4.2. Let x be an element of a convex set S. Assume that x1 =x + €ip E S andX2 = x – €2p E S, where p # 0 and e1, €2 > 0. Prove that x is a convexcombination of x1 and x2. That is, prove thatx = ax1 + (1– a)x2,

Answered: Image /qna-images/answer/0eb3f340-13cb-4654-b0a0-82320eb5d1ca.jpg

Answered: 2. Let x be an element of a convex set…

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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Given that S, T CR are nonempty, bounded subsets of the set of real numbers. State whether True or False. Justify your answer in each case. (a) Then sup (SNT) < sup S. (b) Let U = {s +t:s € S,t ET}, then sup U = sup S + sup T {1+ " : n e N}, and T := {1 - )" : n e N}, then inf S = 0 (c) Suppose that S:= and sup T = 2.
c) given an arbitrary convex function wc) :R"-tR, prove that is a convex set.
2. Let a,, az,., a, be numbers. Show that there is a single number r such that foo: (z-a) +(2 – a,)° +..+ (x – a,) is a minimum and find this number.
2 Consider the set S = {x² + 1, 2x − 2, −x² + x}. The set S spans P2. If t(x) = 5x² + x + 2 € P₂, then t can be written as a linear combination of elements of S. That is, t(x) = Determine a, b, and c a = = a · (x² + 1) + b · (2x − 2) + c · (–x² + x). b = C = Question Help: Video Message instructor Post to forum
Exercise 3.3.15. Prove or find a counterexample to the following statement. Let A, B and C be sets. Then (A-B) UC = (AUBUC) - (ANB).
Q. Let A Then A is = {{x,y), y = 22, xe Z, y € R}, a. connected and pathwise connected space. b. connected but not pathwise connected space. c. connected but not pathwise connected space. d.neither connected nor pathwise connected space. a b d
The conditions for affine dependence are stronger than those for linear dependence, so an affinely dependent set is automatically linearly dependent. Also, a linearly independent set cannot be affinely dependent and therefore must be affinely independent. Construct two linearly dependent indexed sets S1 and S2 in R2 such that S1 is affinely dependent and S2 is affinely independent. In each case, the set should contain either one, two, or three nonzero points.
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Exercise. Suppose that X is a non-empty set, and that a function p: X x X → R satisfies the following two conditions: a) For every x, y E X, we have p(x, y) = 0 if and only if r = y. b) For every x, Y, z E X, we have p(x, y) < p(x, z) + p(y, z). Prove that p is a metric on X. Exercise, Suppose that (xn)nEN and (yn)nɛN are two sequences in a metric space (X, p) such that xn x and yn y as n → 0. Prove that the real sequence (p(xn, Yn))neN Converges to p(x, y) as n → 0. Exercise, a) For every ,x, y E R, let p(x, y) =| x² – y² |. Is pa metric
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1. Let = (1, 2). a. Explain graphically the set S = {(x, y) : (x − 3, y − 1) · √ = 0}. b. Draw the the points in set S in a Cartesian plane.
Recall the definition of Linear Independence: The set {V1,..., Vn} is linearly independent if, whenever a₁ V₁ +...+ an Vn = 0, it must be that a₁ = = an = 0. Suppose that {v2, V3} is linearly independent. Briefly describe what is wrong with the following "proof" that {V1, V2, V3} is linearly independent (where V₁ is some nonzero vector): Since (v2, V3} is linearly independent, a2V2 + α3 V3 = 0. Then, if a₁V₁ + a2 V2 + a3 V3 = 0, we have that a₁ V₁ +0=0 so that a₁ = 0. Therefore, {V1, V2, V3} is linearly independent.

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2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p # 0 and e1, €2 > 0. Prove that x is a convex combination of x1 and x2. That is, prove that x = ax1 + (1 – a)x2,