# 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**.