Linear mapping/Examples/Introduction/Section

We are interested in mappings between two vector spaces that respect the structures, that is, they are compatible with addition and with scalar multiplication.

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 ${}K$-linear. 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 of a linear mapping. 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.  Instead of linear mapping, we also say homomorphism.

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“.

Many important functions, in particular from ${}\mathbb {R}$ to ${}\mathbb {R}$, are not linear. For example, the squaring ${}x\mapsto x^{2}$, the square root, the trigonometric functions, the exponential function, the logarithm is not linear. But also for such more complicated functions there are, in the framework of differential calculus, linear approximations, which help to understand these functions.

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.

If you buy ten times this stuff, you have to pay ten times as much. In the linear world, there is no rebate.

Example

In a shop, there are ${}n$ different products to buy, and the price of the ${}i$-th product (with respect to a certain unit) is ${}a_{i}$. A purchase is described by the ${}n$-tuple

$\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right),$

where ${}x_{i}$ is the amount of the ${}i$-th product bought. The price for the purchase is hence ${}\sum _{i=1}^{n}a_{i}x_{i}$. The price mapping

$\mathbb {R} ^{n}\longrightarrow \mathbb {R} ,\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}x_{i},$

is linear. This means, for example, that if we do first the purchase ${}\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)$ and then, a week later, the purchase ${}\left(y_{1},\,y_{2},\,\ldots ,\,y_{n}\right)$, then the price of the two purchases together is the same as the price of the purchase ${}\left(x_{1}+y_{1},\,x_{2}+y_{2},\,\ldots ,\,x_{n}+y_{n}\right)$.

Example

The mapping

$K^{n}\longrightarrow K^{m},{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto M{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{j=1}^{n}a_{1j}s_{j}\\\sum _{j=1}^{n}a_{2j}s_{j}\\\vdots \\\sum _{j=1}^{n}a_{mj}s_{j}\end{pmatrix}},$

which is given (see example) by an ${}m\times n$-matrix ${}M=(a_{ij})_{1\leq i\leq m,\,1\leq j\leq n}$, is linear.

Definition

Let ${}V$ be a vector space over a field ${}K$. For ${}a\in K$, the linear mapping

$\varphi \colon V\longrightarrow V,v\longmapsto av,$

is called homothety (or dilation)

with scaling factor ${}a$.

For a homothety, the domain space and the target space are the same. The number ${}a$ is called scaling factor. For ${}a=1$, we get the identity, for ${}a=-1$, we get the point reflection at the origin.

Example

Let ${}C^{0}(\mathbb {R} ,\mathbb {R} )$ denote the space of continuous functions from ${}\mathbb {R}$ to ${}\mathbb {R}$, and let ${}C^{1}(\mathbb {R} ,\mathbb {R} )$ denote the space of continuously differentiable functions. Then the mapping

$D\colon C^{1}(\mathbb {R} ,\mathbb {R} )\longrightarrow C^{0}(\mathbb {R} ,\mathbb {R} ),f\longmapsto f',$

which assigns to a function its derivative, is linear. In analysis, we prove that

${}(af+bg)'=af'+bg'\,$

holds for ${}a,b\in \mathbb {R}$ and another function ${}g\in C^{1}(\mathbb {R} ,\mathbb {R} )$.

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.

$\Box$