 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/10c/b/solution

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

In the

A

basis, the transformations whose columns are the new basis vectors are

U_{B} = (1/2 3 /2 3 /2 - 1/2), U_{C} = (1/2 - 3 /2 3 /2 1/2) .

Thus

U_{C}

is a rotation through

- π /3

, while

U_{B}

is a real orthogonal reflection, equivalently a rotation with one basis-vector phase reversed.

For any real orthogonal matrix

U

with columns

∣ u_{j} ⟩

\sum_{j} ∣ u_{j} u_{j} ⟩ = \sum_{j, k, l} U_{kj} U_{l j} ∣ a_{k} a_{l} ⟩ = \sum_{k, l} (U U^{T})_{k l} ∣ a_{k} a_{l} ⟩ = \sum_{k} ∣ a_{k} a_{k} ⟩ .

Applying this to

U_{B}

and

U_{C}

and normalizing proves

\frac{∣ a _{0} a _{0} ⟩ + ∣ a _{1} a _{1} ⟩}{2} = \frac{∣ b _{0} b _{0} ⟩ + ∣ b _{1} b _{1} ⟩}{2} = \frac{∣ c _{0} c _{0} ⟩ + ∣ c _{1} c _{1} ⟩}{2} .

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