# Definition:Scalar Field (Linear Algebra)

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

*This page is about Scalar Field in the context of Linear Algebra. For other uses, see Scalar Field.*

## Definition

Let $\struct {G, +_G, \circ}_K$ be a vector space, where:

- $\struct {K, +_K, \times_K}$ is a field

- $\struct {G, +_G}$ is an abelian group $\struct {G, +_G}$

- $\circ: K \times G \to G$ is a binary operation.

Then the field $\struct {K, +_K, \times_K}$ is called the **scalar field** of $\struct {G, +_G, \circ}_K$.

If the **scalar field** is understood, then $\struct {G, +_G, \circ}_K$ can be rendered $\struct {G, +_G, \circ}$.

## Also known as

A **scalar field**, as used in this context, is also known as a **ground field**.

## Also see

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**vector space** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem