 0, prove that if S is a convex set then rS is a convex set as well.

3. For a nonempty set S C R" and a positive real number r E R (and r > 0), detine the set rS as follows: rS = {z € R" | z = rx, x E S}. Here rx is the multiplication of the vector x E R" by the scalarr e R; in your more familiar notation, rã. The set rS is the set of all points that are obtained by multiplying r with the vectors x E S. For a given nomempty set SC R" and a given positive number r > 0, prove that if S is a convex set then rS is a convex set as well.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. For a nonempty set S C R" and a positive real number r E R (and r > 0), define the set rS as follows: rS := {z € R" | z = rx, x E S}. Here rx is the multiplication of the vector x E R" by the scalarr € R; in your more familiar notation, rĩ. The set rS is the set of all points that are obtained by multiplying r with the vectors x E S. For a given nomempty set SC R" and a given positive number r > 0, prove that if S is a convex set then rS is a convex set as well.
Transcribed Image Text:3. For a nonempty set S C R" and a positive real number r E R (and r > 0), define the set rS as follows: rS := {z € R" | z = rx, x E S}. Here rx is the multiplication of the vector x E R" by the scalarr € R; in your more familiar notation, rĩ. The set rS is the set of all points that are obtained by multiplying r with the vectors x E S. For a given nomempty set SC R" and a given positive number r > 0, prove that if S is a convex set then rS is a convex set as well.
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