Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 4 / b / i

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- Solution i

Solution

Bilinearity of the real inner product gives, for the specified linear combinations,

⟨ u, v ⟩ = ⟨ i = 1 \sum n x_{i} v_{i}, j = 1 \sum n y_{j} v_{j} ⟩ = i = 1 \sum n j = 1 \sum n x_{i} y_{j} ⟨ v_{i}, v_{j} ⟩ = i = 1 \sum n j = 1 \sum n x_{i} A_{ij} y_{j} = x^{T} A y .

The last equality is the entrywise rule for matrix multiplication, with

x^{T}

the transpose of the column vector

x

.

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