Vector spaces must meet the following rules:
Addition to be commutative:
\( x + y = y + x \)
Addition to be distributive:
\( (x+y)+z = x + (y + z) \)
Scalar multiplication to be distributive with respect to vector addition:
\( \alpha\left(x + y \right) = \alpha x + \alpha y\)
Scalar multiplication to be distributive with respect to vector the addition of field scalars:
\( \left( \alpha + \beta \right) x = \alpha x + \beta y \)
Scalar multiplication to be associative:
\( \alpha\left(\beta x \right) = \left(\alpha \beta \right) x \)
It must exist a null element:
\( \exists 0 \in V \ \ | \ \ x + 0 = 0 + x = x \)
It must exist an inverse element for every element in the vector space:
\( \forall x \in V \exists (-x)\ \ | \ \ x + (-x) = 0\)