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

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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.

## Theorem

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be ${\displaystyle {}K}$-vector spaces. Let ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, denote a basis of ${\displaystyle {}V}$, and let ${\displaystyle {}w_{i}}$, ${\displaystyle {}i\in I}$, denote elements in ${\displaystyle {}W}$. Then there exists a unique linear mapping

${\displaystyle f\colon V\longrightarrow W}$

with

${\displaystyle f(v_{i})=w_{i}{\text{ for all }}i\in I.}$

### Proof

This proof was not presented in the lecture.
${\displaystyle \Box }$

## Example

The easiest linear mappings are (beside the null mapping) the linear maps from ${\displaystyle {}K}$ to ${\displaystyle {}K}$. Such a linear mapping

${\displaystyle \varphi \colon K\longrightarrow K,x\longmapsto \varphi (x),}$

is determined (by fact, but this is also directly clear) by ${\displaystyle {}\varphi (1)}$, or by the value ${\displaystyle {}\varphi (t)}$ for an single element ${\displaystyle {}t\in K}$, ${\displaystyle {}t\neq 0}$. In particular, ${\displaystyle {}\varphi (x)=ax}$, with a uniquely determined ${\displaystyle {}a\in K}$. In the context of physics, for ${\displaystyle {}K=\mathbb {R} }$, and if there is a linear relation between two measurable quantities, we talk about proportionality, and ${\displaystyle {}a}$ is called the proportionality factor. In school, such a linear relation occurs as "rule of three“.