Definition:Scalar Field (Linear Algebra)

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.