rank of a linear mapping

The rank of a linear mapping $L:U\to V$ 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 $u\in U$ imposed by the equation

$$L(u)=0.$$

Properties

1. If $V$ is finite-dimensional, then $\mathrm{rank}L=dimV$ if and only if $L$ is surjective.

2. If $U$ is finite-dimensional, then $\mathrm{rank}L=dimU$ if and only if $L$ is injective.

3. Composition of linear mappings does not increase rank. If $M:V\to W$ is another linear mapping, then

   $$\mathrm{rank}ML\le \mathrm{rank}L$$

   and

   $$\mathrm{rank}ML\le \mathrm{rank}M.$$

   Equality holds in the first case if and only if $M$ is an isomorphism, 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