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Let Pn denote the vector space of polynomials in the variablex 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) E P3. Is D a linear transformation? Let p(x) = azx³ + a₂x² + ₁x + a₁ and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 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) = P3? 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 P3? choose c. Is D a linear transformation? choose (Enter a3 as a3, etc.) + +

Let Pn denote the vector space of polynomials in the variablex 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) E P3. Is D a linear transformation? Let p(x) = azx³ + a₂x² + ₁x + a₁ and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 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) = P3? 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 P3? 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 P, 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) E P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + ₁x + ao and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 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) E P3? 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 P3? choose c. Is D a linear transformation? choose (Enter a3 as a3, etc.) + +
Transcribed Image Text:Let P, 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) E P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + ₁x + ao and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 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) E P3? 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 P3? choose c. Is D a linear transformation? choose (Enter a3 as a3, etc.) + +
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