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**1. 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).

**2. Homework Equations**

Gram-Schmidt orthogonalization process.

**3. 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?

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