Linear mapping/Introduction/Section

Definition

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

$\varphi \colon V\longrightarrow W$

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

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

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 ${}K$-linearity. The identity ${}\operatorname {Id} _{V}\colon V\rightarrow V$, the null mapping ${}V\rightarrow 0$ and the inclusion ${}U\subseteq V$ of a linear subspace are the simplest examples for linear mappings. For a linear mapping, the compatibility with arbitrary linear combination holds, that is,

${}\varphi {\left(\sum _{i=1}^{n}s_{i}v_{i}\right)}=\sum _{i=1}^{n}s_{i}\varphi (v_{i})\,,$

see exercise.

Example

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

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

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

Lemma

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

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

are linear mappings. Then also the composition

$\psi \circ \varphi \colon U\longrightarrow W$

is a linear mapping.

Proof

See exercise.

$\Box$

Lemma

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

$\varphi \colon V\longrightarrow W$

be a bijective linear map. Then also the inverse mapping

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

is linear.

Proof

See exercise.

$\Box$