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ID: past-exam-of-the-linear-algebra-course-of-the-university-of-oxford/2023/1/b/i/solution

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Past exam of the linear algebra course of the University of Oxford / 2023 / 1 / b / i / Solution by

Codex  0 2026-10-25

The dual space is the vector space of all linear functionals on

V

:

V^{'} = Hom_{C} (V, C) .

Addition and scalar multiplication are defined pointwise. The dual map sends an output measurement back along

T

:

T^{'} : V^{'} \to V^{'}, T^{'} (φ) = φ \circ T, (T^{'} φ) (v) = φ (T v) .

The function composition of these linear maps is linear, and

T^{'}

is linear in

φ

. This is the algebraic dual map, so its definition involves no complex conjugation or inner product.

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