Linear algebra (Osnabrück 2024-2025)/Part I/Lecture 2

A main focus of mathematics is to study how a certain variable (describing a size or a magnitude) depends on another variable (or several variables). For example, how does the area of a square depend on the length of the side, how does the price depend on the commodities bought, how does the size of a population grow with time. Such dependencies are expressed with the concept of a mapping.

Definition

Let ${}L$ and ${}M$ denote sets. A mapping ${}F$ from ${}L$ to ${}M$ is given by assigning to every element of the set ${}L$ exactly one element of the set ${}M$. The unique element that is assigned to ${}x\in L$ is denoted by ${}F(x)$. For the mapping as a whole, we write

$F\colon L\longrightarrow M,x\longmapsto F(x).$

If a mapping ${}F\colon L\rightarrow M$ is given, then ${}L$ is called the domain (or domain of definition) of the map, and ${}M$ is called the codomain (or target range) of the map. For an element ${}x\in L$, the element

${}F(x)\in M\,$

is called the value of ${}F$ at the place (or argument) ${}x$.

Two mappings ${}F\colon L_{1}\rightarrow M_{1}$ and ${}G\colon L_{2}\rightarrow M_{2}$ are equal if and only if their domains coincide, their codomains coincide, and if for all ${}x\in L_{1}=L_{2}$ the equality ${}F(x)=G(x)$ in ${}M_{1}=M_{2}$ holds. So the equality of mappings is reduced to the equalities of elements in a set. Mappings are also called functions. However, we will usually reserve the term function for mappings where the codomain is a number set like the real numbers ${}\mathbb {R}$.

For every set ${}L$, the mapping

$L\longrightarrow L,x\longmapsto x,$

which sends every element to itself, is called the identity (on ${}L$). We denote it by ${}\operatorname {Id} _{L}$. For another set ${}M$ and a fixed element ${}c\in M$, the mapping

$L\longrightarrow M,x\longmapsto c$

that sends every element ${}x\in L$ to the constant value ${}c$ is called the constant mapping (with value ${}c$). It is usually again denoted by ${}c$.

There are several ways to describe a mapping, like a value table, a bar chart, a pie chart, an arrow diagram, or the graph of the mapping. In mathematics, a mapping is most often given by a mapping rule that allows computing the values of the mapping for every argument. Such rules are, e.g., (from ${}\mathbb {R}$ to ${}\mathbb {R}$) ${}x\mapsto x^{2}$, ${}x\mapsto x^{3}-e^{x}+\sin x$, etc. In the sciences and in sociology, also empirical functions are important that describe real movements or developments. But also for such functions, one wants to know whether they can be described (approximately) in a mathematical manner.

${}x$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$
${}\pi (x)$	${}2$	${}4$	${}6$	${}5$	${}3$	${}1$

${}\cdot$	${}0$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$
${}0$	${}0$	${}0$	${}0$	${}0$	${}0$	${}0$	${}0$
${}1$	${}0$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$
${}2$	${}0$	${}2$	${}4$	${}6$	${}1$	${}3$	${}5$
${}3$	${}0$	${}3$	${}6$	${}2$	${}5$	${}1$	${}4$
${}4$	${}0$	${}4$	${}1$	${}5$	${}2$	${}6$	${}3$
${}5$	${}0$	${}5$	${}3$	${}1$	${}6$	${}4$	${}2$
${}6$	${}0$	${}6$	${}5$	${}4$	${}3$	${}2$	${}1$

Injective and surjective mappings

Definition

Let ${}L$ and ${}M$ denote sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

be a mapping. Then ${}F$ is called injective if for two different elements ${}x,x'\in L$, also ${}F(x)$ and ${}F(x')$

are different.

Definition

Let ${}L$ and ${}M$ denote sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

be a mapping. Then ${}F$ is called surjective if, for every ${}y\in M$, there exists at least one element ${}x\in L$, such that

${}F(x)=y\,.$

Definition

Let ${}M$ and ${}L$ denote sets, and suppose that

$F\colon M\longrightarrow L,x\longmapsto F(x),$

is a mapping. Then ${}F$ is called bijective if ${}F$ is injective as well as

surjective.

These concepts are fundamental!

The question, whether a mapping ${}F\colon L\rightarrow M$ has the properties of being injective or surjective, can be understood looking at the equation

${}F(x)=y\,$

(in the two variables ${}x$ and ${}y$). The surjectivity means that for every ${}y\in M$ there exists at least one solution

${}x\in L\,$

for this equation; the injectivity means that for every ${}y\in M$ there exists at most one solution ${}x\in L$ for this equation. The bijectivity means that for every ${}y\in M$ there exists exactly one solution ${}x\in L$ for this equation. Hence, surjectivity means the existence of solutions, and injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics, and they can also be interpreted as surjectivity or injectivity of suitable mappings.

In order to show that a certain mapping is injective, we often use the following strategy: One shows for any two given elements ${}x$ and ${}x'$ using the condition ${}F(x)=F(x')$ that ${}x=x'$ holds. This method is often easier than showing that ${}x\neq x'$ implies ${}F(x)\neq F(x')$.

Example

The mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

is neither injective nor surjective. It is not injective because the different numbers ${}2$ and ${}-2$ are both sent to ${}4$. It is not surjective because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

is injective, but not surjective. The injectivity can be seen as follows: If ${}x\neq y$, then one number is larger, say

${}x>y\geq 0\,.$

But then also ${}x^{2}>y^{2}$, and in particular ${}x^{2}\neq y^{2}$. The mapping

$\mathbb {R} \longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$

is not injective, but surjective, since every nonnegative real number has a square root. The mapping

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$

is injective and surjective.

Definition

Let ${}F\colon L\rightarrow M$ denote a bijective mapping. Then the mapping

$G\colon M\longrightarrow L$

that sends every element ${}y\in M$ to the uniquely determined element ${}x\in L$ with ${}F(x)=y$,

is called the inverse mapping of ${}F$.

The inverse mapping is usually denoted by ${}F^{-1}$.

We discuss two classes of mappings that are in the framework of linear algebra very important. They are both so-called linear mappings.

Example

Let ${}a\in \mathbb {R}$ be fixed. This real number defines a mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax.$

For ${}a=0$, this is the constant zero mapping. For ${}a\neq 0$, we have a bijective mapping; the inverse mapping is

$y\longrightarrow {\frac {1}{a}}y.$

Here, the inverse mapping has a similar form as the mapping itself.

Example

Let an ${}m\times n$-matrix

${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}$

be given, where the entries ${}a_{ij}$ are real numbers. Such a matrix defines a mapping

$\varphi \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m},$

by sending an ${}n$-tuple ${}x={\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}\in \mathbb {R} ^{n}$ to the ${}m$-tuple

${}\varphi (x)={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{j=1}^{n}a_{1j}x_{j}\\\sum _{j=1}^{n}a_{2j}x_{j}\\\vdots \\\sum _{j=1}^{n}a_{mj}x_{j}\end{pmatrix}}\,.$

The ${}i$-th component of the image vector is

${}y_{i}=\left(a_{i1},\,a_{i2},\,\ldots ,\,a_{in}\right){\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}=\sum _{j=1}^{n}a_{ij}x_{j}\,.$

So one has to apply the ${}i$-th row of the matrix to the column vector ${}x$ in the described way.

It is a goal of linear algebra to determine, in dependence of the entries ${}a_{ij}$, whether the mapping defined by the matrix is injective, surjective, or bijective, and how, in the bijective case, the inverse mapping looks like.

Example

A healthy breakfast starts with a fruit salad. The following table shows how much vitamin C, calcium, and magnesium various fruits have (in milligrams with respect to 100 grams of the fruit).

	apple	orange	grapes	banana
vitamin C	${}12$	${}53$	${}4$	${}9$
calcium	${}7$	${}40$	${}12$	${}5$
magnesium	${}6$	${}10$	${}8$	${}27$

This table yields a mapping, which assigns to a ${}4$-tuple ${}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}$, representing the used fruits, the content of the resulting fruit salad with respect to vitamin C, calcium, and magnesium, in the form of a ${}3$-tuple ${}{\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}$. This mapping can be described with the matrix

${\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}$

using matrix multiplication as

${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}\longmapsto {\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}={\begin{pmatrix}12x_{1}+53x_{2}+4x_{3}+9x_{4}\\7x_{1}+40x_{2}+12x_{3}+5x_{4}\\6x_{1}+10x_{2}+8x_{3}+27x_{4}\end{pmatrix}}={\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}.$

Composition of mappings

Definition

Let ${}L,\,M$ and ${}N$ denote sets, let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

and

$G\colon M\longrightarrow N,y\longmapsto G(y),$

be mappings. Then the mapping

$G\circ F\colon L\longrightarrow N,x\longmapsto G(F(x)),$

is called the composition of the mappings

${}F$ and ${}G$.

So we have

${}(G\circ F)(x):=G(F(x))\,,$

where the left-hand side is defined by the right-hand side. If both mappings are given by functional expressions, then the composition is realized by plugging in the first term into the variable of the second term (and to simplify the expression, if possible).

The composition of

$F\colon \mathbb {R} \longrightarrow \mathbb {R} ,t\longmapsto t^{3},$

and

$G\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}-x,$

is given by

${}(G\circ F)(t)=(t^{3})^{2}-t^{3}=t^{6}-t^{3}\,.$

However,

${}(F\circ G)(x)=(x^{2}-x)^{3}=x^{6}-3x^{5}+3x^{4}-x^{3}\,.$

Hence, the composition of two mappings depends on the ordering.

For a bijective mapping ${}\varphi \colon M\rightarrow N$, the inverse mapping ${}\varphi ^{-1}\colon N\rightarrow M$ is characterized by the conditions

${}\varphi \circ \varphi ^{-1}=\operatorname {Id} _{N}\,$

and

${}\varphi ^{-1}\circ \varphi =\operatorname {Id} _{M}\,.$

Lemma

Let ${}L,M,N$ and ${}P$ be sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

$G\colon M\longrightarrow N,y\longmapsto G(y),$

and

$H\colon N\longrightarrow P,z\longmapsto H(z),$

be mappings. Then

${}H\circ (G\circ F)=(H\circ G)\circ F\,$

holds.

Proof

Two mappings ${}\alpha ,\beta \colon L\rightarrow P$ are the same if and only if the equality ${}\alpha (x)=\beta (x)$ holds for every ${}x\in L$. So let ${}x\in L$. Then

${}{\begin{aligned}(H\circ (G\circ F))(x)&=H((G\circ F)(x))\\&=H(G(F(x)))\\&=(H\circ G)(F(x))\\&=((H\circ G)\circ F)(x).\end{aligned}}$ 

$\Box$

Graph, image and preimage of a mapping

Definition

Let ${}L$ and ${}M$ be sets, and let

$F\colon L\longrightarrow M$

be a mapping. Then the set

${}\Gamma =\Gamma _{F}={\left\{(x,F(x))\mid x\in L\right\}}\subseteq L\times M\,$

is called the graph of the mapping ${}F$.

The graph is a concept of set theory. Whether it is possible to "visualize“ it in a picture depends on whether we can visualize the product set ${}L\times M$.

Definition

Let ${}L$ and ${}M$ be sets, and let

$F\colon L\longrightarrow M$

be a mapping. For a subset ${}S\subseteq L$, we call

${}F(S)={\left\{y\in M\mid {\text{there exists an }}x\in S{\text{ such that }}F(x)=y\right\}}\,$

the image of ${}S$ under ${}F$. For ${}S=L$,

${}F(L)=\operatorname {Im} F\,$

is called the image of the mapping.

Definition

Let ${}L$ and ${}M$ be sets, and let

$F\colon L\longrightarrow M$

be a mapping. For a subset ${}T\subseteq M$, we call

${}F^{-1}(T)={\left\{x\in L\mid F(x)\in T\right\}}\,$

the preimage of ${}T$ under ${}F$. For a subset ${}T=\{y\}$ with one element, we call

$F^{-1}(\{y\})$

also the preimage of ${}y$.

For the mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

the image of ${}[1,2]$ is the set of all squares of real numbers between ${}1$ and ${}2$, this is thus ${}[1,4]$. The preimage of ${}[1,4]$ consists of all real numbers whose square is between ${}1$ and ${}4$. This is ${}[-2,-1]\cup [1,2]$.

For two given sets ${}L$ and ${}M$, we denote the set of mappings from ${}L$ to ${}M$ by

${}\operatorname {Map} \,{\left(L,M\right)}={\left\{f:L\rightarrow M\mid f{\text{ mapping}}\right\}}\,.$

Binary operations

The natural addition assigns to two real numbers another real number, its structure is

$+\colon \mathbb {R} \times \mathbb {R} \longrightarrow \mathbb {R} ,(x,y)\longmapsto x+y.$

Such binary operations play an important role in mathematics.

Definition

An operation (or binary operation) ${}\circ$ on a set ${}M$ is a mapping

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto \circ (x,y)=x\circ y.$

A binary operation assigns to a pair

${}(x,y)\in M\times M\,$

another element

${}x\circ y\in M\,.$

Many mathematical constructions are captured by this concept: addition, difference, multiplication, division of numbers, the composition of mappings, the intersection or the union of sets, etc. Basically, any symbol can be used to denote a binary operation, like ${}\circ ,\cdot ,+,-,\oplus ,\clubsuit ,\heartsuit$. Depending on the symbol, we call the binary operation also multiplication or addition, but this does not mean that we are referring to any natural multiplication. Important structural properties of a binary operation are listed in the following definitions.

Definition

A binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

on a set ${}M$ is called commutative if for all ${}x,y\in M$ the equality

${}x\circ y=y\circ x\,$

holds.

Definition

A binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

on a set ${}M$ is called associative if for all ${}x,y,z\in M$ the equality

${}(x\circ y)\circ z=y\circ (x\circ z)\,$

holds.

Definition

Let a set ${}M$ and a binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

be given. An element ${}e\in M$ is called neutral element of the operation if, for all ${}x\in M$, the equalities ${}x\circ e=x=e\circ x$

hold.

In the commutative case, it is enough to check only one property of the neutral element.

Definition

Let a set ${}M$ with a binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

and a neutral element ${}e\in M$ be given. For an element ${}x\in M$, an element ${}y\in M$ is called inverse element (for ${}x$) if the equalities

${}x\circ y=e=y\circ x\,$

hold.

Example

Let ${}L$ be a set, and let

${}M=\operatorname {Map} \,{\left(L,L\right)}\,$

be the set of all mappings from ${}L$ to itself. The composition of mappings gives a binary operation on ${}M$, which is associative, due to Lemma 2.11 . In general, it is not commutative. The identity on ${}L$ is the neutral element. A mapping ${}f\colon L\rightarrow L$ has an inverse element if and only if it is bijective; the inverse element is just the inverse mapping.

Footnotes

<< | Linear algebra (Osnabrück 2024-2025)/Part I | >> PDF-version of this lecture Exercise sheet for this lecture (PDF)