Notes

A vector space, or linear space, is a set of vectors. It's possible to scale by numbers, called scalars, and add these vectors together.

The operations of vector addition or scalar multiplication in a vector space $X$ must safisty the following axions:
 Additive axions. For every $\vec{x}$, $\vec{y}$, and $\vec{z}$ in $X$, we have:
 $\vec{x} + \vec{y} = \vec{y} + \vec{x}$.
 $(\vec{x} + \vec{y}) + \vec{z} = \vec{x} + (\vec{y} + \vec{z})$.
 $\vec{0} + \vec{x} = \vec{x} + \vec{0} = \vec{x}$.
 $(\vec{x}) + \vec{x} = \vec{x} + (\vec{x}) = \vec{x}$.
 Multiplicative axions. For every $\vec{x}$ in $X$, and real numbers $c$ and $d$, we have:
 $0\vec{x} = 0$.
 $1\vec{x} = \vec{x}$.
 $(cd)\vec{x} = c(d\vec{x})$.
 Distributive axions. For every $\vec{x}$ and $\vec{y}$ in $X$, and real numbers $c$ and $d$, we have:
 $c(\vec{x} + \vec{y}) = c\vec{x} + c\vec{y}$.
 $(c + d)\vec{x} = c\vec{x} + d\vec{x}$.
 Additive axions. For every $\vec{x}$, $\vec{y}$, and $\vec{z}$ in $X$, we have: