Question

Let B be the standard basis of the space P2 of polynomials. Use coordinate vectors to test whether the following set of polynomials span P2. Justify your conclusion. 1-3t+ 2t?, - 4 + 9t-22, -1 + 412, + 3t - 6t2 Does the set of polynomials span P2? O A. Yes, since the matrix whose columns are the B-coordinate vectors of each polynomial has a pivot position in each row, the set of coordinate vectors spans R3. By isomorphism between R3 and P2, the set of polynomials spans P2 OB. No, since the matrix whose columns are the B-coordinate vectors of each polynomial does not have a pivot position in each row, the set of coordinate vectors does not span R3. By isomorphism between R3 and P2, the set of polynomials does not span P2 O C. Yes; since the matrix whose columns are the B-coordinate vectors of each polynomial has a pivot position in each row, the of coordinate vectors spans R2. By isomorphism between R2 and P2 , the set of polynomials spans P2 OD. No; since the matrix whose columns are the B-coordinate vectors of each polynomial does not have a pivot position in each row, the set of coordinate vectors does not span R2. By isomorphism between R2 and P2, the set of polynomials does not span P2

Answer 1

1. 
   21 1 The given set of polynomials is 34 + 2+", -4+9- 2t, -1+4+?, standard basis for P2 is {2, t, t?} 1-34 +2 +2 = 3t-6+²} 3 1 1.1 3.t +2.tk -2.t? - 4 + 9 + -2+2 = -4.1 +got -1 + 4t² sil toot +4.t2 3 t - 6t? + 3.t -6.t2 Matrix 1 - 4 - 1 0 - 3 g 3 -6 2 -2 4 R₂ → R2 + 3R 1 -4 -1 3 -3 -3 2 -2 4 -6 -> R3 2R V R 1 -4 -1 om -3 -3 3 - 6 0 6 6 J Rz > R₂ +2R 1 -4 -3 R2 » -1/3 22 -3 3 0 *** 0 1 -4 -1 1 3 RR +4R2 -4 -1 1 O o O The matrix has turo pivots The third row of the matria only. do es not have any pivot
   
   row . so it So, the B-coordinate vectors of each polynomial doen not have pivot position in each does not span By Isomorphism theorem between R² and P, it does not span P₂ TR3 3 :. Answer option