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

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

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

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

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

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

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

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

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.