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Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D:₂ → ₂ be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) EQ. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + ao and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in and c ER. 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) E₂? 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) = ₂? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.) .

Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D:₂ → ₂ be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) EQ. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + ao and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in and c ER. 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) E₂? 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) = ₂? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.) .

Advanced Engineering Mathematics
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 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.)
Transcribed Image Text: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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