---
type: "concept"
title: "Vector space"
slug: "vector-space"
language: "en"
translationGroupId: "cmr3j4pxl0005o601ajt74mnx"
domain: "Algebra"
status: "usable"
aliases: []
lastEditedBy: "ancient-tree"
---

A vector space over a [[Field|field]] $\mathbb{K}$ is a set $E$ equipped with two operations :
- Vector addition $+:E\times E \rightarrow E$
- Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$

such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:
1) $E$ is an [[Abelian group|abelian group]] with $+$.
2) Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
3) Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$
4) Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$
5) Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.