Axioms of linear algebra

1. $\vec{u}+(\vec{v}+\vec{w})=(\vec{u}+\vec{v})+\vec{w}$
2. $\vec{v}+\vec{w}=\vec{w}=\vec{v}$
3. There is a vector $0$ such that $0+\vec{v}=\vec{v}$ for all $\vec{v}$
4. For every vector $\vec{v}$ there is a vector $-\vec{v}$ so that $\vec{v}+(-\vec{v})=0$
5. $a(b\vec{v})=(ab)\vec{v}$
6. $1\vec{v}=\vec{v}$
7. $a(\vec{v}+\vec{w})=a\vec{v}+a\vec{w}$
8. $(a+b)\vec{v}=a\vec{v}+b\vec{v}$

This is basically just a checklist for Linearity

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References