nLab basis of a vector space

Definition

For $k$ a field and $V$ a free $k$-vector space, a basis for $V$ is a basis of a free module for $V$ regarded as a free module over $k$. In functional analysis, a basis in this sense is called a Hamel basis.

Properties

The basis theorem asserts that, with the axiom of choice, every vector space admits a basis, hence that every module over a field is a free module.

Examples

In representation theory:

• Gelfand-Tsetlin basis

Specifically in representation theory of the symmetric group:

• seminormal basis.

Related concepts

• orthogonal basis

• basis in functional analysis

• dual basis

• basis

• mutually unbiased bases