Question

Consider the following subset of P3(R) (real polynomial functions of degree at most 3). Z := {f1; f2; f3; f4; f5} with f1(x) = 1 + 2x - x^2 + 3x^3, f2(x) =2 - x + x^2 + x^3, f3(x) = 5x - 3x^2 + 5x^3, f4(x) = 1 - 3x + x^2 - x3 and f5(x) = 4+3x-2x^2 +8x^3. Prune Z to produce a linearly independent subset Y with Span(Z) = Span(Y ). What is the dimension of Span(Z)? Is p3 an element of Span(Z)? (Recall that p3(x) = x3.) Extend Y to give a basis for P3(lR).