Vector Space

A *vector space* or a *linear space* consists of a set $$V$$ equipped with operations of addition and scalar multiplication. Assume that the following rules are satisfied:
 * 1) $$\forall \mathbf{v,w} \in V$$, we have that $$\mathbf{v+w = w+v}$$ (commutative)
 * 2) $$\forall \mathbf{v,w,u} \in V$$, we have that $$\mathbf{(v+w)+u = w+(v+u)}$$ (associative)
 * 3) There exists an element called 0 in $$V$$, such that $$\forall \mathbf{v} \in V$$, we have that $$\mathbf{(v+0) = v}$$ (0 is additive identity)
 * 4) For each $$\mathbf{v} \in V$$, there exists an element, called $$-\mathbf{v}$$, such that \mathbf{v+(-v)} = 0 (existence of additive inverse)
 * 5) For all scalars $$\alpha, \beta \in \mathbb{R}$$, $$\alpha(\beta \mathbf{v}) = (\alpha\beta) \mathbf{v}$$ (scalar multiplication is associative)
 * 6) For all $$\alpha, \beta \in \mathbb{R}$$ and all $$\mathbf{v} \in V$$, we have that $$(\alpha+\beta) \mathbf{v} = \alpha \mathbf{v} + \beta \mathbf{v}$$ (left distribution rule)
 * 7) For all $$\alpha \in \mathbb{R}$$ and all $$\mathbf{v,w} \in V$$, we have that $$\alpha (\mathbf{v}+\mathbf{w}) = \alpha \mathbf{v} + \alpha \mathbf{w}$$ (right distribution rule)
 * 8) For all $$\mathbf{v} \in V$$, we have that $$1 \mathbf{v} = \mathbf{v}$$

Then, the space $$V$$, equipped with the operations of addition and scalar multiplication, forms a vector space .