rankf

Rankf is a notational convention used in mathematics to denote the rank of a function f, most often a linear transformation between finite-dimensional vector spaces. The rank, written rank f, is the dimension of the image (range) of f. If f: V → W is linear and V, W are finite-dimensional, then rank f = dim(Im f) and is equal to the rank of any matrix representing f with respect to a chosen basis.

Properties: 0 ≤ rank f ≤ min(dim V, dim W). The rank-nullity theorem states that dim V = rank f + nullity f, where nullity f = dim(Ker f). The rank is invariant under a change of basis, i.e., it does not depend on the particular representation of f.

For composition, if g: U → V and f: V → W are linear, rank(f ∘ g) ≤ min(rank f, rank g). If f is injective, rank f = dim V; if f is surjective, rank f = dim W. Special cases: the zero map has rank 0; the identity on V has rank dim V; a projection from V onto a subspace of dimension r has rank r.

Applications include linear algebra, differential equations, and data science, where rank is used to assess independence and dimension of images in systems of equations and algorithms. In broader algebra, rank concepts generalize to modules and operators, with analogous theorems.

See also: matrix rank; rank-nullity theorem; linear transformation.