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? choosec. Isf a linear transformation? choose+++

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Answered: Letf: 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 f : R² → R be defined by f((x, y)) = 8x +2y + 2. Is ƒ a linear transformation? a. f((x₁, y₁) + (x2, Y2)): b. f(c(x, y)) = f((x1, y₁)) + f((x2, 2)): + Does f((x1, y₁) + (x2, Y2)) = f((x₁, Y₁)) + f((x2, y₂)) for all (x₁, Y₁), (X2, Y2) € R²? choose = c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c = R and all (x, y) = R²? choose . (Enter ₁ as x1, etc.) c. Is f a linear transformation? choose <
If L: R² R³ be defined by L(x, y) = (x, y, 1), show that it is a linear transformation.
Letf : R → R³ be defined by f(x) = (-7x, -3x, 8x). Is f 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 ER? choose c. Isf a linear transformation? choose
Show solution on paper please!
Let T : P3 → P3 be the linear transformation such that T(-2x²) = 3x² + 3x, T(-0.5x + 3) = 3x? + 4x – 1, T(5x² + 1) = -3x + 1. Find T(1), T(x), T(x²), and T(ax? + bx + c), where a, b, and c are arbitrary real numbers. T(1) = T(x) = T(x²) = T(ax? + bx + c) =
Let f: R²R be defined by f((x, y)) = 8y - 4x + 1. Is ƒ a linear transformation? a. f((x1, y₁) + (x2, Y2)) (Enter x₁ as x1, etc.) f((x₁, y₁)) + f((x2, y₂)) = + Does f((x1, y₁) + (x2, 2)) = f((x₁, y₁)) + f((x2, ₂)) 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 CER and all (x, y) = R²? choose c. Is f a linear transformation? choose
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
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 T1 : P2 → R² and T2 : R? → R2x2 be linear transformations defined as follows. a +b T1(ax? + bx + c) = b — с 4x1 2x1 T2 %3D -8x2 -2x2 x2 Ex: 42 (Т, о Ti) (-За? + 4г + 3)

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