Vector Space Axioms

Definition

Let $V$ be an arbitrary nonempty set of objects on which the following operations are defined:

• Addition: $\mathbf{u}+\mathbf{v},\forall \mathbf{u},\mathbf{v}\in V$ (sum of $\mathbf{u}$ and $\mathbf{v}$)
• Scalar multiplication: $k\mathbf{u},\forall \mathbf{u}\in V,k\text{ scalar}$ (scalar multiple of $\mathbf{u}$ by $k$)

If $\forall \mathbf{u},\mathbf{v},\mathbf{w}\in V;\forall k,m\text{ scalar}$

1. $\mathbf{u},\mathbf{v}\in V⟹\mathbf{u}+\mathbf{v}\in V$
2. $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$
3. $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{v}+(\mathbf{u}+\mathbf{w})$
4. $\exists 0\in V\ni (0+\mathbf{v})=(\mathbf{v}+0)=\mathbf{v},\forall \mathbf{v}\in V$
5. $\forall \mathbf{v}\in V\exists (-\mathbf{v})\ni (-\mathbf{v})+\mathbf{v}=\mathbf{v}+(-\mathbf{v})=0\in V$
6. $k\text{ scalar},\mathbf{u}\in V⟹k\mathbf{u}\in V$
7. $k(\mathbf{u}+\mathbf{v})=k\mathbf{u}+k\mathbf{v}$
8. $(k+m)\mathbf{u}=k\mathbf{u}+m\mathbf{u}$
9. $k(m\mathbf{u})=(km)\mathbf{u}$
10. $1\mathbf{u}=\mathbf{u}$

Then

• $V$ is a vector space
• All objects in $V$ are vectors