PROBLEM

Let B = {1,−1 + x,1 + x^2}

(a) Show that B is a basis for P2.

(b) Find the coordinates for p(x) = a + bx + cx^2 with respect to the basis B.

(c) Hence write down the coordinates, with respect to B, for q(x) = 3 − 4x^2 and r(x) = 4 − x + 2x^2

ATTEMPT AT SOLUTION

(a) B as a matrix: $\displaystyle \begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$ which row-reduces to $\displaystyle \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Each row has a leading entry, so B is linearly independent and thus is a basis for P2. Also, B has 3 polynomials and the dimension of P2 is 3, so B spans P2

(b) I know this is incorrect, but I got:

p(x) = a(1,0,0) + b(-1,1,0) + c(1,0,1)

a=b=c=1 (from (a))

Therefore $\displaystyle p(x) = 1 + x + x^2$

(c) No idea?

Please help!