Definition:Scalar Multiplication
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Definition
$R$-Algebraic Structure
Let $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ be an $R$-algebraic structure with $n$ operations, where:
- $\struct {R, +_R, \times_R}$ is a ring
- $\struct {S, *_1, *_2, \ldots, *_n}$ is an algebraic structure with $n$ operations
The operation $\circ: R \times S \to S$ is called scalar multiplication.
Module
Let $\struct {G, +_G, \circ}_R$ be an module (either a left module or a right module or both), where:
- $\struct {R, +_R, \times_R}$ is a ring
- $\struct {G, +_G}$ is an abelian group.
The operation $\circ: R \times G \to G$ is called scalar multiplication.
Vector Space
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.
The operation $\circ: K \times G \to G$ is called scalar multiplication.
Vector Quantity
Let $\mathbf a$ be a vector quantity.
Let $m$ be a scalar quantity.
The operation of scalar multiplication by $m$ of $\mathbf a$ is denoted $m \mathbf a$ and defined such that:
- the magnitude of $m \mathbf a$ is equal to $m$ times the magnitude of $\mathbf a$:
- $\size {m \mathbf a} = m \size {\mathbf a}$
Also known as
Some sources refer to scalar multiplication as exterior multiplication.
Also see
- Definition:Dot Product, also known in some sources as scalar product.