* 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.

Answered: Image /qna-images/answer/e524ef05-3f13-4315-9974-f4cab5fe52bd.jpg

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

See similar textbooks

Similar questions

3. If z = f(x, y) with the relation xy²z³ + x®y²z+1= x + y+ z. Find at (1, 1, 1).
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).
1. Find the following for the function f(x, y) = 3x³y² - 5x+26y - 9. a. f(1,2)= b. fy(x, y) = c. fx(x, y) = d. fx(1,2)= e. fyy(x, y) = f. fxx (x, y) = g. fxy(x, y) = h. fyx (1,2)=
9.
2. For a linear function f: R" → Rm, which of the following statements is/are true? Select all that apply. ☐ Take n = 3, m = 1 and v₁ = [1,1, 1]¹, v₂ = [1,0, 1], v3 = [2, 1, 2] T. Suppose f(v₁) = 1 and f(v₂) = -3. We must have f(v3) = -1 There exists matrix A in Rmxn so that for any x in R", f(x) = Ax. For zero vectors On in R" and Om in Rm, we have f (On) = Om. ☐ For any x in R", define g(x) = 2x + c, where c is a non-zero vector in R". Then g is a linear function defined on g : R" → Rn.
18. Let A = = {W,Y, Z}, C = {r, V2, V5}. Find examples of two functions f: A→ B, and g : B → C, such that {1,2, 3}, B || (a) f is one-to-one, and yet g of is not.
a)Suppose f: R → Z, where f(x) = [x + ¹] i. Graph the function f(x). ii. If A = {x: 1 ≤ x ≤ 3}, find f(A). iii. Find f-¹({0}) iv. Find f¹({-1,0,1)) b) Let the functions f, g and h be defined as follows: f:R Rf(x) = 4x - 3 →>> g: RR; g(x) = x² + 1 h: RR;h(x) = (1 if x ≥ 0 if x < 0 lo Find the rules for the following functions: i fog ii. gof iii. foh iv. hof
I. If z = In(xyz) implicitly defines a function, then ∂z/∂y = z / (z−y) II. Any function of the form z= f(x+t) + g(x−t) with f,g functions of one variable and of class C2 satisfies∂2z / ∂x2 =d2f / du2+ d2g / dv2 and ∂z/ ∂t = df / du - dg / dvwhich is correct, incorrect or both
5.) Find fog and go f, and their domains. f(x) = VT – 4, 9(x) = x²
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.
Prove the following

SEE MORE QUESTIONS

Recommended textbooks for you

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

* 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.