Axioms (Vector Space)

To qualify as a vector space, a set $V$ and its associated operations of addition ($+$) and multiplication/scaling ($\cdot$) must adhere to the below:

Associativity

$$\begin{equation} \mathbf{u}+(\mathbf{v}+\mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w} \end{equation}$$

Commutivity

$$\begin{equation} \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \end{equation}$$

Identity of Addition

There exists and element $\mathbf{0}\,\in\,V$, called the zero vector, such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for all $\mathbf{v}\,\in\,V$.

Inverse of Addition

For every $\mathbf{v}\,\in\,V$, there exists an element $\mathbf{-v}\,\in\,V$, such that $\mathbf{v}\,+\,(\mathbf{-v})\,=\,\mathbf{0}$.

Compatibility

$$\begin{equation} a(b\mathbf{v})\,=\,(ab)\mathbf{v} \end{equation}$$

Scalar/Multiplication Identify

$$\begin{equation} 1\mathbf{v}\,=\,\mathbf{v} \end{equation}$$

where 1 denotes the scalar identity

Distributivity

$$\begin{equation} a(\mathbf{u}\,+\,\mathbf{v})\,=\,a\mathbf{u}\,+\,a\mathbf{v} \end{equation}$$