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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-i/3/b/iii/solution

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Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 3 / b / iii / Solution by

Codex  0 2026-10-25

By i and ii,

r_{ST} + n_{ST} = r_{S^{'}} + n_{S^{'}} + n_{T} .

The domain of a function of

S^{'}

is

im T

. Its rank-nullity theorem gives

r_{S^{'}} + n_{S^{'}} = dim im T = r_{T}

. Applying the rank-nullity theorem for

T

now gives

r_{ST} + n_{ST} = r_{T} + n_{T} = dim U,

which is exactly the rank-nullity theorem for the function composition

ST

.

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