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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21. (a) Prove that for every positive integer n, one can find n +1 linearly independent vectors in F(-∞, ∞o). [Hint: Look for polynomials.] (b) Use the result in part (a) to prove that F(-∞, ∞) is infinite- dimensional. (c) Prove that C(-∞, ∞), C (-∞o, co), and C (-∞, ∞) are infinite-dimensional.
Suppose T: M22→P2 is a linear transformation whose action on a basis for M2 2 is as follows: 2 -2 0 0 0 3 11 -4x2+12x-4 T| x2 4x 3x2 – 12x+3 T = 2x2 - 12x-1 0 -2 1 0 0 3 12 Describe the action of T on a general matrix, using x as the variable for the polynomial and a, b, c, and d as constants. Use the 'A' character to indicate an exponent, e.g. ax^2-bx+c. a = 0 lса
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find a basis for the range from r2 to r3 as defined by [3x1+4x2; x1-2x2; 4x1]

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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)