Definition:Ordered Basis

Definition

Let $R$ be a ring with unity.

Let $G$ be a free $R$-module.

An ordered basis of $G$ is a sequence $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ of elements of $G$ such that $\set {a_1, \ldots, a_n}$ is a basis of $G$.

Also see

• Definition:Change of Basis Matrix
• Definition:Coordinate System

• Results about ordered bases can be found here.

Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases
• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.): Appendix $\text B$. Review of Tensors