# rank of a linear mapping

The rank of a linear mapping $L\colon U\to V$ is defined to be the $\dim L(U)$, 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. 1.

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

2. 2.

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

3. 3.

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

 $\operatorname{rank}ML\leq\operatorname{rank}L$

and

 $\operatorname{rank}ML\leq\operatorname{rank}M.$

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 RankOfALinearMapping 2013-03-22 12:24:03 2013-03-22 12:24:03 yark (2760) yark (2760) 13 yark (2760) Definition msc 15A03 rank Nullity RankNullityTheorem2