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Let P, denote the vector space of polynomials in the variable of degree or less with real coefficients. Let D: P3 → P₂ be the function that sends a polynomial to its derivative. That is, D(p(z)) = p/(x) for all polynomials p(z) P3. Is D a linear transformation? Let p(x) = 3x³ + a₂z² + ª₁ + ª² and q(z) = b¸x³ + b₂x² + b₁z+bbe any two polynomials in P3 and c ER. a. D(p(z) +q(z)) = D(p(x)) + D(g(x)) = [ + Does D(p(z) + q(z)) = D(p(x)) + D(q(1)) for all p(x), g(x) = P3? choose b. D(cp(z)) = c(D(p(z))) = Does D(cp(z)) = c(D(p(z))) for all c R and all p(z) € P3? choose c. Is D a linear transformation? choose V (Enter as as a3, etc.)

Let P, denote the vector space of polynomials in the variable of degree or less with real coefficients. Let D: P3 → P₂ be the function that sends a polynomial to its derivative. That is, D(p(z)) = p/(x) for all polynomials p(z) P3. Is D a linear transformation? Let p(x) = 3x³ + a₂z² + ª₁ + ª² and q(z) = b¸x³ + b₂x² + b₁z+bbe any two polynomials in P3 and c ER. a. D(p(z) +q(z)) = D(p(x)) + D(g(x)) = [ + Does D(p(z) + q(z)) = D(p(x)) + D(q(1)) for all p(x), g(x) = P3? choose b. D(cp(z)) = c(D(p(z))) = Does D(cp(z)) = c(D(p(z))) for all c R and all p(z) € P3? choose c. Is D a linear transformation? choose V (Enter as 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 P₁, denote the vector space of polynomials in the variable 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(z)) = p/(z) for all polynomials p(z) = P3. Is D a linear transformation? Let p(x) = ªx³ + ª² + ª₁ + ª₁ and g(x)=b²x³ +b₂x² +b₁+bbe any two polynomials in P3 and C E R. a. D(p(x) +q(x)) = D(p(x)) + D(q(z)) = + Does D(p(x) + g(x)) = D(p(x)) + D(g(x)) for all p(x), q(z) = P3? choose b. D(cp(z))= c(D(p(z))) = | Does D(cp(x)) = c(D(p(z))) for all c = R and all p(x) = P3? choose c. Is D a linear transformation? choose (Enter as as a3, etc.)
Transcribed Image Text:Let P₁, denote the vector space of polynomials in the variable 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(z)) = p/(z) for all polynomials p(z) = P3. Is D a linear transformation? Let p(x) = ªx³ + ª² + ª₁ + ª₁ and g(x)=b²x³ +b₂x² +b₁+bbe any two polynomials in P3 and C E R. a. D(p(x) +q(x)) = D(p(x)) + D(q(z)) = + Does D(p(x) + g(x)) = D(p(x)) + D(g(x)) for all p(x), q(z) = P3? choose b. D(cp(z))= c(D(p(z))) = | Does D(cp(x)) = c(D(p(z))) for all c = R and all p(x) = P3? choose c. Is D a linear transformation? choose (Enter as as a3, etc.)
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