nullities

Nullity is a term used in linear algebra to denote the dimension of the kernel of a linear transformation. If T: V → W is a linear map between vector spaces over a field, the nullity of T is dim(ker T). For a matrix A, the nullity is the dimension of the solution space of the homogeneous system A x = 0, i.e., the dimension of the null space of A.

The rank-nullity theorem gives a relation with the rank of A: nullity(A) + rank(A) = n, where A is an m × n matrix. Consequently, nullity equals n minus the rank. The nullity measures how many independent solutions exist to A x = 0; it is zero exactly when A has full column rank, meaning the linear map is injective on its domain.

In infinite-dimensional vector spaces, the nullity can be infinite, reflecting an infinite-dimensional kernel.

In graph theory, the term nullity also appears as the cyclomatic number or circuit rank of a graph, defined as nullity = m − n + c, where m is the number of edges, n the number of vertices, and c the number of connected components. This nullity equals the dimension of the cycle space and equals the number of independent cycles in the graph.

Nullity is a fundamental invariant tied to the structure of linear maps and systems of equations, and its computation is central in solving homogeneous systems and in characterizing injectivity.