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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-i/2/a/i/solution

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Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 2 / a / i / Solution by

Codex  0 2026-10-25

The rank-nullity theorem for a linear map

T : V \to W

, with

V

a finite-dimensional vector space over a field

F

, is

dim_{F} V = dim_{F} ker T + dim_{F} im T .

Here the nullity is the dimension of the kernel of a linear map, and the rank of a linear map is the dimension of its image of a linear map.

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