Question

(a). Determine whether the set is linearly dependent or independent. Further, if it is linearly dependent, express one of the polynomials as a linear combination of others. (b). Determine whether the set can be considered as a basis of the vector space P2, which is the set of all polynomials of degree not more than 2 under addition and scalar multiplication. (1). B = {1 – 2,1 – 22, x – x2} (Hint: Similar to the matrix case in last problem ) (Answer: (a). Linearly dependent, x – 22 = (1 – x) + (x – x2); (b). Not a basis )

Answer 1

B=441-6), (-20%) 7 (2-x)} - Det Q, BAHEIR sit (1-) +B(1)+(ax-xe" JE O X-dort B-82²+ fx-8²-0 (X+B) – (-1)2 = (str)x=0 = x+8 =0 ,a-p=0, pt p=0 - = B =-X, Y = Q =) *=BXY & Let a=t, teIR or let a=1 3) B=C), Ya__ Bis linearly dependent $10-20) = 1 (1-2) +(2-6)=0 ie (1-x²) = 1 (1-x)+1(01-2²)

b For some set B to be a basis of P2 ß must be linearly findependent 4 B should span ? But here, B= 21-0, 1-2%, x-x²4 is linearly dependent + 1B1= 3 & dimension (P2) = 3 => B is not basis of h