Knowlpedia

Rank–nullity theorem

For a linear map on a finite-dimensional space, dimension equals rank plus nullity.

Rank–nullity theorem: Let T:V→WT:V\to W be a between finite-dimensional . Define

ker⁡T={v∈V:T(v)=0},im⁡T={T(v):v∈V}.\begin{gathered} \ker T=\{v\in V:T(v)=0\},\\ \operatorname{im}T=\{T(v):v\in V\}. \end{gathered}

Then

dim⁡V=dim⁡(ker⁡T)+dim⁡(im⁡T).\dim V=\dim(\ker T)+\dim(\operatorname{im}T).
Rank and nullity

In particular, the rank rank⁡(T)=dim⁡(im⁡T)\operatorname{rank}(T)=\dim(\operatorname{im}T) and the nullity nullity⁡(T)=dim⁡(ker⁡T)\operatorname{nullity}(T)=\dim(\ker T) satisfy dim⁡V=rank⁡(T)+nullity⁡(T)\dim V=\operatorname{rank}(T)+\operatorname{nullity}(T).

Remarks

The set im⁡T\operatorname{im}T is the of the underlying function, and ker⁡T\ker T is the of {0}\{0\}. This identity is the basic dimension bookkeeping behind the structure of solution spaces to linear equations.