Vector space

A vector space over a field $F$ is a set $V\neq 0$ with two binary operations called addition (denoted here as $+$) and multiplication (denoted here without a symbol between objects) satisfying, for all $u,v,w\in V,a,b\in F$

• There exists an additive identity $0\in V$, the zero vector
• Commutativity of addition: $u+v=v+u$
• Associativity of addition: $u+(v+w)=(u+v)+w$
• There exists an additive inverse for each vector: $(-v)+v=v+(-v)=0$
• There is a scalar $1\in F$ that acts as a multiplicative identity
• Distributivity of scalar multiplication over vector addition: $a(u+v)=au+av$
• Distributivity of scalar multiplication over scalar addition: $(a+b)v=av+bv$
• Scalar multiplication compatible with vector multiplication: $(ab)v=a(bv)$

A vector space over a field $F$ is called an $F$-vector space. In particular we may refer to a real vector space when $F=\mathbb {R}$ or a complex vector space when $F=\mathbb {C}$.

Appearances in group theory

As this wiki is about group theory, we shall discuss where vector spaces appear when discussing groups.

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• In the representation theory of groups, a representation is a homomorphism $\rho :G\to \mathrm {GL} (V)$ where $V$ is some vector space, $G$ is a group, and $\mathrm {GL} (V)$ denotes the general linear group over the vector space.