Generating set

In mathematics, a subset S of a algebraic structure G is a generating set of G, or G is "generated" by S, if the smallest set containing S having the same algebraic structure is G itself. For example, if G is a group and itself is the smallest group containing S, then S is the generating set of G.

Examples

• The integers as a group under addition has 1 as a generating set. The element 2 is not a generating set, as the odd numbers will be missing. The two-elements subset {3, 5} is a generating set.

• In linear algebra, S is a generating set or spanning set of a vector space V if V is the linear span of S.

• Continuous Functions on the interval. Polynomials are a generating set, because closure under limits forms the entire space. (we need the concept of closure under a given topology here)