Definition:Basis (Linear Algebra)

Definition
Let $R$ be a ring with unity.

Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module.

A basis of $G$ (plural: bases) is a linearly independent subset of $G$ which is a generator for $G$.

Alternatively, a basis is a maximal linearly independent subset of $G$.

The two definitions are equivalent, as shown on Equivalence of Definitions of Basis (Linear Algebra).

Linguistic Note
The pronunciation of bases in this context is bay-seez, not bay-siz.

Also known as
The phrase basis for $G$ can also be seen instead of basis of $G$.

Also see

 * Definition:Ordered Basis
 * Expression of Vector as Linear Combination from Basis is Unique
 * Free Module is Isomorphic to Free Module Indexed by Set