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rank of a linear mapping (Definition)

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. $

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
    $\displaystyle \operatorname{rank}ML \le \operatorname{rank}L$
    and
    $\displaystyle \operatorname{rank}ML \le \operatorname{rank}M.$
    Equality holds in the first case if and only if $ M$ is an isomorphism, and in the second case if and only if $ L$ is an isomorphism.



"rank of a linear mapping" is owned by yark. [ full author list (3) | owner history (2) ]
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See Also: nullity, rank-nullity theorem

Other names:  rank

Attachments:
determining rank of matrix (Algorithm) by Algeboy
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Cross-references: isomorphism, equality, composition, injective, surjective, finite-dimensional, equation, independent, number, image, mapping's, dimension, linear mapping
There are 22 references to this entry.

This is version 10 of rank of a linear mapping, born on 2002-02-19, modified 2007-01-18.
Object id is 2236, canonical name is RankLinearMapping.
Accessed 7957 times total.

Classification:
AMS MSC15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)

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