##  [Null Space (Kernel)](/null-space-kernel-0) 

 Definition

For a linear map L: V → W between vector spaces, the null space is the subspace of V consisting of all vectors x with L x = 0; it is also called the kernel or solution space of the homogeneous equation L x = 0.

 

 

 

 

 

 





## Principle

Principle

Linearity implies the null space is a linear subspace; its dimension (nullity) measures the degrees of freedom of homogeneous solutions and participates in rank–nullity relations for finite-dimensional maps.

 

 

 

 

 





## Demonstration

Demonstration

For a matrix A∈R^{m×n}, the null space is the set of vectors x in R^n satisfying A x = 0; computing a basis via Gaussian elimination yields the parametric form of all homogeneous solutions.

 

 

 

 

## Misapplication

Misapplication

Confusing the null space of L with the orthogonal complement of the image of L*: these are related by the fundamental theorem of linear algebra only in inner-product spaces and require attention to domain and codomain.

 

 

 

 

 





## Consequence

Consequence

Characterizing the null space gives existence and multiplicity information for linear systems; its dimension enters counts of independent constraints and informs invertibility (trivial null space = injectivity).

 

 

 

 

## Reversal

Reversal

The dual notion is the image (range) of L: while the null space measures where L annihilates vectors, the image measures which targets are attainable by L.

 

 

 

 

 





## Boundary

Boundary

Null space is defined only for linear maps (or linear operators); nonlinear maps have solution sets to f(x)=0 but these are not linear subspaces in general and lack rank–nullity structure.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Null space versus kernel terminology: the two are synonymous in linear algebra, but 'kernel' also has other meanings in analysis and probability, so context disambiguates.

 

 

 

 

 





## Synthesis

Synthesis

The null space is the linear subspace of inputs mapped to zero by a linear map, central to solvability, dimension counts, and the structure of linear systems.