T: C[-1,1] → R is defined as T(p)Prove that statement!; T(p) = √₁₁ p(x)dx is a linear transformation.

Given is the transformation Tp=∫-11p(x)dx A transformation is linear if it satisfies the following…

Answered: T: C[-1,1] → R is defined as T (p) = ¹₁…

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 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.)
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
A certain radioactive substance has a half-life of 3.6 years. Find how long it takes (in years) for the substance to decompose to 10% of its initial amount.
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

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T: C[-1,1] → R is defined as T (p) = ¹₁ p(x)dx is a linear transformation. Prove that statement!