the dimension theorem.
* A generating set is a basis of if and only if it is minimal, that is, no proper subset of is also a generating set of .
* A linearly independent set is a basis if and only if it is maximal, that is, it is not a proper subset of any linearly independent set.

If is a vector space of dimension , then:
* A subset of with elements is a basis if and only if it is linearly independent.
* A subset of