Let ƒ : R → R³ be defined by ƒ(x) = (x, −6x², −6x). 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 E R? chooseb. f(cx) =+c(f(x)) ==Does f(cx) = c(f(x)) for all c, x € R? choosec. Isf a linear transformation? choose>

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Answered: Let ƒ : R → R³ be defined by f(x) = (x,…

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 : R → R³ be defined by f(x) = (−5x, 6x, 5x). Is ƒ a linear transformation? a. f(x + y) = -5x-5y, 6x+6y, 5x+5y f(x) + f(y) = -5x, 6x, 5x + -5y,6y, 5y Does f(x + y) = f(x) + f(y) for all x, y E R? Yes, they are equal b. f(cx) = -5cx, 6cx, 5cx c(f(x)) = C -5x, 6x, 5x Does f(cx) = c(f(x)) for all c, x ER? Yes, they are equal c. Is f a linear transformation? f is a linear transformation
Suppose y = f (x) is transformed to the function y = 2f [- 1/3 (x + 2) ] - 4. Write this as a mapping transformation.
4. Describe the transformations of y=-2f[3(x– 4)]–5.
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
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 + + +
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
given f(x) = x5 a) Determine a mapping rule that can be applied to f(x) = x5 to obtain the transformed function of y=3f (-1/2x - 1) + 4 b) Determine the image point of f(1) on the transformed graph. Show all your work

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Let ƒ : R → R³ be defined by f(x) = (x, −6x², −6x). 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 ER? choose b. f(cx) = c(f(x)) = Does f(cx) = c(f(x)) for all c, x € R? choose c. Isf a linear transformation? choose