 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2018/10/b/solution

v = \sum_{j} v_{j} e_{j}

. Using linearity in the second slot and the Kronecker delta gives

⟨ e_{i}, v ⟩ = j \sum v_{j} ⟨ e_{i}, e_{j} ⟩ = v_{i} .

Expanding both vectors and using conjugate linearity in the first slot therefore gives

⟨ u, v ⟩ = i, j \sum \overline{u_{i}} v_{j} ⟨ e_{i}, e_{j} ⟩ = i \sum \overline{u_{i}} v_{i} .

An orthonormal set that is not a basis would recover only the coordinates of the orthogonal projection onto its linear span; here the

n

C^{n}

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