Let f: R² → R be defined by f((x, y)) = 8x +9y. Is f a linear transformation?a. f((x1, y₁) + (x2, Y₂)) =f((x1, y₁)) + f((x2, Y2)) =+Does f((x1, y₁) + (x2, Y₂)) = ƒ((X₁,Y₁)) + f((x2, y₂)) for all (x₁, y₁), (2, Y2) € R²? chooseEb. f(c(x, y)):c(f((x, y))) =Does f(c(x, y)) = c(f((x, y))) for all c = R and all (x, y) = R²? choosec. Is f a linear transformation? choose. (Enter ₁ as x1, etc.)

Answered: Image /qna-images/answer/4afcd609-e24d-4af9-b1f0-a03dd99e3cd9.jpg

Answered: Let f: R² → R be defined by f((x, y)) =…

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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Let ƒ : R² → R be defined by ƒ((x, y)) = −6x − 5y + 8. Is ƒ a linear transformation? a. f((x₁, y₁) + (x₂, y2)) = f((x₁, y₁)) + f((x₂, y2)) = + Does f((x₁, y₁) + (x₂, y₂)) = f((x₁, y₁)) + f((x₂, y₂)) for all (x₁, y₁), (x₂, y2) E R²? choose . (Enter x₁ as x1, etc.) b. f(c(x, y)) = c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c E R and all (x, y) = R²? choose c. Isf a linear transformation? choose +
- Let ƒ : R² → R be defined by f((x, y)) = 3x + 8y − 1. Is ƒ a linear transformation? a. f((x₁, y₁) + (x₂, y₂)) = f((x₁, y₁)) + f((x2, y₂)) = = + Does ƒ((x₁, y₁) + (x2, y2)) = f((x₁, y₁ )) + ƒ((x2, y2)) for all (x1, Y1 ), (x2, Y2 ) E R² ? choose b. f(c(x, y)) : = c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c E R and all (x, y) = R²? choose c. Is f a linear transformation? choose . (Enter x₁ as x1, etc.)
Letf : R² → R be defined by ƒ((x, y)) = −3x – 7y. Is ƒ a linear transformation? a. f((x₁, y₁) + (x₂, y2)) = f((x₁, y₁)) + f((x₂, y2)) = Does f((x₁, y₁) + (x₂, y₂)) = f((x₁, y₁)) + f((x₂, y₂)) for all (x₁, y₁), (x₂, y₂) E R²? choose . (Enter x₁ as x1, etc.) b. f(c(x, y)) = c(f((x, y))) = = Does f(c(x, y)) = c(f((x, y))) for all c E R and all (x, y) = R²? choose c. Isf a linear transformation? choose
Letf: R2. →R be defined by f((x, y)) = 4y - 3x. Is f a linear transformation? a. f((x₁, y₁) + (x2, y2)) = b. f(c(x, y)) = f((x₁, y₁)) + f((x2, y2)) = + Does f((x1, y1) + (x₂, y₂)) = f((x₁, y₁)) + f((x₂, y₂)) for all (x₁, y₁), (x2, y₂) E R²? choose (Enter x₁ as x1, etc.) c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c ER and all (x, y) E R²? choose c. Is f a linear transformation? choose ⇒
Let f: R → R³ be defined by f(x) = (8x, -7x, -9x). Is ƒ a linear transformation? a. f(x + y) = f(x) + f(y) + Does f(x + y) = f(x) + f(y) for all x, y = R? choose b. f(cx) = = c(f(x)) = Does f(cx) = c(f(x)) for all c, x = R? choose c. Is f a linear transformation? choose
Letf: R² → R be defined by f((x, y)) = -6x - 8y. Isf a linear transformation? a. f((x₁, y₁) + (x2, y₂)) = b. f(c(x, y)) = f((x₁, y₁)) + f((x₂, y₂)) = + Does f(x,y1) + (x₂, y2)) =f((x₁, y₁)) + f((x2, y2)) for all (x₁, y₁), (x₂, y2) E R²? choose (Enter x₁ as x1, etc.) c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c ER and all (x, y) E R²? choose c. Isf a linear transformation? choose + + +
Let T:P3→P3 be the linear transformation satisfying T(1)=4x2+7, T(x)=4x+3, T(x2)=−3x2+x−6. Find the image of an arbitrary quadratic polynomial ax2+bx+c.
Let f: R² → R be defined by f((x, y)) = −9x – 7y — 9. Is f a linear transformation? a. f((x₁, y₁) + (x2, Y₂)) = f((x₁, y₁)) + f((x2, Y2)) = + Does f((x1, y₁) + (x2, y2)) = f((x1, y₁)) + f((x2, y2)) for all (x₁, y₁), (x2, y2) € R²? choose b. f(c(x, y)) = c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c = R and all (x, y) = R²? choose c. Is f a linear transformation? choose . (Enter x₁ as x1, etc.)
Let f : R? → R be defined by f((x, y)) = 5x + 7y. Is f a linear transformation? a. f({x1,Yı) + (x2, Y2)) : f({x1, Y1)) + f({x2, Y2)) = + Does f((x1,Y1) + (x2, Y2)) = f(x1, Y1)) + f((x2, Y2)) for all (01, Y1), (x2, Y2) E R?

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