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.

Theorem

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

$f\colon V\longrightarrow W$

with

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

Proof

This proof was not presented in the lecture.

$\Box$

The graph of a linear mapping from ${}\mathbb {R}$ to ${}\mathbb {R}$, the mapping is determined by the proportionality factor ${}k$ alone.

Example

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

$\varphi \colon K\longrightarrow K,x\longmapsto \varphi (x),$

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