 ##  [Vector Space](/vector-space) 

  ##  [Vector Space](https://natural.quantumdictionary.io/vector-space-0) 

  
 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)


**Natural &amp; Formal Sciences Dictionary**

 
Definition

A set V equipped with two operations (vector addition and scalar multiplication by elements of a field) satisfying axioms: associativity, commutativity of addition, identity and inverse for addition, distributivity of scalar multiplication over field addition and vector addition, compatibility of scalar multiplication, and identity scalar acting as identity; vectors can be combined linearly.

 
 ##  [Vector Space](https://mathlogic.quantumdictionary.io/vector-space-1) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

An algebraic structure V over a field F consisting of an abelian group (V, +) and a scalar multiplication F × V → V satisfying distributivity, associativity with field multiplication, and identity scalar action; vectors can be added and scaled by field elements.

 
 ##  [Vector Space](https://puremath.quantumdictionary.io/vector-space-2) 

  
 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

A set V equipped with an abelian group operation + and a scalar multiplication by elements of a field F, satisfying distributivity, associativity of scalar multiplication, and 1·v = v; equivalently, a module over a field.

 
 ##  [Vector Space](https://algebra.quantumdictionary.io/vector-space-3) 

  
 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 
Definition

A collection of objects called vectors forming an abelian group under addition together with scalar multiplication by elements of a field such that the usual axioms (distributivity, associativity, identity) hold; equivalently, a module over a field.