Nullspace

The nullspace, also known as the kernel, of a matrix or a linear transformation is the set of all vectors that map to the zero vector under that transformation. For an \(m \times n\) matrix \(A\), the nullspace is defined as \(\{x \in \mathbb{R}^n \mid Ax = 0\}\). It is a subspace of the domain of \(A\) and contains at least the zero vector. The dimension of the nullspace is called the nullity of \(A\). By the rank–nullity theorem, the rank of \(A\) plus its nullity equals the number of columns \(n\).

The nullspace is important in solving linear systems. A system \(Ax = b\) has a solution only if \(b\) lies in the column space of \(A\). When \(b = 0\), every vector in the nullspace provides a solution, giving infinitely many solutions if the nullity is greater than zero. The general solution to \(Ax = b\) can be expressed as a particular solution plus any vector from the nullspace.

The structure of the nullspace can be studied via row reduction. A basis for the nullspace can be found from the reduced row echelon form by expressing leading variables in terms of free variables. The nullspace also appears in differential equations, where it represents the set of equilibrium solutions. In functional analysis, the kernel of a linear operator plays a similar role, capturing elements mapped to zero.

In summary, the nullspace is a fundamental concept that describes all vectors annihilated by a linear map, and it is central to the analysis of linear equations, linear independence, and operator theory.