 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-1/5b/a/solution

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

For any

h

∥ A (c^{*} + h) - y ∥_{2}^{2} = ∥ A c^{*} - y ∥_{2}^{2} + 2 h^{T} A^{T} (A c^{*} - y) + ∥ A h ∥_{2}^{2} .

A^{T} A c^{*} = A^{T} y

holds, the last two terms reduce to

∥ A h ∥^{2} \geq 0

, so

c^{*}

minimizes. Conversely, at a minimizer the directional derivative in every

h

vanishes, forcing

A^{T} (A c^{*} - y) = 0

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