Vector space - Knowlpedia

A vector space over a field $\mathbb F$ is a set $V$ with operations (functions) $+:V\times V\to V$ and $\cdot:\mathbb F\times V\to V$ , and an element $0\in V$ , satisfying the following for all $u,v,w\in V$ and $a,b\in\mathbb F$ :

Commutativity: $u+v=v+u$ .
Associativity: $(u+v)+w=u+(v+w)$ .
Additive identity: $v+0=v$ .
Additive inverse: for each $v$ , there is $-v\in V$ with $v+(-v)=0$ .
Distributivity over vector addition: $a\cdot(u+v)=a\cdot u+a\cdot v$ .
Distributivity over scalar addition: $(a+b)\cdot v=a\cdot v+b\cdot v$ .
Scalar associativity: $(ab)\cdot v=a\cdot(b\cdot v)$ .
Scalar identity: $1\cdot v=v$ .

Context

Vector spaces are the basic objects studied via linear maps and invariants including the determinant and eigenvalues of operators. A vector space over $\mathbb F$ is the same thing as a module over the field $\mathbb F$ .

Examples

The zero vector space, consisting only of $0$ , is allowed. The field is part of the structure: the same additive group can be a vector space over more than one field, but the scalar multiplication must be specified.

$\mathbb{R}^n$ with componentwise addition and scalar multiplication is a vector space over $\mathbb{R}$ .
The set of polynomials $\mathbb{F}[x]$ with the usual addition and scalar multiplication is a vector space over $\mathbb{F}$ .
The set of $m\times n$ matrices over $\mathbb{F}$ is a vector space over $\mathbb{F}$ .