Problem 4. Let U be the subspace of P₁ (R) defined asUJ = {p(2) € Pall(a) Find a basis of U.= P₁(R) : [₁ p(x)dx =p(x)dx = p(0)x = p(0)}.(b) Extend the basis in part (a) to a basis of P₁(R)(c) Find a subspace W of P₁ (R) such that P₁ (R) = U + W.

Given that U be the subspace of P4ℝ defined as U=px∈P4ℝ: ∫-11p(x)dx=p(0) We have to find the basis…

Answered: Problem 4. Let U be the subspace of P₁…

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) Let V be a subspace of Rn and T: Rn → Rm be a linear transformation. Show that T(V) = {T(v) : 7 € V} is a subspace of Rm and dim(V) ≥ dim(T(V)). (Hint: let {vi,...,} be a basis for V.)
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, P;(x) = 1+x + + x³, •..) is a basis of the vector space F[x] of all polynomials with coeficients in F. (b) What are the coordinates of the vector x" relative to the basis (po(x),P1(x),..., P; (x),...)?
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), ...)?
at most 2. Find a basis {p(x), q(x)} for the vector space {f(x) = P₂[x] | f'(-8) = f(1)} where P₂[x] is the vector space of polynomials in x with degree You can enter polynomials using notation e.g., 5+3xx for 5 + 3x². p(x) = , 9(x) =
2. Show all the steps clearly please
(c) How many binary strings of length 5 have at least 2 adjacent bits that are the same (“00" or “11") somewhere in the string?
3) Let V = R[t]<2, W = M2x2 (R) with standard basis B = (1, t, t²), C = (m₁, m2, m3, m4). Suppose : V → W is some linear map such that, under the complexified basis Bc, Cc we have (a) (b) (c) [yc]c = 1 2 0 0 1 -1 22 2 1 4 0 What is yc(10 t²) What is p(1+t) (not the complexified function, the original) What is [y]
3. Let T = 1 7 W= 1 and x = 0 (a) Are vectors u, V, and w linearly independent? (b) Do u, v', and w span the entire R³? (c) Do u, v', and w form a basis of R³? abc po
(a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 4 9 -1 X1 + 2x2 - x3 X1 X1 X1 T| x2 4x1 + 9x2 - x3 i.e. T x2 X2 2x1 + 7x2 + 7x3) X3 2 7 7 X3 (i) Find a basis for the kernel of T. (ii) Find a basis for the image of T. (iii) Does the image of T include every vector in R'? Justify your answer. (You are not required to verify that your answers to (a) (i), (a) (i) are bases.) (b) Consider the following subset of R: X1 S = x2 ER: x2 + 4x3 = 2 Does there exist a real linear map, from R to R' say, such that the set S is the image of the real linear map? Justify your answer.
3. Let W1 = (1, 1, 0, 1), w₂ = (−1, 1, 0, 0), w3 = = (1, 2,0,0). Let W = Span(w₁, W2, W3 (a) Use the Gram-Schmidt process to obtain an orthogonal basis {V₁, V2, V3} for W. (b) What is Projw (3, 3, 3, 3), the orthogonal projection of the vector (3,3,3,3) onto W?
show that a set of vectors{et cos(2t), e¯t sin(2t)} form a basis of solution space for x" + 2x' + 5x = 0

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Problem 4. Let U be the subspace of P₁ (R) defined as (a) Find a basis of U. U = {p(x) € P.(1 : P.(R) : [", p(x)dx = p(0)} . (b) Extend the basis in part (a) to a basis of P₁(R) (c) Find a subspace W of P₁ (R) such that P₁(R) = UW.