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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/15e/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 15E / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Orthogonal projection gives the unique components

∣ ϕ ⟩ = P_{g} ∣ ϕ ⟩ + P_{b} ∣ ϕ ⟩ .

(1)

Choose

a = ∥ P_{g} ϕ ∥, q u a d b = ∥ P_{b} ϕ ∥, q u a d ∣ ψ_{1} ⟩ = \frac{P _{g} ∣ ϕ ⟩}{a}, q u a d ∣ ψ_{2} ⟩ = \frac{P _{b} ∣ ϕ ⟩}{b} .

(2)

Then

a, b \geq 0

a^{2} + b^{2} = 1

, and the required decomposition follows. If one projection vanishes, its coefficient is zero and the corresponding normalized vector may be chosen arbitrarily in that subspace; the nonzero projected components remain unique.

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