 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017-resit/9/b/solution

Use the convention that the Hermitian inner product is conjugate-linear in its first argument and linear in its second. In orthonormal coordinates,

⟨ v, w ⟩ = \sum_{i} \overline{v_{i}} w_{i}

. It satisfies conjugate symmetry, sesquilinearity and positive definiteness. Expanding in the orthonormal basis gives

v = \sum_{i} e_{i} ⟨ e_{i}, v ⟩, w = \sum_{j} e_{j} ⟨ e_{j}, w ⟩ .

⟨ e_{i}, e_{j} ⟩ = δ_{ij}

yields

⟨ v, w ⟩ = i \sum \overline{⟨ e_{i}, v ⟩} ⟨ e_{i}, w ⟩ = i \sum ⟨ v, e_{i} ⟩ ⟨ e_{i}, w ⟩ .

This identifies exactly where complex conjugation enters; replacing the first factor by

⟨ e_{i}, v ⟩

would generally be wrong under this convention.

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