Def/Vector space

Definition of Vector space: Suppose that $F$ is a field, such as $\RR$ or $\CC$, for example. An $F$-vector space (for example, a real or complex vector space) is an ordered triple $(V, +, \cdot)$, where:

• $V$ is a set; the elements of $V$ are called vectors.
• $+$ is a binary operation on $V$, usually called vector addition.
• $\cdot$ is a function from $F \times V$ to $V$, usually called scaling or scalar multiplication.

These operations are required to satisfy the following axioms:

• The pair $(V, +)$ satisfies the axioms of an abelian group.
• Scalar multiplication is associative, in the sense that, for all $x,y \in F$, and all $v \in V$, $x \cdot (y \cdot v) = (x \cdot y) \cdot v$.
• Scalar multiplication is distributive, in the sense that, for all $x,y \in F$, and all $v \in V$, $(x + y) \cdot v = (x \cdot v) + (y \cdot v)$.
• Scalar multiplication is distributive, in the sense that, for all $x \in F$, and all $v,w \in V$, $x \cdot (v + w) = (x \cdot v) + (x \cdot w)$.