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}