Vector space

Gist

Mathematical definition

The mathematical structure that has a set of vectors (V) and a field of scalars (F) where it has these properties
1. Commutatively ( $x + y = y + x$ )
2. Associatively $(x + (y + z) = (x + y) + z)$
3. There exists an element in the vector space where x+0 = x (there exists a Null vector)
4. There is an inverse $- x \in V$ for each $x \in V$
for all vectors, $x, y, z \in V$

And $F \times V \to V$ which satisfies

$a (b x) = (a b) x$ (associativity)
$a (x + y) = a x + a y)$ (distributivity)
$1 x = x$

Properties

Every vector space has a basis
Not every vector space is infinite.

References

https://www.math.toronto.edu/gscott/WhatVS.pdf