3.Given a nonstandard basis of a vector space of a second-order polynomial asB = { 1 + x, -x + x², 1 + x − x²}. If a transformation T: P₂ R³ is given asT(¹+x) - ().T-*+ **)-().ra+x-)-(3)=; T(−x x²) = 2;T(1 x²) =Determine T(1 - x + x²)

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Answered: 3. Given a nonstandard basis of a…

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), ...)?
Let V be a vector space, and T :V →V a linear transformation such that T(5v1 + 3v2) = 401 – 402 and T(301 + 202) = 2v1 + 572. Then T(7) T(52) T(201 – 202) =
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a) Verify the given sequence B is a basis ; and b) compute the images of the standard basis vectors with respect to the given linear transformation. 15. fi = 1+2x – 2x2, f2 = 2 + 3x – 4x2, f3 = 1 + 4x – x², B = (f1, f2, f3). - 1 T(fi) = (; 10)- 4 T(fi) = (; ).T ,T(f2) : 3 8
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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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3. Given a nonstandard basis of a vector space of a second-order polynomial as R³ is given as B = {1 + x, -x + x²,1 + x - x²}. If a transformation T: P₂ T(1 + x) = >-();r-x+x) - ()+*-*- (1) (19) |; x²) = x²) = ; T(1 + x - x²) = Determine T(1-x+x²)

3. Given a nonstandard basis of a vector space of a second-order polynomial as R³ is given as B = {1 + x, -x + x²,1 + x - x²}. If a transformation T: P₂ T(1 + x) = >-();r-x+x) - ()+-- (1) (19) |; x²) = x²) = ; T(1 + x - x²) = Determine T(1-x+x²)