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rank of a linear mapping
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(Definition)
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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$$ L(u)=0.$$
- If $V$ is finite-dimensional, then $\rank L=\dim V$ if and only if $L$ is surjective.
- If $U$ is finite-dimensional, then $\rank L=\dim U$ if and only if $L$ is injective.
- Composition of linear mappings does not increase rank. If $M\colon V\to W$ is another linear mapping, then$$\rank ML \le \rank $$ and$$\rank ML \le \rank M$$ 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.
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"rank of a linear mapping" is owned by yark. [ full author list (3) | owner history (2) ]
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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 9746 times total.
Classification:
| AMS MSC: | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) |
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Pending Errata and Addenda
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