2. (a) Show that the list {p o(x),p 1(x), p 2(x),... ,P n(x),...,} wherePo(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),...)?

Answered: Image /qna-images/answer/ba29158c-e6f7-4b51-8890-e2401fc4f2ca.jpg

Answered: (a) Show that the list {p o(x), p 1(x),…

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 V = {(x, 8) | x E R}. You are given that (V, 0, O) is a vector space with (x1, 8) O (x2, 8) = (x1+ x2, 8) and c O (x, 8) = (ca, 8). What is the zero element of V? What is the additive inverse of (5, 8) in V?
Find a basis {p(x), g(x)} for the vector space {f(x) = P₂(1) | ƒ'(−2) = f(1)} where P₂(x) is the vector space of polynomials in with degree less than or equal to 2. |p(x) = , q(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), ...)?
(b) Is the following set of vectors S basis for vector space V? Where (i) S = {(1,2), (0, 3), (1, 5)} and V = R? (ii) S = {(-1,3, 2), (6, 1, 1)} and V = R³ If not explain the reason.
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) =
(b) Show that the set {V₁, V2, V3, V₁, V₁} = {1+x, 1-x, x² + x³, 1-x², x² + x²} is a basis for P4 (R).
Find a basis {p(r), q(x)} for the vector space {f(x) e P2[z] | f' (-8) = f(1)} where P2[T] is the vector space of polynomials in x with degree at most 2.
1. Vectors v1 = (1,1,1,1,0), v2 = (-1,1,0,0,0), v3 = (1,1,-1,-1,0), v4 = (0,0,2,-2,0) and v5 = (0,0,0,0,2) form an orthogonal basis for a of R5 and X = (-3, -2,5,1,-4). Find the coordinates of X with respect to the given basis.
Answer the following questions: (a) () Find the 12-norms of the vectors, x = (1,2,3)™ and y = (1,2,3)T and y = (9, 11, 13). (b) ma Show that the set, ( is orthonormal. T T 5-{(2,-1,0)", (1,2-5), (2.4.2)"} 2, 4, = S = T 1 30 √24 (c) i- Let f(x) sin x and consider co = 4, x1 9 and x2 = = after rounding. Now, Find the polynomial using Least Square Approximation for n = 1. -6. For f(x) take up to 3 decimal places
12. Let B = {1+2x, x - x²,x + x²}. (a) Show that B is a basis for P2. (b) Let p(x) = 1+ 3x + x². Compute the coordinate [p(x)]g of p(x).
(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.

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