Generating set

In group theory, a generating set is a set of elements of the group, such that whenever you take the closure of that set under group multiplication and inverses, you get the entire group.

In general, a generating set is a subset where you, when you perform all the legal operations, you get the entire space.

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.

• Linear Algebra. Any set whose span is the entire space is a generating set. A basis is a linearly independent generating set. (But it needs to be closed under the operation of multiplying by a field elements as well, which is a different definition from the one above)

• 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)