 Solution

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

⟨ u_{i}, w_{j} ⟩ = δ_{ij}

. Expanding each vector in its specified basis and using real bilinearity gives

⟨ a, b ⟩ = ⟨ i \sum a_{i} u_{i}, j \sum b_{j} w_{j} ⟩ = i \sum a_{i} b_{i} .

The simple sum-of-products formula holds because the two coordinate vectors use paired reciprocal bases. It does not require either basis to be orthogonal, and would generally fail if both sets of components were taken in the same nonorthogonal basis.

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