# rank-nullity theorem

The sum of the rank and the nullity of a linear mapping gives the dimension of the mapping’s domain. More precisely, let $T:V\rightarrow W$ be a linear mapping. If $V$ is a finite-dimensional, then

 $\dim V=\dim\mathop{\mathrm{Ker}}T+\dim\mathop{\mathrm{Img}}T.$

The intuitive content of the Rank-Nullity theorem is the principle that

Every independent linear constraint takes away one degree of freedom.

The rank is just the number of independent linear constraints on $v\in V$ imposed by the equation

 $T(v)=0.$

The dimension of $V$ is the number of unconstrained degrees of freedom. The nullity is the degrees of freedom in the resulting space of solutions. To put it yet another way:

The number of variables minus the number of independent linear constraints equals the number of linearly independent solutions.

 Title rank-nullity theorem Canonical name RanknullityTheorem Date of creation 2013-03-22 12:24:09 Last modified on 2013-03-22 12:24:09 Owner rmilson (146) Last modified by rmilson (146) Numerical id 8 Author rmilson (146) Entry type Theorem Classification msc 15A03 Classification msc 15A06 Related topic Overdetermined Related topic Underdetermined Related topic RankLinearMapping Related topic Nullity Related topic UnderDetermined Related topic FiniteDimensionalLinearProblem