PlanetMath: rank of a linear mapping

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rank of a linear mapping

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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

$\displaystyle L(u)=0.$

If is finite-dimensional, then $\operatorname{rank}L=\dim V$ if and only if is surjective.
If is finite-dimensional, then $\operatorname{rank}L=\dim U$ if and only if is injective.
Composition of linear mappings does not increase rank. If $M\colon V\to W$ is another linear mapping, then
$\displaystyle \operatorname{rank}ML \le \operatorname{rank}L$
and
$\displaystyle \operatorname{rank}ML \le \operatorname{rank}M.$
Equality holds in the first case if and only if is an isomorphism, and in the second case if and only if is an isomorphism.

"rank of a linear mapping" is owned by yark. [ full author list (3) | owner history (2) ]

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