Homework SourseGiven ordered basis B 1 x 5 z2 x 4 z3 x2 2x 1 for P3
Given ordered basis B = [1, x + 5, z^2 + x - 4, z^3 + x^2 + 2x + 1] for P^3 (x) and vector v element P^3 (x) having presentation v = (-13 17 -10 8)_B with respect to this basis find coefficients b_1, b_2, b_3, b_4 element R such that v = b_1 + b_2 z + b_3 x^2 + b_4 z^3 element P^3 (x). b_1 = b_2 = b_3 = b_4 = ![Given ordered basis B = [1, x + 5, z^2 + x - 4, z^3 + x^2 + 2x + 1] for P^3 (x) and vector v element P^3 (x) having presentation v = (-13 17 -10 8)_B with resp](/WebImages/33/given-ordered-basis-b-1-x-5-z2-x-4-z3-x2-2x-1-for-p3-1096195-1761578100-0.webp "Given ordered basis B = [1, x + 5, z^2 + x - 4, z^3 + x^2 + 2x + 1] for P^3 (x) and vector v element P^3 (x) having presentation v = (-13 17 -10 8)_B with resp")
Solution
We have v = -13(1)+17(x+5)-10(x2+x-4)+8(x3+x2+2x+1)= -13 +17x +85 -10x2-10x+40+ 8x3+ 8x2+16x+8 = 8x3-2x2+23x+120. Thus, b1 = 8 ,b2=-2, b3=23 and b4 = 120.
