Vector_Space

Definition

A vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on $V$ such that the following properties hold

  1. Commutativity: $u + v = v + u$ for all $u,v \in V$

  2. Associativity $\left. (u + v) \right. + w = u + \left. (v + w) \right.$ and $\left. (ab) \right.v = a\left. (bv) \right.$ for all $u,v,w \in V$ and all $a,b \in \mathbb{F}$

  3. Additive identity: There exists an element $0 \in V$ such that $v + 0 = v$ for all $v \in V$

  4. Additive inverse: For every $v \in V$, there exists $w \in V$ such that $v + w = 0$

  5. Multiplicative identity: $1v = v$ for all $v \in V$

  6. Distributive properties: $a\left. (u + v) \right. = au + av$ and $\left. (a + b) \right.v = av + bv$ for all $a,b \in \mathbb{F}$ and all $u,v \in V$