DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-03

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Rank and nullity of a linear map with finite-dimensional domain

Definition

Let $T:V\to W$ be a linear map whose domain $V$ is finite-dimensional over $F$ . Its nullity and rank are

$\operatorname{nullity}T:=\dim_F(\ker T),\qquad \operatorname{rank}T:=\dim_F(\operatorname{im}T).$

Both dimensions are defined. The kernel is a linear subspace of the finite-dimensional space $V$ , so it is finite-dimensional by The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial and If $\dim_F V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_F U \le n$ , and $\dim_F U = n$ if and only if $U = V$ . The image is finite-dimensional because Extending a basis of the kernel to a basis of the domain gives a basis of the image constructs a finite basis of it from a basis of $V$ . Thus rank and nullity are natural numbers, including when $V$ is the zero space.

Depends on

Used by

The rank of a matrix equals the rank of the linear map x↦ Ax Corollary
Rank-nullity: dim_F V=nullityT+rankT Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 86 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

UCLA Algebra Notes, rank and nullity (standard reference, not scraped)
Axler, Linear Algebra Done Right, 4th ed., Chapter 3 (standard reference, not scraped)