rank of a linear mapping

The rank of a linear mapping L:UV is defined to be the dimL(U), the dimension of the mapping’s image. Speaking less formally, the rank gives the number of independent linear constraints on uU imposed by the equation



  1. 1.

    If V is finite-dimensional, then rankL=dimV if and only if L is surjectivePlanetmathPlanetmath.

  2. 2.

    If U is finite-dimensional, then rankL=dimU if and only if L is injectivePlanetmathPlanetmath.

  3. 3.

    CompositionMathworldPlanetmathPlanetmath of linear mappings does not increase rank. If M:VW is another linear mapping, then




    Equality holds in the first case if and only if M is an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and in the second case if and only if L is an isomorphism.

Title rank of a linear mapping
Canonical name RankOfALinearMapping
Date of creation 2013-03-22 12:24:03
Last modified on 2013-03-22 12:24:03
Owner yark (2760)
Last modified by yark (2760)
Numerical id 13
Author yark (2760)
Entry type Definition
Classification msc 15A03
Synonym rank
Related topic Nullity
Related topic RankNullityTheorem2