# Definition:Scalar Multiplication

## 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 $\mathbf a$:
$\size {m \mathbf a} = m \size {\mathbf a}$
the direction of $m \mathbf a$ is the same as the direction of $\mathbf a$.

## Also known as

Some sources refer to scalar multiplication as exterior multiplication.