Null Space

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Definition

\[\text{null}(A)=\{x\in\mathbb{R}^n : Ax=0\}\]

Examples

Example 1. Null space of [[1,2],[2,4]].

Solution. x+2y=0 → x=−2y. Null space: span{(−2,1)}.

In Depth

The null space (kernel) of a matrix \(A\) is \(\text{Null}(A)=\{\mathbf{x}:A\mathbf{x}=\mathbf{0}\}\). It is a subspace of the domain. The rank-nullity theorem: \(\text{rank}(A)+\text{nullity}(A)=n\) (number of columns). The null space captures all solutions to the homogeneous system.

The null space is computed by row-reducing \(A\) to reduced row echelon form (RREF). Free variables (columns without pivots) parameterize the null space. Each free variable contributes one basis vector to the null space.

The null space of \(A^T\) is the left null space of \(A\). The four fundamental subspaces of \(A\) (column space, null space, row space, left null space) are orthogonally complementary in pairs: \(\text{Col}(A)\perp\text{Null}(A^T)\) and \(\text{Row}(A)\perp\text{Null}(A)\).

A matrix has trivial null space (only \(\mathbf{0}\)) iff it has full column rank iff its columns are linearly independent iff \(A\mathbf{x}=\mathbf{b}\) has at most one solution for any \(\mathbf{b}\). Square matrices with trivial null space are invertible.

In differential equations, the null space of a linear differential operator \(L\) is the solution space of \(Lu=0\). For \(L=d^2/dx^2+1\), the null space is \(\{c_1\cos x+c_2\sin x\}\), a 2-dimensional space. The dimension of the null space equals the order of the ODE.

Key Properties & Applications

The null space is used in data compression: if \(A\mathbf{x}=\mathbf{b}\) has many solutions, the null space parameterizes the degrees of freedom. In control theory, the null space of the controllability matrix determines which states are unreachable.

Sparse null spaces arise in network analysis: the null space of the incidence matrix of a graph encodes the cycle space. Each basis vector of the null space corresponds to a fundamental cycle. The dimension of the null space equals the circuit rank (cyclomatic number) of the graph.

In machine learning, the null space of the feature matrix \(X\) contains directions in feature space that have no effect on predictions. Regularization (ridge regression) effectively shrinks components in the null space direction, improving generalization.