Rank–nullity theorem

The theorem asserts that if T: V → W is a linear map and V is finite-dimensional, then dim(V) = rank(T) + nullity(T).¹ A standard proof constructs a basis of the kernel, extends it to a basis of V, and shows that the images of the extension vectors form a basis of the image.² The theorem holds for any field, and it generalizes to infinite-dimensional spaces only with cardinal arithmetic, where the sum is replaced by cardinal addition.

The rank–nullity theorem is a cornerstone of linear algebra, used to determine whether a linear system has solutions, to prove the invertibility of square matrices, and to analyze the dimension of solution spaces of differential equations.³ In functional analysis, it underpins the Fredholm alternative for compact operators. In graph theory, it relates the rank of an incidence matrix to the number of connected components of a graph.⁴

The theorem was implicit in the work of James Joseph Sylvester on the nullity of matrices in the 1880s, and it was explicitly formulated by Frobenius in the context of matrix rank.⁵ The term 'nullity' was coined by Sylvester in 1884.⁵ The theorem is sometimes called the dimension theorem or the fundamental theorem of linear maps.

For infinite-dimensional spaces, the theorem fails in general; for example, the shift operator on a sequence space has rank equal to the dimension but nullity 1, violating the finite-dimensional formula.⁶ The theorem also has a categorical interpretation: it expresses the exactness of the sequence 0 → ker T → V → im T → 0.² In numerical linear algebra, the rank–nullity theorem underlies the concept of numerical rank and the sensitivity of rank to perturbations.⁷

Rank

The dimension of the image of a linear map.

Nullity

The dimension of the kernel of a linear map.

Kernel

The set of vectors mapped to the zero vector.

Image

The set of all outputs of the linear map.