Question 7 (Chapters 6-7) 2+2+2+3+2+4+4-19 mark Let 0メs c Rn and fix r' E S. For a R" consider the followi...
Question 8 (Chapters 6-7) 12+2+2+3+2+4+4-19 marks] Let 0メS C Rn and fix E S. For a E R consider the following optimization problem: (Pa) min a r, and define the set K(S,x*) := {a E Rn : x. is a solution of (PJ) (a) Prove that K(S,'). Hint: Check 0 (b) Prove that K(S, r*) is a cone. (c) Prove that K(S,) is convex d) Let S C S2 and fix eS. Prove that K(S2, ) cK(S, (e) Ifx. E...
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n 2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n
2. Let {xn}nEN be a sequence in R converging to x 0. Show that the sequence R. Assume that x 0 and for each n є N, xn converges to 1. 3. Let A C R". Say that x E Rn is a limit point of A if every open ball around x contains a point y x such that y E A. Let K c Rn be a set such that every infinite subset of K has a limit...
Exercice 1 We consider the function f(x) = 2 #0 and for r > 0. let S, = {€ C/2 = r} with positive orientation. For 0 < <R, we denote by r the curve consisting of SRUT-R,-€) US, UL, R), where S = {z E C/121 = } with negative orientation. 1. Prove that o = [513)dz = [5(=)dz + [s()de – [ (dz + 1" $(x)dr.
6. Let S : R + R3 be the linear transformation which satisfies |(1,0,0) = (1,0,–3), S(0,1,0) = (0,-1,0) and S(0,0,1) = (1,-1, -2). Give an expression for S(x, y, z). 4 Marks] Let S be the basis (1,0,0), (0,1,0), (0,0,1) for R3 and let T be the basis (0,0,1), (0,1,1), (1,1,1) for R. Compute the change of basis matrix s[1]7. (b) Compute the matrices s[S]s and s[ST. 18 Marks)
(7) Let 0くa 〈 b 〈 c 〈 d for a,b,c,d R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation (a) Sketch D in the x-y plane for the case ad - bc > 0. (a) Sketch D in the z-y plane for the case ad-bc 〈 0. (c) Calculate the area of D. Show all working. (7) Let 0くa 〈 b 〈 c 〈...
EASY 4.0 UTM IMAT102 WINTER 2019 ASY GROUP WEEKLY PACKAGE 2 PROBLEM SET C 2.5.2. Let S = {2,3} × {-2,-1,0), and let T be the set of all ordered pairs (z, y) E Z × Z such that -2 + 2y 3, Prove that S T. Does equality hold? Explain. 2.5.8. Let A = [-1, 1], B (-r,r), C = [2,00) and U = R be the universal set. Find the sets Anc, AnB, (Bnc)nz and B\A Use the...
In the vector space R, let 8 {(1,3,0), (1, -3, 0), (0, 2, 2)}. (a) (6 points) Show that y is a basis of R3. (b) (7 points) Find the matrix [I,where I is the identity transform R3 R3 (c) (7 points) Using the matrix [I, convert the vector (r, y, z) into coordinates with respect to y instead of B. In other words, find ((x, y, z)] {(1,0,0), (0, 1,0), (0,0, 1)} be the standard basis, and let
Question 3 (Chapter 6) 13+2+3+6 14 marks Fix p EN and consider the following set: : T1 (a) Prove that Cp is convex. (b) Prove that C, is a cone. (c) Compute Ci and C2. (d) Show that x = 0 is an extreme point of CP. Question 3 (Chapter 6) 13+2+3+6 14 marks Fix p EN and consider the following set: : T1 (a) Prove that Cp is convex. (b) Prove that C, is a cone. (c) Compute Ci...
7. Let E C R be nonempty, n E N, and K, L E Z such that K/n is an upper bound for E, but L/n is not an upper bound for E. (a) Show that there exists an for E, but (m - 1)/n is not an upper bound for E. (Hint: Prove by contradiction, and use induction. Drawing a picture might help) m < K such that m/n is an upper bound integer L (b) Show that m...