rank-nullity theorem

The sum of the rank and the nullityMathworldPlanetmath of a linear mapping gives the dimensionPlanetmathPlanetmath of the mapping’s domain. More precisely, let T:VW be a linear mapping. If V is a finite-dimensional, then


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

Every independentPlanetmathPlanetmath linear constraint takes away one degree of freedom.

The rank is just the number of independent linear constraints on vV imposed by the equation


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 independentMathworldPlanetmath 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