**Definition:** A **vector space** over a field $F$ is a set $V$ together with two operations:
1. **Vector addition:** $+: V \times V \to V$
2. **Scalar multiplication:** $\cdot: F \times V \to V$
satisfying the following axioms for all $u, v, w \in V$ and $a, b \in F$:
- **A1:** $u + v = v + u$ (commutativity of addition)
- **A2:** $(u + v) + w = u + (v + w)$ (associativity of addition)
- **A3:** $\exists 0 \in V$ such that $v + 0 = v$ (additive identity)
- **A4:** $\forall v, \exists (-v)$ such that $v + (-v) = 0$ (additive inverse)
- **M1:** $a(bv) = (ab)v$ (associativity of scalar multiplication)
- **M2:** $1 \cdot v = v$ (scalar identity)
- **D1:** $a(u + v) = au + av$ (distributivity over vectors)
- **D2:** $(a + b)v = av + bv$ (distributivity over scalars)
**Terminology:**
- Elements of $V$ are called **vectors**
- Elements of $F$ are called **scalars**
- $F$ is typically $\mathbb{R}$ (real numbers) or $\mathbb{C}$ (complex numbers)