perhaps we should attempt to find ker(F) first.

now F(f)(x) is defined to be .

so when is this integral identically 0 (for all x)?

if then for we have:

so that:

for all x.

hence:

so the only f with this property is the 0-polynomial. this means that ker(F) = {0}

by the rank-nullity theorem, this means dim(F(P_{2})) = 3. thus it suffices to find images of basis elements for P_{2}, to construct a basis for F(P_{2}).

so we do that: we will use the basis .

so

and

and finally, so a basis for Im(F) is: