You are given the following data points: (i) Construct a Lagrange basis of P_2 u

Question

You are given the following data points: (i) Construct a Lagrange basis of P_2 using the x values from the data set. (ii) Hence find the unique polynomial in P_2 that fits the data exactly. (iii) Express x^2 as a linear combination of the Lagrange basis vectors that you found in part (i). (b) Let B = (1 2 0 1 1 0 1 -1 0 -1 3 -4) and let T = {(1 1 0), (2 0 -1), (0 1 3), (1 -1 -4)}. (i) Show that the system of equations Bx = b is consistent for all b element R^3. (ii) Hence (or otherwise) show that T spans R^3. (iii) Explain why T is not a basis for R^3. (iv) Find a subset of T that is a basis for R^3, or explain why no such subset exists. (c) Let X = {f elementof F | f(x) = c for all x elementof R, where c is a real number}. Determine whether or not X is a subspace of F. Justify your answer.

Explanation / Answer

lagrange interpolation basis function is

i)

Lo(x) = (x-x1) (x-x2)/(x0-x1) (xo-x2)

= (x - 0) (x-2) /(-2-0) (-2-2)

= x(x-2) / 8 = (x^2-2x) / 8

L1(x) = (x-x0) (x-x2)/(x1-x0) (x1-x2)

= (x+2)(x-2)/(0+2) (0-2)

= x^2-4/-4 = (4-x^2)/4

L2(x) = (x-x0) (x-x1)/(x2-x0) (x2-x1)

= (x+2) (x-0) /(2+2) (2-0)

= (x^2+2x)/8

ii)

unique polynomial is

P2(x) = y0 * L0 + y1 * L1 + y2 * L2

= 4 * (x^2 - 2x) /8 + 0 * L1 + 20 * (x^2+2x)/8

= x^2/2 - x + 5x^2/2 + 5x

= 6x^2/2 + 4x

= 3x^2 + 4x

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