# Category:Scalar Multiplication

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This category contains results about **Scalar Multiplication**.

Definitions specific to this category can be found in Definitions/Scalar Multiplication.

### $R$-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**.

## Pages in category "Scalar Multiplication"

The following 9 pages are in this category, out of 9 total.

### S

- Scalar Multiplication by Zero gives Zero Vector
- Scalar Multiplication of Vectors is Associative
- Scalar Multiplication of Vectors is Distributive over Scalar Addition
- Scalar Multiplication of Vectors is Distributive over Vector Addition
- Scalar Multiplication on Locally Convex Space is Continuous
- Scalar Multiplication on Normed Vector Space is Continuous
- Scalar Product of Magnitude by Unit Vector Quantity