Definition:Basis of Vector Space/Definition 2

Definition

Let $R$ be a division ring.

Let $\struct {G, +_G, \circ}_R$ be an vector space over $R$.

A basis is a maximal linearly independent subset of $G$.

Also known as

A basis of a vector space can also be referred to as a basis for a vector space.

Some sources refer to it as an algebraic basis.

A basis of a vector space over a subfield of $\C$ may also be known as a Hamel basis, for Georg Karl Wilhelm Hamel, to contrast with Schauder basis.

Also see

• Equivalence of Definitions of Basis of Vector Space

• Results about bases of vector spaces can be found here.

Linguistic Note

The plural of basis is bases.

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

Sources

• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE
• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem