Question

2008-2. Please provide clear justified step-by-step solutions (preferably handwritten) for the following question. The answers have been provided.

Questions:

(a) Let X be the set of functions f : R → R in F such that the graph of y - f(x) passes through the point (0, 0). Show that X is a subspace of F (b) For any real number m, define pm E P1 by Pm(x)-mx +1. (c) Let Z P, P2, p3) C P2, where pi(x) -3, 2(x)3+x and (d) You are given the following data points Show that the set Y-{ m 1 m є R} is not closed under addition p3(x) 3x +2r2. Determine whether or not Z is a basis for P2 r1 0 2 yl-6 0 12 0 (i) Construct a Lagrange basis of Ps using the a values from the data (ii) Hence find the unique polynomial in Ps that fits the data exactly. set.

Answers:

2. (a) See linear mathematics assignment Question 2. (c) Z is a basis for P2. r(x 1( -2) (x1)(x 1) (x -2) (r+1)x(x - 2) (x+1)x(x-2) -6 -2

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