Definition:Scalar Field (Linear Algebra)

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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

• Definition:Scalar Ring:
  • Definition:Scalar Ring of Module
  • Definition:Scalar Ring of Unitary Module
  • Definition:Scalar Division Ring

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