# Definition:Basis (Linear Algebra)

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## 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 $G$.

## Linguistic Note

The plural of basis is bases.

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