Determine which of the following forms a subspace of P3 For

Determine which of the following forms a subspace of P^3. For the one that does, find a basis and determine the dimension. W_1 = { p(x) | p(1) = 0 and p\'(0) - 1} W_2 = {p(x) | p(0) = 0 and p\'(1) = 0}

Solution

It is assumed that P³ is the space of all polynomials of degree less than or equal to 3.

A) W₁ is not a subspace as 0 does not belong to it(if f(x) =0 its derivative at any point is 0}

B) W₂ is a subspace .

If p(x),q(x) are in W₂, then , by linearity

            (p+q)(0) =0 , (p+q)\'(1) = p\'(1)+q\'(1) =0

and is clearly closed under scalar multiplication.

Basis for W₂:

p(x) = ax³ +bx² +cx+d is in W₂ , iff

                        d =0 and 3a+2b+c=0

These conditions define a 2 dimensional subspace with basis (for example)

              x³ -3x and x²-2x