A vector space over a field \((S,\cdot,+)\) is a set V with two operations, addition \({\oplus : V \times V \rightarrow V}\) and scalar multiplication \({\odot : S \times V \rightarrow V}\), that satisfy the axioms:

- associativity of addition
$$\forall_{v, w, u \in V}: (v \oplus w) \oplus u = v \oplus ( w \oplus u)$$

- commutativity of addition
$$\forall_{v, w \in V}: v \oplus w = w \oplus v$$

- identity element of addition
$$\exists_{\mathrm{0} \in V}\forall_{v \in V}: \mathrm{0} \oplus v = v$$

- inverse elements of addition
$$\forall_{v \in V} \exists_{(-v)\in V}: v \oplus (-v) = \mathrm{0}$$

- identity element of scalar multiplication
$$\exists_{\mathrm{1} \in S}\forall_{v \in V}: \mathrm{1} \odot v = v$$

- compatibility of scalar multiplication with field multiplication
$$\forall_{v \in V; \alpha, \beta \in S}: \alpha \odot ( \beta \odot v) = (\alpha \cdot \beta) \odot v$$

- scalar multiplication is distributive over vector addition
$$\forall_{v \in V; \alpha, \beta \in S}: \alpha \odot ( v \oplus w ) = (\alpha \odot v ) \oplus (\alpha \odot w )$$

- scalar multiplication is distributive over field addition
$$\forall_{v \in V; \alpha, \beta \in S}: (\alpha + \beta) \odot v = (\alpha \odot v) \oplus (\beta \odot v) $$