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4. A convex combination of the vectors x,, x2, ..., x, is a linear com- bination of the form «,x, + ¤z x2 + +a, x, where a; 2 0, for each i; and a, + az +…+¤, = 1. Given a set S in a vector space, let K be the set of vectors consisting of all convex combinations from S. Show that K = co(S). 5. Let Cand D be convex cones in a vector space. Show that Cn D and C+ D are convex cones.

4. A convex combination of the vectors x,, x2, ..., x, is a linear com- bination of the form «,x, + ¤z x2 + +a, x, where a; 2 0, for each i; and a, + az +…+¤, = 1. Given a set S in a vector space, let K be the set of vectors consisting of all convex combinations from S. Show that K = co(S). 5. Let Cand D be convex cones in a vector space. Show that Cn D and C+ D are convex cones.

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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4. A convex combination of the vectors x1, x2, ..., X, is a linear com- bination of the form a,x1 + ¤z x2 ++ a, x, where a, 2 0, for each i; and a, + az +… + ¤, = 1. Given a set S in a vector space, let K be the set of vectors consisting of all convex combinations from S. Show that K = co(S). 5. Let Cand D be convex cones in a vector space. Show that C O D and C+ D are convex cones.
Transcribed Image Text:4. A convex combination of the vectors x1, x2, ..., X, is a linear com- bination of the form a,x1 + ¤z x2 ++ a, x, where a, 2 0, for each i; and a, + az +… + ¤, = 1. Given a set S in a vector space, let K be the set of vectors consisting of all convex combinations from S. Show that K = co(S). 5. Let Cand D be convex cones in a vector space. Show that C O D and C+ D are convex cones.
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