# Definition:Basis (Linear Algebra)

## Module

Let $R$ be a ring with unity.

Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module.

### Definition 1

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

### Definition 2

Let $\BB = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$.

Let $\Psi: R^{\paren I} \to M$ be the homomorphism given by Universal Property of Free Module on Set.

Then $\BB$ is a **basis** if and only if $\Psi$ is an isomorphism.

## Vector Space

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

The phrase **basis for $G$** can also be seen instead of **basis of $G$**.

## Also see

- Equivalence of Definitions of Basis (Linear Algebra)
- Vector Space has Basis
- Definition:Ordered Basis
- Definition:Dimension (Linear Algebra)
- Expression of Vector as Linear Combination from Basis is Unique
- Definition:Change of Basis Matrix

## Linguistic Note

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

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

## Sources

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