 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-3/9g/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

Fix

y \in V

. The map

x \mapsto θ (x, y)

β (y)

such that

θ (x, y) = ⟨ x, β (y)⟩ (x \in V) .

λ, μ

⟨ x, β (λ y + μ z)⟩ = θ (x, λ y + μ z) = \overline{λ} θ (x, y) + \overline{μ} θ (x, z) = ⟨ x, λ β (y) + μ β (z)⟩ .

β (λ y + μ z) = λ β (y) + μ β (z),

β

is linear.

For

α \in End (V)

, apply this result to

θ (x, y) = ⟨ αx, y ⟩ .

It produces a unique linear map

α^{*}

satisfying

⟨ αx, y ⟩ = ⟨ x, α^{*} y ⟩,

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