User:SirMeowMeow/sandbox/Linear Algebra

Linear Algebra is the study of vector spaces and the maps which preserve the properties of vector spaces.

Field

A field is a set of elements upgraded with closed binary operators addition $+$ and multiplication $\times$, such that:

$\alpha +0=\alpha$	Identity element
$\alpha \times 1=\alpha$
$\alpha +\beta =\beta +\alpha$	Commutative
$\alpha \times \beta =\beta \times \alpha$
$x+(y+z)=(x+y)+z$	Associative
$x\times (y\times z)=(x\times y)\times z$
$x+(-x)=0$	Inverse element
$x\times x^{-1}=1$

List

A list is a collection of elements ordered by the naturals, and a list with precisely $n\in \mathbb {N}$ elements is known as of length $n$. The set of all possible lists of $n$ elements drawn from a field ${\mathsf {F}}$ may be written ${\mathsf {F}}^{n}$.