---
title: "vector space"
type: concept
related: [Additive Identity, Commutivity, Multiplicative Identity, Group, Inverses]
source: https://www.jemoka.com/posts/kbhvector_space/
confidence: high
status: active
---

A vector space is an object between a field and a group; it has two ops&mdash;addition and scalar multiplication. Its not quite a field and its more than a group.
constituents A set \(V\) An addition on \(V\) An scalar multiplication on \(V\) such that&hellip;
requirements commutativity in add.: \(u+v=v+u\) associativity in add. and mult.: \((u+v)+w=u+(v+w)\); \((ab)v=a(bv)\): \(\forall u,v,w \in V\) and \(a,b \in \mathbb{F}\) distributivity: goes both ways \(a(u+v) = au+av\) AND!! \((a+b)v=av+bv\): \(\forall a,b \in \mathbb{F}\) and \(u,v \in V\) additive identity: \(\exists 0 \in V: v+0=v \forall v \in V\) additive inverse: \(\forall v \in V, \exists w \in V: v+w=0\) multiplicative identity: \(1v=v \forall v \in V\) additional information Elements of a vector space are called vectors or points. vector space &ldquo;over&rdquo; fields Scalar multiplication is not in the set \(V\); instead, &ldquo;scalars&rdquo; \(\lambda\) come from this magic faraway land called \(\mathbb{F}\). The choice of \(\mathbb{F}\) for each vector space makes it different; so, when precision is needed, we can say that a vector space is &ldquo;over&rdquo; some \(\mathbb{F}\) which contributes its scalars.
Therefore:
A vector space over \(\mathbb{R}\) is called a real vector space A vector space over \(\mathbb{C}\) is called a real vector space
