Finding an Orthogonal Basis of Polynomials Using Gram-Schmidt Process

[cookiesyum]

Homework Statement

Let P2 denote the space of polynomials in k[x] and degree < or = 2. Let f, g exist in P2 such that

f(x) = a2x^2 + a1x + a0
g(x) = b2x^2 + b1x + b0

Define

<f, g> = a0b0 + a1b1 + a2b2

Let f1, f2, f3, f4 be given as below

f1 = x^2 + 3
f2 = 1 - x
f3 = 2x^2 + x + 1
f4 = x + 1

Find an orthogonal basis of Span(f1, f2, f3, f4).

Homework Equations

Gram-Schmidt orthogonalization process.

The Attempt at a Solution

Span(f1, f2, f3, f4) = Span(w1, w2, w3, w4)

Take S = {f, 1}

w1 = f1
w2 = 1 - (<1, f1>/<f1, f1>)*f1
w3 = 1 - (<1, f2>/<f2, f2>)*f2 - (<1, f1>/<f1, f1>)*f1
w4 = 1 - (<1, f3>/<f3, f3>)*f3 - (<1, f2>/<f2, f2>)*f2 - (<1, f1>/<f1, f1>)*f1

Is this the correct procedure? Can I take g = 1 like that?

 

[Mark44]

Looks good to me. There's nothing wrong with starting with g(x) = 1.

 

[cookiesyum]

Looks good to me. There's nothing wrong with starting with g(x) = 1.

Thanks a bunch!