# Linear mapping/Introduction/Section

## Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be ${\displaystyle {}K}$-vector spaces. A mapping

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

is called a linear mapping, if the following two properties are fulfilled.

1. ${\displaystyle {}\varphi (u+v)=\varphi (u)+\varphi (v)}$ for all ${\displaystyle {}u,v\in V}$.
2. ${\displaystyle {}\varphi (sv)=s\varphi (v)}$ for all ${\displaystyle {}s\in K}$ and ${\displaystyle {}v\in V}$.

Here, the first property is called additivity and the second property is called compatibility with scaling. When we want to stress the base field, then we say ${\displaystyle {}K}$-linearity. The identity ${\displaystyle {}\operatorname {Id} _{V}\colon V\rightarrow V}$, the null mapping ${\displaystyle {}V\rightarrow 0}$ and the inclusion ${\displaystyle {}U\subseteq V}$ of a linear subspace are the simplest examples for linear mappings.

## Example

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}K^{n}}$ be the ${\displaystyle {}n}$-dimensional standard space. Then the ${\displaystyle {}i}$-th projection, this is the mapping

${\displaystyle K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{i-1},\,x_{i},\,x_{i+1},\,\ldots ,\,x_{n}\right)\longmapsto x_{i},}$

is a ${\displaystyle {}K}$-linear mapping. This follows immediately from componentwise addition and scalar multiplication on the standard space. The ${\displaystyle {}i}$-th projection is also called the ${\displaystyle {}i}$-th coordinate function.

## Lemma

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}U,V,W}$ denote vector spaces over ${\displaystyle {}K}$. Suppose that

${\displaystyle \varphi :U\longrightarrow V\,\,{\text{ and }}\,\,\psi :V\longrightarrow W}$

are linear mappings. Then also the composition

${\displaystyle \psi \circ \varphi \colon U\longrightarrow W}$

is a linear mapping.

### Proof

${\displaystyle \Box }$

## Lemma

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be ${\displaystyle {}K}$-vector spaces. Let

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

be a bijective linear map. Then also the inverse mapping

${\displaystyle \varphi ^{-1}\colon W\longrightarrow V}$

is linear.

### Proof

${\displaystyle \Box }$