 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-1/8f/a/i/solution

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

α^{*}

is defined by

⟨ αv, w ⟩ = ⟨ v, α^{*} w ⟩ (v, w \in V) .

Since

V

α

v

λ

. Normality gives

∥ (α - λ I) x ∥^{2} = ∥ (α^{*} - \overline{λ} I) x ∥^{2}

for every

x

, by expanding both sides and using

α α^{*} = α^{*} α

. Taking

x = v

yields

αv = λ v, α^{*} v = \overline{λ} v .

 New to topics? Read the docs here!