Let P₂ denote the vector space of polynomials of degree up to 2. Which of the following subsets of P₂ are subspaces of P₂?A. {p(t) | p(−t) = p(t) for all t}B. {p(t) | p(5) = 0}c. {p(t) | p(2) =1}D. {p(t) | p (1) = p(0)}Ē. {p(t) | p' (t) + 9p(t) + 5 = 0}F. {p(t) | p' (t) is constant }

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Answered: Let P₂ denote the vector space of…

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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2. (a) Show that the list {p o(x), p 1(x), p 2(x),...,p n(x),...,} where Po(x) = 1, P₁(x) = 1 + x, pj(x) = 1 + x + · +x², is a basis of the vector space F[x] of all polynomials with coeficients in F. (b) What are the coordinates of the vector xn relative to the basis (po(x), p₁(x),..., pj(x), ...)?
3. Let V = P3, the vector space of polynomials of degree at most 3. Let B {1, t, t², t³} and C = {t³,t,t², 1} be bases for P3. If f(t) = (1+ t)(t² + 4t + 1). Find [f(t)]8 and [f(t)]c
(. 4 Let B be the basis of R² consisting of the vectors P = and and let R be the basis consisting of -2 [B] 1 3 and 3 [B] -2 Find a matrix P such that [x] = P[x]g for all x in R². [88]
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Let P₂ denote the vector space of polynomials of degree up to 2. Which of the following subsets of P₂ are subspaces of P₂? A. {p(t) | p(−t) = p(t) for all t} B. {p(t) | p(5)=0} c. {p(t) | p(2) = 1} ] D. {p(t) | p′ (1) = p(0)}