Let f : R² → R be defined byƒ((x, y)) = 7x − 5y. Is ƒ a linear-transformation?a. f((x₁, y₁) + (x₂, Y2)) ==(Enter x₁ as x1, etc.)f((x1, y₁)) + f((x2, Y2)) =+Doesf((x1, y1₁) + (x2, Y2)) = f((x1, Y₁)) + f((x2, Y2))for all (x1, y₁), (x2, Y₂) = R²?chooseb. f(c(x, y)) =c(f((x, y))) =Does f(c(x, y)) = c(ƒ((x, y))) for allcЄR and all (x, y) = R²?choosec. Is f a linear transformation?choose

Answered: Image /qna-images/answer/457bf43a-aa8d-4ca0-b44d-ed8da7ddf591.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 f((x, y)) = 7x + 7y + 1. Isf 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) ER2? 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) = R²? choose c. Is f a linear transformation? choose Note: In order to get credit for this problem all answers must be correct. →
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
7. Let T : R² → R³ be a linear transformation with T(x1, x2) = (2x1 X2, -3x1 + - x2, 2x1 – 3x2). Can you find X (x1, x2) such that T(x1, x2) = (1,2, 3)?
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 ⇒
2 Let T₁: P₂ → R² and T₂: R² → R2x2 be linear transformations defined as follows. Ti (az² + bx + c) = [a+b] ¹([2])-[ (T₂0 T₁) (2x² + 4x + 5) = 5х1 2x1 -9x2 -8x2 [ Ex: 42
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 f: R R be defined by f((x, y)) = 7y - 3x - 3. Is f a linear transformation? a. f((x₁, y₁) + (x₂, y₂)) = etc.) 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 R ²? choose b. f (c(x, y)) = c(f((x, y))) = Does f(c(x, y)) = c(f ((x, y))) for all c ER and all (x, y) ER ²? choose c. Is f a linear transformation? choose . (Enter x₁ as x1, <
Letf : R² → R be defined by f((x, y)) = −7x − 9y. Is ƒ a linear transformation? a. f((x₁, y₁) + (x2, Y₂)) = as x1, etc.) f((x₁, y₁)) + f((x2, 3/2)) = Does f((x₁, y₁) + (x2, Y2 )) = f((x₁, y₁ )) + ƒ((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 E R and all (x, y) = R²? choose c. Isf a linear transformation? choose . (Enter X₁
Let f: R² → R be defined by f((x, y)) = -8-8y +6. Is ƒ a linear transformation? a f((₁, ₁) + (12, Y2)) = f((₁, 1)) + f((F2, Y2)) = Does f((11, y1) + (12, Y2)) = f((F1, Y₁)) + f((2, 2)) for all (T₁, Y₁), (2, 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 ₁ as x1, etc.)

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