* 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).
2. Find the degree of homogeneity, if there is one, for each of the following functions: (xjXzxz)² 1 1 1 (a) f(x1,X2, X3) = xf + x$ + x (++ X2 X2 (b) the CES function: x(v1, v,, ..., v,) = A (8, v,° + 8,v, + + 8,v,")
4. Let X an Y be sets with A, B C X and let f: X→Y be a function. Recall that f(A) = {f(x) EYE A), and f-¹(A) = {re A: f(x) = A}. (a) Show that A CS-¹(f(A)). (b) Give an example of a function f : X→ Y and the set A such that f(f(A)) ‡ A.
5. Find f og and g•f , where f(x) = x² + 1 and g(x) = x + 2, are functions from R to R.
= Evaluate [[2₁,22,23] ƒ (2) dz where 21 = −i, 22 = 2 + 5i, 23 5i and f(x + y) = x² + iy.
5.) Find fog and go f, and their domains. f(x) = VT – 4, 9(x) = x²
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}.
Suppose that f(x, y) is a differentiable function of two variables defined on the entire xy–plane, with the following property, (i) f(x, y) is a strictly increasing function of x for each fixed value of y, and (ii) f(x, y) is a strictly increasing function of y for each fixed value of x. Let g(t) be the function of a single variable defined by g(t) = f(t, t). A. Give an example of a function f(x, y) that has the indicated property, consider the associated function g(t), and analyze its increasing/decreasing behavior. B. Make a conjecture regarding the behavior of the function g(t) in the most general setting, and then prove your conjecture.

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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.