Exercise 1.13 In the vector space R[X]<3 = {p(X) = ao + a₁ + a₂X² + a3X³|a, € R} of polynomials of degree less or equal to 3, consider the subsets U = {p(X) €R[X]<3 P(0)=0}; a) Determine the values & R such that W is a subspace of R[X]53. b) For the values & R found in a), write down a basis of UnWk. c) For the values k € R found in a), determine U+W. Is it a direct sum? W₁ = {p(X) €R[X]<3[p(−1) = k}. d) For the values k R found in a), write, if possible, the polynomial 1-X in two different ways as a sum of a polynomial in U and one in Wk- e) Compute dim(W) for all k € R.
Exercise 1.13 In the vector space R[X]<3 = {p(X) = ao + a₁ + a₂X² + a3X³|a, € R} of polynomials of degree less or equal to 3, consider the subsets U = {p(X) €R[X]<3 P(0)=0}; a) Determine the values & R such that W is a subspace of R[X]53. b) For the values & R found in a), write down a basis of UnWk. c) For the values k € R found in a), determine U+W. Is it a direct sum? W₁ = {p(X) €R[X]<3[p(−1) = k}. d) For the values k R found in a), write, if possible, the polynomial 1-X in two different ways as a sum of a polynomial in U and one in Wk- e) Compute dim(W) for all k € R.
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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![Exercise 1.13 In the vector space R[X]<3 = {p(X) = ao+a₁X + a₂X² + a3X³|a; € R} of
polynomials of degree less or equal to 3, consider the subsets
U = {p(X) €R[X]<3 P(0)=0};
a) Determine the values & € R such that W, is a subspace of R[X] 53.
b) For the values & R found in a), write down a basis of UnWk.
c) For the values k € R found in a), determine U+Wk. Is it a direct sum?
W₁ = {p(X) € R[X]<3] P(-1) = k}.
d) For the valuesk R found in a), write, if possible, the polynomial 1
different ways as a sum of a polynomial in U and one in Wk.
e) Compute dim(Wk) for all k € R.
X in two](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedbd0a5d-0e43-41b1-a3de-4c5dcfca7a91%2F0c8d6c1c-aca1-45af-a302-a982778ac97e%2Frw6boai_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 1.13 In the vector space R[X]<3 = {p(X) = ao+a₁X + a₂X² + a3X³|a; € R} of
polynomials of degree less or equal to 3, consider the subsets
U = {p(X) €R[X]<3 P(0)=0};
a) Determine the values & € R such that W, is a subspace of R[X] 53.
b) For the values & R found in a), write down a basis of UnWk.
c) For the values k € R found in a), determine U+Wk. Is it a direct sum?
W₁ = {p(X) € R[X]<3] P(-1) = k}.
d) For the valuesk R found in a), write, if possible, the polynomial 1
different ways as a sum of a polynomial in U and one in Wk.
e) Compute dim(Wk) for all k € R.
X in two
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