Definition: A vector space over a field (\mathbb{F}) is a set (V) together with two operations --- vector addition ((+: V \times V \to V)) and scalar multiplication ((\cdot: \mathbb{F} \times V \to V)) --- satisfying the following ten axioms for all (\mathbf{u}, \mathbf{v}, \mathbf{w} \in V) and (a, b \in \mathbb{F}):
**Definition:** A *vector space* over a field \(\mathbb{F}\) is a set \(V\) together with two operations --- vector addition \((+: V \times V \to V)\) and scalar multiplication \((\cdot: \mathbb{F} \times V \to V)\) --- satisfying the following ten axioms for all \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in V\) and \(a, b \in \mathbb{F}\):
**Addition axioms:**
1. **Closure:** \(\mathbf{u} + \mathbf{v} \in V\)
2. **Commutativity:** \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\)
3. **Associativity:** \((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\)
4. **Zero vector:** There exists \(\mathbf{0} \in V\) such that \(\mathbf{u} + \mathbf{0} = \mathbf{u}\)
5. **Additive inverse:** For each \(\mathbf{u}\), there exists \(-\mathbf{u} \in V\) such that \(\mathbf{u} + (-\mathbf{u}) = \mathbf{0}\)
**Scalar multiplication axioms:**
6. **Closure:** \(a \cdot \mathbf{u} \in V\)
7. **Distributivity over vectors:** \(a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}\)
8. **Distributivity over scalars:** \((a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u}\)
9. **Compatibility:** \(a(b\mathbf{u}) = (ab)\mathbf{u}\)
10. **Identity:** \(1 \cdot \mathbf{u} = \mathbf{u}\)
**Example:** \(\mathbb{R}^3\) with standard operations satisfies all ten axioms and is therefore a vector space over \(\mathbb{R}\).