rank-nullity theorem

Let V and W be vector spacesMathworldPlanetmath over the same field. If ϕ:VW is a linear mapping, then


In other words, the dimensionPlanetmathPlanetmath of V is equal to the sum (http://planetmath.org/CardinalArithmetic) of the rank (http://planetmath.org/RankLinearMapping) and nullityMathworldPlanetmath of ϕ.

Note that if U is a subspacePlanetmathPlanetmathPlanetmath of V, then this (applied to the canonical mapping VV/U) says that


that is,


where codim denotes codimension.

An alternative way of stating the rank-nullity theoremMathworldPlanetmath is by saying that if


is a short exact sequenceMathworldPlanetmathPlanetmath of vector spaces, then


In fact, if


is an exact sequenceMathworldPlanetmathPlanetmathPlanetmathPlanetmath of vector spaces, then


that is, the sum of the dimensions of even-numbered terms is the same as the sum of the dimensions of the odd-numbered terms.

Title rank-nullity theorem
Canonical name RanknullityTheorem
Date of creation 2013-03-22 16:35:40
Last modified on 2013-03-22 16:35:40
Owner yark (2760)
Last modified by yark (2760)
Numerical id 7
Author yark (2760)
Entry type Theorem
Classification msc 15A03
Related topic RankLinearMapping
Related topic Nullity