Does S_2 span P_2 ? Solution lets calculate the dimension of S_2 It is clear S2 is a subspace of P2 since every element of S2 is in P2. So if the dimension of S2 is 3 i.e. all elements of S2 are linearly independent then S2 spans P2 and it is a basis for P2 otherwise Not. Now lets see if all elements are LI if c1 (x^2 +1) + c2(x^2 + x) + c3(x+6) = 0 (zero polynomial) then c1 = -c2 c1 = -6c3 c2 = -c3 which gives all c1,c2,c3 = 0 so they are linearly independent and hence its a basis for P2..

1. Does S_2 span P_2 ?
Solution
lets calculate the dimension of S_2
It is clear S2 is a subspace of P2 since every element of S2 is in P2.
So if the dimension of S2 is 3 i.e. all elements of S2 are linearly independent then S2 spans P2
and it is a basis for P2
otherwise Not.
Now lets see if all elements are LI
if c1 (x^2 +1) + c2(x^2 + x) + c3(x+6) = 0 (zero polynomial)
then
c1 = -c2
c1 = -6c3
c2 = -c3
which gives all c1,c2,c3 = 0
so they are linearly independent and hence its a basis for P2.