Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics I / 4 / c / i

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- Solution i

Solution

If

T

is an orthogonal transformation and

e_{1}, e_{2}, e_{3}

is an orthonormal basis, then

⟨ T e_{i}, T e_{j} ⟩ = ⟨ e_{i}, e_{j} ⟩ = δ_{ij} .

The images form an orthonormal set, hence are linearly independent and form an orthonormal basis in dimension three.

Conversely, suppose the images

T e_{1}, T e_{2}, T e_{3}

of one chosen orthonormal basis form an orthonormal basis. Write

u = \sum_{i} a_{i} e_{i}

and

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

. By linearity of

T

and bilinearity,

⟨ T u, T v ⟩ = \sum_{i, j} a_{i} b_{j} ⟨ T e_{i}, T e_{j} ⟩ = \sum_{i} a_{i} b_{i} = ⟨ u, v ⟩ .

Thus mapping one orthonormal basis to an orthonormal basis is equivalent to preserving every inner product. In the converse, a single chosen orthonormal basis suffices.

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