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rank-nullity theorem
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(Theorem)
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The sum of the rank and the nullity of a linear mapping gives the dimension of the mapping's domain. More precisely, let $T:V\rightarrow W$ be a linear mapping. If $V$ is a finite-dimensional, then $$\dim V = \dim \mathop{\mathrm{Ker}} T + \dim \mathop{\mathrm{Img}} T.$$
The intuitive content of the Rank-Nullity theorem is the principle that
Every independent linear constraint takes away one degree of freedom.
The rank is just the number of independent linear constraints on $v\in V$ imposed by the equation $$T(v)=0.$$ The dimension of $V$ is the number of unconstrained degrees of freedom. The nullity is the degrees of freedom in the resulting space of solutions. To put it yet another way:
The number of variables minus the number of independent linear constraints equals the number of linearly independent solutions.
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"rank-nullity theorem" is owned by rmilson.
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Cross-references: linearly independent, variables, solutions, equation, number, degree of freedom, independent, finite-dimensional, domain, mapping's, dimension, linear mapping, nullity, rank, sum
There are 2 references to this entry.
This is version 5 of rank-nullity theorem, born on 2002-02-19, modified 2002-02-22.
Object id is 2238, canonical name is RankNullityTheorem.
Accessed 11393 times total.
Classification:
| AMS MSC: | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) | | | 15A06 (Linear and multilinear algebra; matrix theory :: Linear equations) |
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Pending Errata and Addenda
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