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 finitedimensional, then $\mathrm{rank}L=dimV$ if and only if $L$ is surjective^{}.

2.
If $U$ is finitedimensional, 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  20130322 12:24:03 
Last modified on  20130322 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 