# Definition:Basis of Vector Space

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It has been suggested that this page be renamed.In particular: Hamel BasisTo discuss this page in more detail, feel free to use the talk page. |

## Definition

Let $K$ be a division ring.

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

### Definition 1

A **basis of $G$** is a linearly independent subset of $G$ which is a generator for $G$.

### Definition 2

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**.

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

## Also see

- Equivalence of Definitions of Basis of Vector Space
- Definition:Dimension of Vector Space
- Bases of Vector Space have Equal Cardinality

### Generalizations

## Linguistic Note

The plural of **basis** is **bases**.

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

## Sources

- 1992: Frederick W. Byron, Jr. and Robert W. Fuller:
*Mathematics of Classical and Quantum Physics*... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.2$ The Resolution of a Vector into Components - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**basis**(*plural***bases**)