denote the vector space of polynomials of degree up to 2. Which of the following subsets of P2 are subspaces of P₂?LET P₂□A. {p(t) | p' (t) is constant }B. {p(t) | p(-t) = -p(t) for all t}c. {p(t) | p(t)dt = 0}D. {p(t) | p' (4) = p(0)}□E. {p(t) | p' (t) + 7p(t) + 5 = 0}F. {p(t) | P(8) = 3}

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Answered: denote the vector space of polynomials…

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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Q # 2(b) Let X (a, b) and Y Prove that X and Y are homeomorphic (i.e. there exists a function f: X Y which is a homeomorphism). (0, 1) be two subspaces of R (with the usual topologies).
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E 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) = ? 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 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
Recall that P₂(R) is the vector space of all polynomials of degree at most 2 with coefficients in R, with usual polynomial addition and scalar multiplication. For p = P₂(R), let p' denote the first derivative of p with respect to x. Show that U = {p(x) = P2(R) | p′(1) = p(0)} is a subspace of P₂(R).
The vector x is in a subspace H with a basis B = {b₁,b₂}. Find the B-coordinate vector of x. 1 -1. b₁ = 5 - 8 MG-18 [X] B b₂ - 1 |-*-[ X: 5 - 1
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
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) is constant } B. {p(t) |p(-t) = -p(t) for all t} c. {p(t) | fp(t)dt = 0} D. {p(t) | p'(4) = p(0)} □E. {p(t) | p' (t) + 7p(t) + 5 = 0} F. {p(t) |p(8) = 3}
Let the vector C represents the coefficients of a polynomial pt=c1+c2t+c3t2+c4t3. Express the conditions p0=1, p'0=2, p1=1, p'1=0 in the form AC=b.
Let V = P² be the vector space of polynomials of degree at most 2, and let H be the span of the vectors f1(t) = t² – 2t + 1 and f2(t) or why not? = 2t2 – t – 1. Is the vector g(t) = t + 1 in the subspace H? Why
Let Pn be the vector space of polynomials of degree at most n and define L:P2 → P1 : p(t) → p'(t), p(t) e P2. (a) Show that L is linear. (b) Compute ker(L) and find dim(im(L)). (c) Is L is injective? Is L surjective? Justify your answers.
Q4. Let V be P2 with the inner product given by evaluation at -2,-1,0,1 and 2. Let p1(t) = 1+3t, P2(t) = 2t2 and p3(t) = t2 – 1. (a) Find the orthogonal projection of p3 onto the subspace spanned by P1 and p2. (b) Compute the lengths of the vectors p1(t) and p2(t).
= 4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ = {ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by Define T: P2 P₂ by (p\q) = p(x)q(x) dx. 0 T(ao + ax + a2x²) = a₁x. (a) Show that T is not Hermitian (i.e. self-adjoint). (b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian. Explain why this does not contradict with result from Part (a).

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denote the vector space of polynomials of degree up to 2. Which of the following subsets of P₂ are subspaces of P₂? LET P₂ A. {p(t) | p' (t) is constant} B. {p(t) | p(−t) = −p(t) for all t} c. {p(t) | f p(t)dt = 0}