##  [Nullity](/nullity-1) 

 Definition

The nullity of a linear transformation is the dimension of its kernel (nullspace), i.e., the subspace of vectors mapped to the zero vector. It measures the number of independent directions collapsed by the map.

 

 

 

 

 

 





## Principle

Principle

Nullity = dim ker(A). For a linear map from an n-dimensional domain, the rank–nullity theorem states rank + nullity = n, so nullity = n − rank.

 

 

 

 

 





## Demonstration

Demonstration

If A: R^3 → R^3 has row-reduced form with two pivot columns, then rank(A) = 2 and nullity(A) = 3 − 2 = 1; there is a 1-dimensional family of homogeneous solutions to Ax = 0.

 

 

 

 

## Misapplication

Misapplication

Assuming nullity equals the number of zero rows after naive row operations without tracking column structure, or equating nullity with algebraic multiplicity of the eigenvalue 0 without distinguishing geometric vs algebraic counts.

 

 

 

 

 





## Consequence

Consequence

Nullity gives the dimension of the solution space to homogeneous linear systems, counts free parameters in general solutions, and indicates degrees of non-injectivity of a linear map.

 

 

 

 

## Reversal

Reversal

The complementary invariant is rank: increasing nullity reduces rank (for fixed domain dimension); a map of nullity zero is injective on the domain.

 

 

 

 

 





## Boundary

Boundary

Standard in finite-dimensional linear algebra; in infinite-dimensional spaces the kernel may be infinite-dimensional and rank–nullity in the simple finite form need not apply. Nullity refers to linear kernels, not to algebraic multiplicities except for eigenvalue zero where nullity equals geometric multiplicity of 0.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with algebraic multiplicity of zero eigenvalue: nullity equals geometric multiplicity of eigenvalue 0, which may be less than its algebraic multiplicity; tension with intuitive notions of 'zero rows' or sparsity.

 

 

 

 

 





## Synthesis

Synthesis

Nullity quantifies how many independent directions are collapsed to zero by a linear map; via rank–nullity it pairs with rank to partition the domain's dimension and determines the structure of solution spaces to homogeneous systems.