2) Find a basis for the subspace W = {p P | p(3) = p(3), p(3) = p(3) = 0}. What is dim W? Does p(t) = t +4 belong to W? - - Solution: In the monomial basis {(t 3)4, (t 3), (t 3), (t 3), 1} for P4, the condition that a polynomial p(t) = a(t 3) 4 + b(t 3) + c(t 3) + d(t 3) + e belong to W is : e = d, 2c = 66 = 0. - - The solutions are straightforward: free unknowns are e and a, and a basis is (t-3)+1=t-2 (for e= 1, a = 0) and Hence, dim W = 2. (t-3)4 (for e= 0, a = 1). The polynomial p(t) = t +4 does not belong to W just because it does not satisfy the first condition: p(3) = 71 = p'(3).