# Definition:Generator of Vector Space

## Definition

Let $K$ be a division ring.

Let $\mathbf V$ be a vector space over $K$.

Let $S \subseteq \mathbf V$ be a subset of $\mathbf V$.

$S$ is a **generator of $\mathbf V$** if and only if every element of $\mathbf V$ is a linear combination of elements of $S$.

## Generated Subspace

### Definition 1

The **subspace generated by $S$** is the intersection of all subspaces of $\mathbf V$ containing $S$.

### Definition 2

The **subspace generated by $S$** is the set of all linear combinations of elements of $S$.

## Also known as

A **generator** of a vector space is also known as a **spanning set**.

Some sources refer to a **generator for** rather than **generator of**. The two terms mean the same.

It can also be said that $S$ **generates $\mathbf V$ (over $K$)**.

Other terms for $S$ are:

- A
**generating set of $\mathbf V$ (over $K$)** - A
**generating system of $\mathbf V$ (over $K$)**

Some sources refer to such an $S$ as a **set of generators of $\mathbf V$ over $K$** but this terminology is misleading, as it can be interpreted to mean that each of the elements of $S$ is itself a **generator** of $\mathbf V$ independently of the other elements.

## Also see

- Results about
**generators of vector spaces**can be found here.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 33$. Definition of a Basis - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): $\text{A}.2$: Linear algebra and determinants: Definition $\text{A}.5$