Definition:Basis (Linear Algebra)

Definition

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).

Comment

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

Also see

• Expression of Vector as Linear Combination from Basis is Unique

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.4$
• Seth Warner: Modern Algebra (1965): $\S 27$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 7.33$
• John F. Humphreys: A Course in Group Theory (1996): $\text{A}.2$: Definition $\text{A}.6$
• For a video presentation of the contents of this page, visit the Khan Academy.