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rank-nullity theorem (Theorem)

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

$\displaystyle \dim V = \dim \mathop{\mathrm{Ker}} T + \dim \mathop{\mathrm{Img}} T.$

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
$\displaystyle T(v)=0.$
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.



"rank-nullity theorem" is owned by rmilson.
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See Also: overdetermined, under determined, rank of a linear mapping, nullity, under determined, finite-dimensional linear problem


Attachments:
proof of rank-nullity theorem (Proof) by rmilson
rank-nullity theorem (Theorem) by yark
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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 10153 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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