* Prove the following (for level LM only):•(a) If fi : R→ R are convex and a > 0 for all i = 1, 2, . . ., m, then the functionf(x) := Σ²±²₁ αi fi(x) is convex.•:=(b) If g : R → R is an affine function and ƒ : Rm → R is convex, then h(x) :=f(g(x)) is convex.

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Answered: * Prove the following (for level LM…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

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Exercise II.2. let F(.) : R² → R³ and G(): R² → R² be given by (fi(x)) f2(x) f3(x)) y = 9/₁ F(x) = where f1(), f2), f3(-) are as in Exercise II.1(a). The notation is x = (a) x = and G(y) = (2y1 + 3y2 y² + y² 21 I₂ Use the Chain Rule to calculate (FG)'(x) for the various choices of x: - (₁²) (b) x = - (-³₁) (c) x = - (4) (d) x = For reference F1,F2, F3 Exercise II.1. Consider the case n = 2. (a) Find the gradients of the following functions from R2 to R: (i) f₁ (22) := x² + x² − 2 = 4x² + (iii) f3 (ii) f₂ := = 9(x₁ − 1)² + (²+¹)² — 2 (iv) fa x1 X2 (=-¹). I1 In x1 X₂ and −1 == ((x₁ + 1)² + (x₂ −2)²
6. Let S = {-1,0, 2, 4, 7}. Find f(S) (image of S under f) where f is from R to R for (a) f(a) = 1 (b) f(a) = 3r + 1 (c) [a2/3] (d) [x/4] 7. Find range of the following real functions f: R→ R. Note that you have to clearly point ou the interval, For example, if given real function f(x) = 2?, then range should be {æ |2 > 0!
21. Let f(x) = 2x, x = [0, 1]. Take P = {0, 1, 1, 1, 1, 1} and set x₁ = 1 x₁ = 16, x²=², x={, x3 = 1. Calculate the following: (a) Lƒ(P). (b) S* (P). (c) Uƒ(P).
Consider the three points (2, 2), (4,4), and (6, —2). (a) Supposed that at (2, 2), we know that fx = fy = 0 and ƒxx 0, function near the point (4,4)? ? fxy fry = 0. What can we conclude about the behavior of this (c) Supposed that at (6, -2), we know that fx = fy = 0 and fxx > 0, fyy < 0, and fxy = 0. What can we conclude about the behavior of this function near the point (6, -2)? ? Using this information, on a separate sheet of paper sketch a possible contour diagram for f.
= Evaluate [[2₁,22,23] ƒ (2) dz where 21 = −i, 22 = 2 + 5i, 23 5i and f(x + y) = x² + iy.
2. Consider a convex function f: R R. Recall that f is convex if f(tx+ (1-t)y) 0. • Prove that for any n f (a1x1 + a2x2 + ..+ anxn) 0. • Prove that – log(x) is convex • Hence or otherwise conclude the Arithmetic Mean-Geoemtric Mean inequality, i.e., if x;'s are non-negative real numbers, then xi + x2 + + xn
4. Let u be a root of ƒ = t³ − t² + t + 2 € Q[t] and K = Q(u). (a) Show that f = mo(u). (b) Express (u²+u+1) (u²-u) and (u-1)-¹ in the form au²+bu+c, for some a, b, c € Q.
2. Show that the following functions are convex by verifying the condition that ² f(x) 20 is satisfied for all a in the domain of f: (a) f(u₁, ₂) In(e" + e"), (b) f(ui, u2, U3, U₁)=-In(1-u₁-u2-uz-ua) over the domain {ue R4 | u₁+U₂+uz+us < 1}.
3. Let A = {1,2,3), let S = P(A) and let RC S x S be a relation defined by the rule (X,Y) E R (XnY=0) ^ (XUY = A). (a) Prove or find a counterexample for the following properties: • antisymmetry. asymmetry. symmetry. irreflexivity/ • reflexivity. • transitivity • uniqueness
2. Show that the following functions are convex by verifying the condition that V² f(x) ≥ 0 is satisfied for all a in the domain of f: (a) f(u₁, u₂) In(e" + e"), (b) f(u₁, U2, U3, U4) = ln(1-u₁-uz-us-u4) over the domain {u E R4|u₁ + ₂ + us+ us ≤ 1}. -

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* Prove the following (for level LM only): • (a) If fi : R→ R are convex and a > 0 for all i = 1, 2, . . ., m, then the function f(x) := Σ²±²₁ αi fi(x) is convex. • := (b) If g : R → R is an affine function and ƒ : Rm → R is convex, then h(x) := f(g(x)) is convex.