Definition:Scalar
Definition
R-Algebraic Structure
Let $\struct {R, +_R, \times_R}$ be the scalar ring of an $R$-algebraic structure $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$.
The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called scalars.
Module
Let $\struct {R, +_R, \times_R}$ be the scalar ring of a module $\struct {G, +_G, \circ}_R$.
The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called scalars.
Vector Space
Let $\struct {K, +_K, \times_K}$ be the scalar field of a vector space $\struct {G, +_G, \circ}_K$.
The elements of the scalar field $\struct {K, +_K, \times_K}$ are called scalars.
Scalar (Matrix Theory)
Let $\map \MM {m, n}$ be a matrix space of order $m \times n$ over an underlying structure $R$.
The elements of $R$ are referred to as scalars.
Scalar Quantity
A scalar quantity is a real-world concept that needs for its model a mathematical object which contains only one (usually numeric) component.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities