Are polynomials of degree exactly 3 U 0 a subspace of polyno

Are {polynomials of degree exactly 3} U {0} a subspace of {polynomials of degree at most 3}? if so, justify.

Thank you.

Solution

Are {polynomials of degree exactly 3} U {0} a subspace of {polynomials of degree at most 3}? if so, justify

Any polynomial in P3(R) is of the form f(x) = a+bx+cx2+dx3 . The given equations imply that a = b a + b + c + d = b + 2c + 3d, a system of two linear equations is four unknowns. Eliminating b from the second equation gives a = c + 2d. Thus we may take c and d to be our independent variables (parameters), and a and b depend on these. A basis for W is given by setting (c, d) as (1, 0) and (0, 1), respectively, giving the basis 1 + x + x 2 , 2 + 2x + x 3 .