 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ib/paper-3/19g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

τ

, each vector decomposes as

u = (u - τu) + τu, τ (u - τu) = 0, τu \in im τ .

w \in ker τ \cap im τ

, write

w = τv

; then

0 = τ w = τ^{2} v = τv = w

For the self-adjoint map in the question,

k \in ker τ

and

w = τv

satisfy

⟨ k, w ⟩ = ⟨ k, τv ⟩ = ⟨ τ k, v ⟩ = 0

U = ker τ \oplus^{⊥} im τ .

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