Null space

What Is Null Space?

Null space is a fundamental concept in linear algebra defined as the set of all vectors that a linear transformation maps to the zero vector. For a matrix A representing a linear map from an n-dimensional domain to an m-dimensional codomain, the null space consists of every vector x satisfying Ax = 0. The null space is always a subspace of the domain: it is closed under vector addition and scalar multiplication, and it always contains the zero vector. Null space is also called the kernel of the linear transformation, and both terms are used interchangeably across mathematics, engineering, and computational science.

The concept arises wherever systems of linear equations are analyzed. Understanding the null space tells an analyst whether a system of equations has a unique solution, infinitely many solutions, or whether a transformation loses information by collapsing a non-trivial subspace of its domain to zero.

Definition and Computation

The null space of an m-by-n matrix A is found by row-reducing the augmented matrix [A | 0] to reduced row echelon form and reading off the free variable solutions. As the Statlect reference on the kernel of a linear map establishes, the key structural fact is that the kernel forms a subspace: any linear combination of null-space vectors is itself in the null space. When a matrix has linearly independent columns, its null space contains only the zero vector, meaning the transformation is injective and the equation Ax = 0 has exactly one solution. When columns are linearly dependent, the null space is non-trivial and Ax = 0 admits infinitely many solutions parameterized by the free variables.

Rank-Nullity Theorem

The relationship between the null space and the range of a linear transformation is expressed by the rank-nullity theorem, also called the dimension theorem. It states that for any linear transformation T from a finite-dimensional vector space V to a vector space W, the dimension of V equals the sum of the rank of T (the dimension of the image) and the nullity of T (the dimension of the null space): dim(V) = rank(T) + nullity(T). As the LibreTexts Linear Algebra treatment of kernel and image explains, this theorem has immediate practical consequences: knowing the rank of a matrix immediately determines the dimension of its null space, and vice versa. For an n-by-n square matrix, the null space is trivial if and only if the matrix has full rank n, which is equivalent to the matrix being invertible.

Null Space in Signal Processing and Control

In engineering, null space has concrete interpretations. In communications and signal processing, a channel matrix H relates transmitted signals to received signals, and the null space of H defines the set of transmitted signals that produce no received output, a critical concern in multiple-input multiple-output (MIMO) systems where null-space beamforming deliberately directs signal energy away from interfering receivers. In control theory, the null space of the observability matrix characterizes the states of a dynamical system that cannot be detected from output measurements, determining whether the system is fully observable. In robotics, the null space of a robot's Jacobian matrix represents the joint-velocity directions that produce no motion of the end effector, enabling redundant manipulators to reconfigure their posture while holding a tool stationary, as detailed in MIT OpenCourseWare numerical methods course notes on null space existence and uniqueness of solutions.

Applications

Null space has applications in a range of fields, including: