Question

Let S={2,3+x,1−x2}, p(x)=2−x−x2 and V=P2

(a) If possible, express p(x)as a linear combination of vectors in S.

(b) By justifying your answer, determine whether the set S is linearly independent or linearly dependent.

(c) By justifying your answer, determine whether the set S is a basis for P2

Please solve it in very detail, and make sure it is correct.

Answer 1

son Q) consider the standcud hasis B = { 1, x,x2} of Pz. Then, the coordinate vectou Weit B are 3 [2]g [3+x3 = 1 [1-x2), = 2 བ བ༠ -- 3 9 0 2 [b]o - [2-*-**]g

we have S= {2, 3+x, 1-2}={v,, V2,3} then consider the malex [vi] B [V₂] B [v3] [bem]8] 2 3 2 0 - 0 -1 O 6 - 3/ /2 RITRA Rz 1-1.R3 -1 2 D 1 0 0 D - 0 12 5/2 - RITR, 3 R₂ - 0 0 0 0 - 12 23 O 0 2 0 0 D 21 RITR / Rz -) 0 As can be seen, we have obtained the identify maudy w the left so, we are done weget 4 = 2 x = L x₂ = 1 this p(x) can be neitten as a lineal combination of recious ins PTT) = rith V2 + 3 لا ولا درخودمه P() = 212)+ (-1)(3+x) +1(1-4)

b) from the reduced mate's reduced mata's we oblained the identity malix to the left. Each collemn of the une mattis correspondings has a pivot element. So, s ú a linearly independent set c) since, s in linearly independent set, the imples dim(s) = 3. = linealy independent number of veclers dim(s) = 3= dim (P₂) s ů a basis for p