Which of the following mappings is not a linear transformation?1. T:C*(-∞,∞) → C(-∞,∞0) such that T'(f) = f +2f"2. V is an inner product space and T:V → V such that T(v) =< v, vo >, vo is anonzero vector in V.3. T:P, → Pe such that T(P(x)) = x²p²(x).4. None of theseO option1O option2O option3option4

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Answered: Which of the following mappings is not…

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 Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D : P3 - P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) = P3. Is D a linear transformation? Let p(x) = a³x³ + ²x² + a₁ + a₁ and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = = Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ✪ c. Is D a linear transformation? choose ◆
Let T : R? → R³ be a linear transformation such that T(x, y) 2y, -x+3y, 3x- 2y). Find a vector x = (x1, x2) such that T(x) (-1,4, 9). (r - (5-5) A) (5,3) B) (8.3) (-4,6) D) E)
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
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A function T:VW between vector spaces which satisfies T(a+b) = T(a) + T(b) for all a, b eV must be a linear transformation. Select one O True False
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, <
Let the linear transformation T : R² → R² be such that T(1, 1) = (1, –2), T(-1,1) = (-4, 1) Find a formula for Т(а, у)аnd use that formula to get T(5, —3) tip : write an arbitrary vector (x, y) as a linear, combinationof (1, 1) and (-1,1))

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