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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 + a₁x + a₂ and q(x) = a. D(p(x) + q(x)) = b. D(cp(x)) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 and c E R. D(p(x)) + D(g(x)) = + Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) E P3? choose c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) = P3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)

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 + a₁x + a₂ and q(x) = a. D(p(x) + q(x)) = b. D(cp(x)) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 and c E R. D(p(x)) + D(g(x)) = + Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) E P3? choose c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) = P3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 22E
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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) E P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + α and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 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) 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.) D
Transcribed Image Text: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) E P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + α and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 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) 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.) D
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