# Properties of vector spaces

Vector spaces must meet the following rules:
$$x + y = y + x$$

$$(x+y)+z = x + (y + z)$$

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:
$$\forall x \in V \exists (-x)\ \ | \ \ x + (-x) = 0$$

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