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.

If $V$ is finite-dimensional, then $\operatorname{rank}L=\dim V$ if and only if $L$ is surjective.
2.

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