Let P_2 denote the vector space of all polynomials in the variable x of degree l

Question

Let P_2 denote the vector space of all polynomials in the variable x of degree less than or equal to 2. Let C = {-3, -3 + 3x, -2 + x + 3^2} be an ordered basis for P_2. Write -6x+ 9x^2 as a linear combination of elements from the basis C. -6x + 9x^2 = _______ (-3) + _________ (-3 + 3x) + __________ (-2 + x +3x^2). Let [q]_C denote the coordinate representation of q relative to the basis C. Find the coordinate vector representation for -6x + 9x^2 relative to the basis C. Your answer should be a vector of the general form [-6x + 9x^2]_C

Explanation / Answer

let (a,b,c) be coordinate representation of [q}c

so

(-6x+9*x^2) = a (-3) + b ( -3+3x) +c(-2 + x + 3 x^2)

so

-3a -3b-2c = 0

3b+c = -6

3c = 9

solving we get (c = 3 , b = -3 , a = 1)

hence

(-6x+9*x^2) = 1* (-3) -3 *( -3+3x) +3*(-2 + x + 3 x^2)

b) (a,b,c) = (1,-3,3)

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