MathGloss

A vector space $V$ over the field $F$ is a set together with two operations “addition” ($+: V\times V \to V$) and “scalar multiplication” ($\cdot: F\times V \to V$) satisfying the following eight axioms:

1. Addition is associative
2. Addition is commutative
3. There exists a vector $0 \in V$ such that $0 + v = v$ for all $v \in V$. ($0$ is the additive identity in $V$)
4. For every $v \in V$, there exists $-v \in V$ such that $v + (-v) = 0$. ($-v$ is the additive inverse of $v$)
5. $a(bv)= (ab)v$ for all $a,b \in F$ and $v \in V$.
6. $1v = v$ for all $v \in V$, where $1$ is the multiplicative identity in $F$.
7. $a(u+v) = au+av$ for all $a \in F$, $u,v \in V$.
8. $(a+b)v = av+bv$ for all $a,b \in F$, $v\in V$.

Axioms 7. and 8. are distributive laws.

Wikidata ID: Q125977

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