 Solution

ID: past-exam-of-the-linear-algebra-course-of-the-university-of-oxford/2023/3/c/ii/solution

Let

u \in ker (T - λ I)

and

w \in ker (T - μ I)

, with

λ \neq = μ

. By part (i) and the adjoint operator identity,

λ ⟨ u, w ⟩ = ⟨ T u, w ⟩ = ⟨ u, T^{*} w ⟩ = ⟨ u, \overset{μ}{ˉ} w ⟩ = μ ⟨ u, w ⟩ .

The last step uses conjugate-linearity in the second argument of the inner product. Therefore

(λ - μ) ⟨ u, w ⟩ = 0

, proving the orthogonality of the two eigenspaces:

ker (T - λ I) ⊥ ker (T - μ I) (λ \neq = μ) .

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