Problem 7. Suppose D E L(P3(R), P₂ (R)) is the differentiation map defined by Dp = p.Find a basis for P3(R) and a basis for P₂(R) such that the matrix of D with respect to thesebases is[1000]1 0 000 0 1 0

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Answered: Problem 7. Suppose D E L(P3(R), P₂ (R))…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 4CM

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Let Pn denote the vector space of polynomials in the variable x of degree ŉ 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) = P3. Is D a linear transformation? Let p(x) = a3x³ + ª²x² + ª₁x + ªº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 and c € 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(æ)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c € R and all p(x) = P3? choose c. Is D a linear transformation? choose (Enter a3 as a3, etc.)

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Problem 7. Suppose D E L(P3(R), P₂ (R)) is the differentiation map defined by Dp = p'. Find a basis for P3(R) and a basis for P2(R) such that the matrix of D with respect to these bases is [1000] 0 1 0 0 0010