rank-nullity theorem

The sum of the rank and the nullity of a linear mapping gives the dimension of the mapping’s domain. More precisely, let $T:V\to W$ be a linear mapping. If $V$ is a finite-dimensional, then

$$dimV=dimKerT+dimImgT.$$

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.

Title	rank-nullity theorem
Canonical name	RanknullityTheorem
Date of creation	2013-03-22 12:24:09
Last modified on	2013-03-22 12:24:09
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	8
Author	rmilson (146)
Entry type	Theorem
Classification	msc 15A03
Classification	msc 15A06
Related topic	Overdetermined
Related topic	Underdetermined
Related topic	RankLinearMapping
Related topic	Nullity
Related topic	UnderDetermined
Related topic	FiniteDimensionalLinearProblem