vector space

def (Vector Space)

A vector space over $F$ is a set $V$ with two operations:

Satisfying the following operations:

[ $+$ ] For all $u, v \in V$ , $u + v = v + u$ (commutativity)
[ $+$ ] For all $u, v, w \in V$ , $(u + v) + w = u + (v + w)$ (associative)
[ $+$ ] $\exists 0 \in V$ s.t. $\forall v \in V$ $0 + v = v + 0 = 0$ (additive identity)
[ $+$ ] $\forall v \in V$ $\exists w \in V$ s.t. $v + w = w + v = 0$ (additive inverse)
[ $\times$ ] $a, b \in F$ $v \in V$ $(a b) v = a (b v)$ (associative)
[ $\times$ ] $v \in V$ , $1 v = v$ (multiplicative identity)
[ $\times$ ] $a, b \in F$ $u, v \in V$ (distributive)
- $a (u + v) = a u + a v$
- $(a + b) v = a v + b v$

Proposition: A vector space has a unique additive identity

Proof: Let $0, 0^{'}$ be two additive identities of $V$

0 = 0 + 0^{'} = 0^{'}

Proposition: Additive inverses are unique

Proof: Let $v \in V$ . Suppose $w, w^{'} \in V$ be two additive inverses of $V$

w = w + 0 = w + (v + w^{'}) = (w + v) + w^{'} = 0 + w^{'} = w^{'}

Proposition: For every

v \in V

0 v = 0

Proof: Note here $0 \in F$ .

\begin{aligned} (0 + 0) v & = 0 v + 0 v \\ 0 v & = 0 v + 0 v \\ 0 & = 0 v \end{aligned}

Proposition: For every

a \in F

a 0 = 0

Proof: $$\begin{array}{rl} a(0 + 0) &= a0 + a0 \ a0 &= a0 + a0 \ 0 &= a0 \end{array}$$

Proposition: For every

v \in V

(- 1) v = - v

Proof: Note here $- 1$ is a scalar and $- v \in V$

\begin{aligned} (- 1) (v + 0) & = (- 1) (v) + (- 1) (0) \\ = - v \end{aligned}