Overview
A vector space is a set of elements {% V %}, called vectors, and a field {% K %}, such that the function (multiplication by a scalar, where the scalar is an element of the field {% K %})
{% f : K \times V \rightarrow V %}
is well defined
and follows the distributive law.
{% a(\vec{A}+ \vec{B}) = a\vec{A} + a\vec{B} %}
When a
basis
exists for the vector space, such that
{% \vec{v} = \sum_i v^i \vec{e}_i %}
Scalar multiplication can be obtained as
{% a\vec{v} = \sum_i a v^i \vec{e}_i %}
Example
For a vector {% \vec{v} \in \mathbb{R}^n %}, written as
{% (x_1, ..., x_n) %}
Multiplication by a scalar {% a \in \mathbb{R} %} is defined as
{% a \vec{x} = (ax_1, ..., ax_n) %}