ALGEBRA 2026-05-15

Rank-Nullity Theorem

Theorem

(Rank-Nullity Theorem)

Let $T:V\to W$ be a linear map between finite-dimensional vector spaces. Then

\dim(V)=\operatorname{rank}(T)+\operatorname{nullity}(T)

where $\operatorname{rank}(T)=\dim(\operatorname{Im}\,T)$ and $\operatorname{nullity}(T)=\dim(\ker T)$ .

This theorem is central to understanding the structure of linear maps. It tells us that the ‘information lost’ (nullity) and ‘information retained’ (rank) must sum to the total dimension of the domain.

Remark

(Consequence)

If $T:V\to V$ is a linear operator on a finite-dimensional space with $\ker T=\{0\}$ , then $T$ is automatically surjective — hence an isomorphism.