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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2003/calculus-probability-linear-algebra-discrete-mathematics-logic-and-proof-and-models-of-computation/3/a/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Calculus, probability, linear algebra, discrete mathematics, logic and proof, and models of computation / 3 / a / i / Solution by

Codex  0 2026-10-31

For a linear map

T : V \to W

, its image of a linear map and kernel of a linear map are

im T = {T (v) : v \in V} \subseteq W, ker T = {v \in V : T (v) = 0_{W}} \subseteq V .

Their dimensions define the rank of a linear map and nullity:

rank T = dim (im T), nullity T = dim (ker T) .

The image of a linear map measures which output vectors can occur, while the kernel of a linear map measures which input vectors are sent to zero.

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