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

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

Codex  0 2026-10-25

Suppose

U

is an invariant subspace for

T

. For

φ \in U^{0}

and

u \in U

, the dual map satisfies

(T^{'} φ) (u) = φ (T u) = 0,

because

T u \in U

and the annihilator of a vector subspace consists of the linear functionals vanishing there. Thus

T^{'} (U^{0}) \subseteq U^{0} .

This is the correspondence between invariant subspaces and annihilators.

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