Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics I / 4 / b

The relevant inner products are

⟨ 1, x ⟩ = \frac{- 1 + 0 + 1}{3} = 0, ⟨ 1, 1 ⟩ = 1, ⟨ x, x ⟩ = \frac{2}{3} .

Hence

1

and

x

f = 1, g = \frac{3}{2} x

U

ii

f, g

U

⟨ f, h ⟩ ⟨ g, h ⟩ = ⟨ f, x^{2} ⟩ - ⟨ f, x^{2} ⟩ ⟨ f, f ⟩ - ⟨ g, x^{2} ⟩ ⟨ f, g ⟩ = 0, = ⟨ g, x^{2} ⟩ - ⟨ f, x^{2} ⟩ ⟨ g, f ⟩ - ⟨ g, x^{2} ⟩ ⟨ g, g ⟩ = 0.

Together with

⟨ f, g ⟩ = 0

, these identities prove pairwise orthogonality. The construction subtracts the orthogonal projection onto a finite-dimensional subspace

U

from

x^{2}

Use

f, g

from i. Their inner products with

x^{2}

are

⟨ f, x^{2} ⟩ = \frac{2}{3}, ⟨ g, x^{2} ⟩ = \frac{3}{2} \frac{- 1 + 0 + 1}{3} = 0.

Thus

h = x^{2} - 2/3

. Its values at

- 1, 0, 1

are

1/3, - 2/3, 1/3

, so

∥ h ∥^{2} = \frac{1}{3} (\frac{1}{9} + \frac{4}{9} + \frac{1}{9}) = \frac{2}{9} .

By ii the three polynomials are orthogonal vectors, and normalizing

h

gives

B = (1, \frac{3}{2} x, \frac{3 x ^{2} - 2}{2}) .

This orthonormal set is a basis of the three-dimensional vector space

V

. It is the result of the Gram-Schmidt process applied to

1, x, x^{2}

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