3. Determine whether each of the following sets is a subspace of the vector space Pn(R) of polyno-mials of degree at most n.(a) U = {ƒ € Pn (R) |f(-x) = f(x)} (b) V = {ƒ Pn(R) | ƒ(1) = 1}.

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Answered: 3. Determine whether each of the…

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 P3[x] be the vector space of all polynomial of order at most 3. Suppose that B = { 1, (x-1), (x-1)2, (x-1)3}. Show that B is a basis for P3[x]. Give p(x) = x3+ x, find [p(x)]B.
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Let V be a finite dimensional vector space. Let T E L(V). Show that there exists an integer n such that Ker(T") = (a) Ker(T+1) = Ker(T"+2) Hint: Ker(Tk) c Ker(Tk+1) (b) .. Show that V = Ker(T") Im(T").
2. Determine whether the following subsets of the given vector spaces are linearly dependent or independent: a. {2 x − 4x², x + 2x², −1 + x + 3x²} ≤ P₂ -3 1 b. {[2] [3] [73]} CM₂ (R)
20. Let f be a continuous linear functional defined on the normed linear space X. Show that M = {xe X\f(x) = 0) is a closed subspace of X.
Let P2(R) be the vector space of polynomials over R up to degree 2. ConsiderB ={1 + x − 2x^2 , −1 + x + x^2, 1 − x + x^2},B' ={1 − 3x + x^2, 1 − 3x − 2x^2, 1 − 2x + 3x^2}.(a) Show that B and B' are bases of P2(R).(b) Find the coordinate matrices of p(x) = 9x^2 + 4x − 2 relative to the bases B and B'.(c) Find the transition matrix P_B=→B'(d) Verify that, [x(p)]B'= P_B→B' [x(p)B].
Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤ b is a vector space, with addition and numerical multiplication defined by (f+g)(x) = f(x) + g(x) and (tf)(x) = tf(x).
4. Let T: U→V be a linear transformation from a vector space U (F) into a vector space V(F) and U(F) be finite dimensional. Show that U and R(T) have the same dimension iff T is nonsingular.
Let fi = 1+2x + 3x², f2 = 2 + 3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao +a1x+azx²[ao – 2a1 +a2 = 0} of R2[r]. Let g = 1+5x + 9x². If [g]g = (,) then s = a) -7 b) -5 c) 5 d) 7 e) g is not in V.
2. Let U and W be a subspaces of an n-dimensional vector space V and suppose further that UNW= {0} and U +W =V. Prove that dim(U)+dim(W)=n.
4. Let V and W be finite dimensional vector spaces. Let T : V –→ W be a linear transformation of V into W. (a) Let H be a subspace of V. Let T(H) be the set of all images of the vectors in H under the transformation T. Show that T(H) is a subspace of W and dim T(H) < dim H (b) Now suppose that T is one-to-one. Show that dim T(H)= dim H.
a) Let P: the vector space of all polynomials of degree ≤ 2. Prove that W= (p = P₂: p(1) = 2 p(0)) is a subspace of P:. b) Let G=Span{(1, 1, 0), (1, 0, 1)). If the vector v = (m. - (1+m), -m) belongs to G. Find the value of m.

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3. Determine whether each of the following sets is a subspace of the vector space Pn(R) of polyno- mials of degree at most n. (a) U = {ƒ € P₂ (R) | f(-x) = f(x)} (b) V = {f € Pn(R) | ƒ(1) = 1}.