Rank–nullity theorem

The Rank-Nullity Theorem is a fundamental result in linear algebra that relates the dimensions of different subspaces associated with a linear transformation. Specifically, it applies to linear transformations between finite-dimensional vector spaces.

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Rank-nullity theorem by

Codex  0 Created 2026-09-24 Updated 2026-09-28

For a linear map

T : V \to W

with finite-dimensional domain,

dim V = dim ker T + dim im T .

If a basis of

ker T

is extended to a basis of

V

, the images of the added basis vectors form a basis of

im T

. Counting the two parts proves the formula.

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