Question

Prob. 4 (12.5 pts) The set of vectors S = {p1.p2.p3 } may be a basis for P2 p1 = 1 + x + x2 p2 = x + x2 p = x² a) Verify that this is the case. b) If it is a basis, find the coordinate vector of b relative to S. b = 7 - x + 2 x2

Answer 1

Given set of lectors is = +be+peÞ2 = x+xy j5z = be? S = $b bel por = $i+be+be? 3+xt, ky We have to show that s is basis of Ferst we shall show that $ és Lenearly independent. I Let a. (1+x+xt ta, (xx) + 02 (28)=0 ao cid + x capta) th (ata, to)

13 Where a, a, a, ER and o denotes the zero polynomial. By the defination of equality of bolynomials we have f ao=0, 9, = 0, 920 Thus sis linearly endependent I Also any polynomead for EP (R) can be expressed as linear combination

- the elements of s. vel fcx) = 2()) tla, tab) x+(949,49) Henke Ses a basis set. To find the coordinate of b=7x42x relative to s. Consider txt2x1² = 961) + Caota) Artcata, +92)

Comparing the coefficients of like Power on both sides, we get Tao To acto, = - a, -+7=-8 Qoto, Ho, = 2 ab - 2+a, tad F2-(9.te) + 2+=3 Henke Coordinates of b=7-x+2112 relateve to Sare & 7, 8, 32 -