Definition and terminology

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)