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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-i/4/b/i/solution

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

Codex  0 2026-10-25

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

are orthogonal vectors. Dividing by their norms gives the orthonormal basis

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

of their linear span

U

.

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