Rangzahl

Rangzahl, often translated as rank, is a fundamental concept in linear algebra and algebraic structures. In German mathematical usage, Rangzahl is the term most commonly used for rank in various contexts, including matrices, linear maps, groups, and modules.

In matrix theory, the rank of an m-by-n matrix over a field is the dimension of its row space (equivalently its column space). It equals the maximum number of linearly independent rows or columns and is denoted rank(A). The Rangzahl satisfies 0 ≤ rank(A) ≤ min(m, n) and is invariant under elementary row and column operations. It also equals the number of nonzero singular values of A.

For a linear map T: V → W, rank(T) is the dimension of its image, the subspace of W that is reached by T. The rank-nullity theorem then gives dim(V) = rank(T) + nullity(T), where nullity is the dimension of the kernel of T.

In group theory and module theory, Rangzahl describes the minimal number of generators of a group (its rank). For finitely generated abelian groups, the rank equals the number of infinite cyclic direct summands and can also be computed as dim(G ⊗ Q) as a vector space over the rationals. For modules over an integral domain, the rank is defined for torsion-free modules as the dimension of their tensor product with the field of fractions.

Examples illustrate the concept: a 3×4 matrix with two independent rows has rank 2; a linear map R^3 → R^2 with rank 2 is surjective.

In numerical contexts, a numerical rank may be defined to account for rounding errors, using a tolerance to distinguish significant from negligible singular values.