ranknullitytheorema

The rank-nullity theorem, sometimes called the dimension theorem, is a fundamental result in linear algebra. Let V be a finite-dimensional vector space over a field F, and let T: V → W be a linear transformation. Then the dimension of V equals the sum of the dimensions of the kernel and the image: dim V = dim Ker T + dim Im T. Equivalently, rank(T) + nullity(T) = dim V, where rank(T) = dim(Im T) and nullity(T) = dim(Ker T).

Intuitively, the kernel consists of vectors that map to zero, while the image captures the directions that actually affect the output. The dimensions of these two subspaces add up to the dimension of the domain, reflecting how many independent degrees of freedom are preserved by T.

In matrix terms, if A is an m×n matrix representing T, then rank(A) + nullity(A) = n. Here, rank(A) is the dimension of the column space (the image), and nullity(A) is the dimension of the solution space to Ax = 0 (the kernel). This theorem provides a bridge between solving linear systems and the structure of linear maps.

Consequences and corollaries include: T is injective if and only if nullity(T) = 0; T is surjective onto its image by definition, and if W is finite-dimensional, T is onto W exactly when rank(T) = dim W. The theorem also yields practical tools, such as the dimension of the solution space to Ax = b often being n − rank(A) when a solution exists.

Proofs typically extend a basis of Ker T to a basis of V and observe that the images of the added basis vectors form a basis for Im T, yielding the dimension relation.