ranknullity

Rank-nullity refers to a fundamental theorem in linear algebra that connects the dimensions of two key subspaces associated with a linear transformation.

Let T be a linear transformation from a finite-dimensional vector space V to another vector space W. The kernel of T is Ker(T) = {v in V | T(v) = 0}, a subspace of V. The image of T is Im(T) = {T(v) | v in V}, a subspace of W. The rank of T is the dimension of Im(T), denoted rank(T). The nullity of T is the dimension of Ker(T), denoted nullity(T).

The rank-nullity theorem states that if V is finite-dimensional, then the dimension of V equals the sum of the rank and the nullity of T: dim(V) = rank(T) + nullity(T). Equivalently, dim(V) = dim(Im(T)) + dim(Ker(T)). This provides a precise accounting of how the domain’s dimension is distributed between the part of V mapped nontrivially into W (the image) and the part sent to zero (the kernel).

Consequences include: rank(T) ≤ dim(V); nullity(T) = dim(V) − rank(T); T is surjective if and only if rank(T) = dim(W); T is injective if and only if nullity(T) = 0. The theorem is closely related to the First Isomorphism Theorem, which gives V/Ker(T) ≅ Im(T).

Examples: the identity map on R^n has rank n and nullity 0; the zero map from R^n to any W has rank 0 and nullity n. A concrete map T: R^3 → R^2 with a rank-2 image has nullity 1.