##  [Nullity](/nullity-2) 

 Definition

The dimension of the kernel (null space) of a linear transformation between finite-dimensional vector spaces; equivalently the number of linearly independent solutions to the homogeneous equation T(v)=0.

 

 

 

 

 

 





## Principle

Principle

Rank–nullity principle: for a linear map T: V → W between finite-dimensional spaces, dim V = rank(T) + nullity(T). Nullity measures the degrees of freedom lost to mapping into zero.

 

 

 

 

 





## Demonstration

Demonstration

Example: a linear map T: R^4 → R^3 whose matrix in a standard basis has rank 2 has nullity 4−2=2; there are two independent free variables in the homogeneous system T(v)=0, so the kernel is a 2-dimensional subspace of R^4.

 

 

 

 

## Misapplication

Misapplication

Treating nullity as the algebraic multiplicity of the eigenvalue 0 without checking that eigenvectors span a kernel, or applying rank–nullity unchanged in infinite-dimensional settings where dimensions can be infinite and subtleties of topological closure occur.

 

 

 

 

 





## Consequence

Consequence

Correct use of nullity identifies the dimension of the solution space of homogeneous linear systems, informs count of free parameters in parametric solutions, and determines noninjectivity: nullity>0 implies T is not injective.

 

 

 

 

## Reversal

Reversal

Invert the perspective to consider codimension of the image (corank): instead of counting kernel dimensions, count how many dimensions of the codomain are missed by the image.

 

 

 

 

 





## Boundary

Boundary

Applies to linear maps between finite-dimensional vector spaces over a field; for infinite-dimensional spaces, or for nonlinear maps, analogous notions require additional structure (topology, Fredholm index, etc.) and may not reduce to a finite integer.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Nullity versus algebraic multiplicity of zero eigenvalue: nullity equals geometric multiplicity but can be less than algebraic multiplicity; also tension between nullity and notions like index or cokernel in dual settings.

 

 

 

 

 





## Synthesis

Synthesis

Nullity is the integer dimension of the kernel of a linear operator; by the rank–nullity relation it complements rank and precisely counts independent homogeneous solutions.