# Linear mapping/Determination theorem/Finite-dimensional/Section

Behind the following statement
(the *determination theorem*),
there is the important principle that in linear algebra
(of finite-dimensional vector spaces),
the objects are determined by finitely many data.

Let be a field, and let and be -vector spaces. Let , , denote a basis of , and let , , denote elements in . Then there exists a unique linear mapping

with

### Proof

The easiest linear mappings are (beside the null mapping) the linear maps from to . Such a linear mapping

is determined
(by
fact,
but this is also directly clear)
by , or by the value for an single element
, .
In particular,
,
with a uniquely determined
.
In the context of physics, for
,
and if there is a linear relation between two measurable quantities, we talk about *proportionality*, and is called the *proportionality factor*. In school, such a linear relation occurs as "rule of three“.