ABSTRACT
In a first course in linear algebra, a rank function is usually defined as the dimension of the row space of a real n × n matrix or the number of nonzero rows in a row echelon form of the matrix. In this chapter, the authors use a general definition of a rank function, incorporating its most basic properties, that being that a rank function is a mapping from an algebraic system with a binary operation to the set of nonnegative integers which must map only the additive identity to 0, and be subadditive, that is the “rank” of a sum must be at most the sum of the “ranks” of the summands. They give some interesting examples of rank functions, both some that are commonly known as well as some that are virtually unknown or are not known as rank functions. The authors provide some of the known equivalences among matrix rank functions and rank functions defined on graphs.
