The Field Axioms of ℝ Notes

This subsection formalizes what it means for a set to be a vector space over a field (like ℝ).

Definition:
A vector space

V over a field

F is a set equipped with:

• Vector addition:

  $+ : V \times V \to V$

  +:V×V→V

• Scalar multiplication:

  $\cdot : F \times V \to V$

  ⋅:F×V→V
  that satisfy the following 8 axioms for all

  $u, v, w \in V$

  u,v,w∈V and

  $a, b \in F$

  a,b∈F:

1. $u + v = v + u$

   u+v=v+u (commutativity)

2. $(u + v) + w = u + (v + w)$

   (u+v)+w=u+(v+w) (associativity of addition)

3. There exists a zero vector

   $0 \in V$

   0∈V such that

   $v + 0 = v$

   v+0=v

4. Each

   $v \in V$

   v∈V has an additive inverse

   $-v \in V$

   −v∈V such that

   $v + (-v) = 0$

   v+(−v)=0

5. $a(u + v) = au + av$

   a(u+v)=au+av (distributivity over vector addition)

6. $(a + b)v = av + bv$

   (a+b)v=av+bv (distributivity over field addition)

7. $a(bv) = (ab)v$

   a(bv)=(ab)v (compatibility of scalar multiplication)

8. $1 \cdot v = v$

   1⋅v=v (identity element of scalar multiplication)