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

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A main focus of mathematics is to study how a certain variable \extrabracket {describing a size or a magnitude} depend on another \extrabracket {or some other} variable \extrabracket {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 grows with time. Such dependencies are expressed with the concept of a mapping.
\inputdefinition { } {
Let \mathkor {L} {and} {M} denote sets. A \definitionword {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 which is assigned to \marelationchain {\relationchain {x } { \in }{L } { }{ } { }{ } { }{ } }{,} is denoted by \mathl{F(x)}{.} For the mapping as a whole, we write \mathdisp {F \colon L \longrightarrow M
If a mapping $F \colon L \rightarrow M$ is given, then $L$ is called the \definitionword {domain} (or domain of definition) of the map and $M$ is called the \definitionword {codomain} (or \keyword {target range}) of the map. For an element \marelationchain {\relationchain {x } { \in }{L } { }{ } { }{ } { }{ } }{,} the element \marelationchaindisplay {\relationchain { F(x) } { \in} { M } { } { } { } { } { } { } } is called the \keyword {value} of $F$ at the \keyword {place} (or \keyword {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 \marelationchain {\relationchain {x } { \in }{L_1 } { = }{L_2 } { }{ } { }{ } } the equality \marelationchain {\relationchain { F(x) } { = }{ G(x) } { }{ } { }{ } { }{ } } in \marelationchain {\relationchain {M_1 } { = }{ M_2 } { }{ } { }{ } { }{ } } holds. So the equality of mappings is reduced to the equalities of elements in a set. Mappings are also called \keyword {functions} {.} However, we will usually reserve the term \keyword {function} for mappings where the codomain is a number set like the real numbers $\R$.
For every set $L$, the mapping \mathdisp {L \longrightarrow L , x \longmapsto x} { , } which sends every element to itself, is called the \keyword {identity} (on $L$). We denote it by $\operatorname{Id}_{ L }$. For another set $M$ and a fixed element \marelationchain {\relationchain {c } { \in }{M } { }{ } { }{ } { }{ } }{,} the mapping \mathdisp {L \longrightarrow M , x \longmapsto c} { , } which sends every element \marelationchain {\relationchain {x } { \in }{L } { }{ } { }{ } { }{ } } to the \keyword {constant value} $c$, is called the \definitionword {constant mapping} \extrabracket {with value $c$} {.} It is usually again denoted by $c$\extrafootnote {Hilbert has said that the art of denotation in mathematics is to use the same symbol for different things} {.} {.}
There are several ways to describe a mapping, like value table, bar chart, pie chart, arrow diagram, the graph of the mapping. In mathematics, a mapping is most often given by a mapping rule, which allows computing the values of the mapping for every argument. Such rules are e.g. \extrabracket {from $\R$ to $\R$} \mathl{x \mapsto x^2}{,} \mathl{x \mapsto x^3- e^x + \sin x}{,} etc. In the sciences and in sociology also \keyword {empirical functions} are important which describe real movements or developments. But also for such functions, one wants to know whether they can be described (approximated) in mathematical manner.
Examples of such mapping rules are \extrabracket {from $\R$ to $\R$} \mathl{x \mapsto x^2}{,} \mathl{x \mapsto x^3- e^x + \sin (x)}{,} etc.
\wertetabellesechsausteilzeilen { $x$ } {\mazeileundfuenf {1} {2} {3} {4} {5} } { {6 } } { $\pi(x)$ } {\mazeileundfuenf {2} {4} {6} {5} {3} } { {1} }
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\tabelleleitsiebenxsieben
\subtitle {Injective and surjective mappings}
\inputdefinition { } {
Let $L$ and $M$ denote sets, and let \mathdisp {F \colon L \longrightarrow M , x \longmapsto F(x)} { , } be a mapping. Then $F$ is called \definitionword {injective}{,} if for two different elements \marelationchain {\relationchain { x,x' } { \in }{ L } { }{ } { }{ } { }{ } }{,} also \mathkor {F(x)} {and} {F(x')}
\inputdefinition { } {
Let $L$ and $M$ denote sets, and let \mathdisp {F \colon L \longrightarrow M , x \longmapsto F(x)} { , } be a mapping. Then $F$ is called \definitionword {surjective}{,} if for every \marelationchain {\relationchain {y } { \in }{M } { }{ } { }{ } { }{ } }{,} there exists at least one element \marelationchain {\relationchain {x } { \in }{L } { }{ } { }{ } { }{ } }{,} such that \marelationchaindisplay {\relationchain {F(x) } { =} {y } { } { } { } { } { } { } }
\inputdefinition { } {
Let $M$ and $L$ denote sets and suppose that \mathdisp {F \colon M \longrightarrow L , x \longmapsto F(x)} { , } is a mapping. Then $F$ is called \definitionword {bijective} if $F$ is injective as well as
This 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 \marelationchaindisplay {\relationchain {F(x) } { =} {y } { } { } { } { } { } { } } (in the two variables $x$ and $y$). The surjectivity means that for every \marelationchain {\relationchain {y } { \in }{ M } { }{ } { }{ } { }{ } } there exists at least one solution \marelationchaindisplay {\relationchain {x } { \in} { L } { } { } { } { } { } { } } for this equation, the injectivity means that for every \marelationchain {\relationchain {y } { \in }{M } { }{ } { }{ } { }{ } } there exist at most one solution \marelationchain {\relationchain {x } { \in }{L } { }{ } { }{ } { }{ } } for this equation, and the bijectivity means that for every \marelationchain {\relationchain {y } { \in }{M } { }{ } { }{ } { }{ } } there exists exactly one solution \marelationchain {\relationchain {x } { \in }{ L } { }{ } { }{ } { }{ } } for this equation. Hence, surjectivity means the existence of solutions, injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics and they also can 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 \mathkor {x} {and} {x'} using the condition \marelationchain {\relationchain { F(x) } { = }{ F(x') } { }{ } { }{ } { } } that \marelationchain {\relationchain {x } { = }{x' } { }{ } { }{ } { } } holds. This method is often easier than to show that \marelationchain {\relationchain {x } { \neq }{x' } { }{ } { }{ } { }{ } } implies \marelationchain {\relationchain {F(x) } { \neq }{F(x') } { }{ } { }{ } { }{ } }{.}
\inputexample {
The mapping \mathdisp {\R \longrightarrow \R , x \longmapsto x^2} { , } is neither injective nor surjective. It is not injective, because the different numbers \mathkor {2} {and} {-2} are both sent to $4$. It is not surjective, because only nonnegative elements are in the image \extrabracket {a negative number does not have a real square root} {.} The mapping \mathdisp {\R_{\geq 0} \longrightarrow \R , x \longmapsto x^2} { , } is injective, but not surjective. The injectivity can be seen as follows: If \marelationchain {\relationchain {x } { \neq }{y } { }{ } { }{ } { }{ } }{,} then one number is larger, say \marelationchaindisplay {\relationchain {x } { >} {y } { \geq} {0 } { } { } { } { } }{.} But then also \marelationchain {\relationchain {x^2 } { > }{ y^2 } { }{ } { }{ } { }{ } }{,} and in particular \marelationchain {\relationchain {x^2 } { \neq }{y^2 } { }{ } { }{ } { }{ } }{.} The mapping \mathdisp {\R \longrightarrow \R_{\geq 0} , x \longmapsto x^2} { , } is not injective, but surjective, since every nonnegative real number has a square root. The mapping \mathdisp {\R_{\geq 0} \longrightarrow \R_{\geq 0} , x \longmapsto x^2} { , } is injective and surjective.
\inputdefinition { } {
Let $F \colon L \rightarrow M$ denote a bijective mapping. Then the mapping \mathdisp {G \colon M \longrightarrow L} { } which sends every element \marelationchain {\relationchain {y } { \in }{M } { }{ } { }{ } { }{ } } to the uniquely determined element \marelationchain {\relationchain {x } { \in }{L } { }{ } { }{ } { }{ } } with \marelationchain {\relationchain {F(x) } { = }{y } { }{ } { }{ } { }{ } }{,}
The inverse mapping is usually denoted by $F^{-1}$.
We discuss two classes of mappings which are in the framework of linear algebra very important. They are both so-called \keyword {linear mappings} {.}
\inputexample {
Let \marelationchain {\relationchain { a } { \in }{ \R } { }{ } { }{ } { }{ } } be fixed. This real number defines a mapping \mathdisp {\R \longrightarrow \R , x \longmapsto ax} { . } For \marelationchain {\relationchain {a } { = }{ 0 } { }{ } { }{ } { }{ } }{,} this is the constant zero mapping. For \marelationchain {\relationchain {a } { \neq }{0 } { }{ } { }{ } { }{ } }{,} we have a bijective mapping, the inverse mapping is \mathdisp {y \longrightarrow { \frac{ 1 }{ a } } y} { . } Here, the inverse mapping has a similar form as the mapping itself.
\inputexample {
Let an $m \times n$-matrix \mathdisp {\begin{pmatrix} a_{11 } & a_{1 2} & \ldots & a_{1 n } \\ a_{21 } & a_{2 2} & \ldots & a_{2 n } \\ \vdots & \vdots & \ddots & \vdots \\ a_{ m 1 } & a_{ m 2 } & \ldots & a_{ m n } \end{pmatrix}} { } be given, where the entries \mathl{a_{ij}} are real numbers. Such a matrix defines a mapping \mathdisp {\varphi \colon \R^n \longrightarrow \R^m} { , } by sending an $n$-tuple \marelationchain {\relationchain { x } { = }{ \begin{pmatrix} x_1 \\x_2\\ \vdots\\x_n \end{pmatrix} } { \in }{ \R^n } { }{ } { }{ } } to the $m$-tuple \marelationchaindisplay {\relationchain { \varphi(x) } { =} { \begin{pmatrix} a_{11 } & a_{1 2} & \ldots & a_{1 n } \\ a_{21 } & a_{2 2} & \ldots & a_{2 n } \\ \vdots & \vdots & \ddots & \vdots \\ a_{ m 1 } & a_{ m 2 } & \ldots & a_{ m n } \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 \marelationchaindisplay {\relationchain { 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.
\inputexample {
A healthy breakfast starts with a fruit salad. The following table shows how much vitamin C, calcium and magnesium various fruits have \extrabracket {in milligram with respect to 100 gram of the fruit} {.}
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\tabelleleitdreixvier This table yields a mapping, which assigns to a $4$-tuple \mathl{\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 \mathl{\begin{pmatrix} y_1 \\y_2\\ y_3 \end{pmatrix}}{.} This mapping can be described with the matrix \mathdisp {\begin{pmatrix} 12 & 53 & 4 & 9 \\ 7 & 40 & 12 & 5 \\ 6 & 10 & 8 & 27 \end{pmatrix}} { }
using matrix multiplication as \mathdisp {\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+9 x_4 \\7x_1 +40x_2 +12x_3+ 5 x_4\\ 6 x_1 + 10 x_2 + 8x_3+ 27 x_4 \end{pmatrix} = \begin{pmatrix} y_1 \\y_2\\ y_3 \end{pmatrix}} { . }
\subtitle {Composition of mappings}
\inputdefinition { } {
Let $L,\, M$ and $N$ denote sets, let \mathdisp {F \colon L \longrightarrow M , x \longmapsto F(x)} { , } and \mathdisp {G \colon M \longrightarrow N , y \longmapsto G(y)} { , } be mappings. Then the mapping \mathdisp {G \circ F \colon L \longrightarrow N , x \longmapsto G(F(x))} { , } is called the \definitionword {composition} of the mappings
So we have \marelationchaindisplay {\relationchain { (G \circ F)(x) } { \defeq} { 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 \extrabracket {and to simplify the expression if possible} {.}
For a bijective mapping $\varphi \colon M \rightarrow N$, the inverse mapping $\varphi^{-1} \colon N \rightarrow M$ is characterized by the conditions \marelationchaindisplay {\relationchain { \varphi \circ \varphi^{-1} } { =} { \operatorname{Id}_{ N } } { } { } { } { } { } { } } and \marelationchaindisplay {\relationchain { \varphi^{-1} \circ \varphi } { =} { \operatorname{Id}_{ M } } { } { } { } { } { } { } }{.}
The composition of \mathdisp {F \colon \R \longrightarrow \R , t \longmapsto t^3} { , } and \mathdisp {G \colon \R \longrightarrow \R , x \longmapsto x^2 -x} { , } is given by \marelationchaindisplay {\relationchain { (G \circ F) (t) } { =} { (t^3)^2 - t^3 } { =} { t^6-t^3 } { } { } { } { } }{.} However, \marelationchaindisplay {\relationchain { (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.
\faktsituation {Let $L, M, N$ and $P$ be sets and let \mathdisp {F \colon L \longrightarrow M , x \longmapsto F(x)} { , } \mathdisp {G \colon M \longrightarrow N , y \longmapsto G(y)} { , } and \mathdisp {H \colon N \longrightarrow P , z \longmapsto H(z)} { , } be mappings.} \faktfolgerung {Then \marelationchaindisplay {\relationchain { H \circ (G \circ F) } { =} { (H \circ G) \circ F } { } { } { } { } { } { } } holds.} \faktzusatz } {
Two mappings $\alpha, \beta \colon L \rightarrow P$ are the same if and only if the equality \marelationchain {\relationchain { \alpha(x) } { = }{\beta(x) } { }{ } { }{ } { }{ } } holds for every \marelationchain {\relationchain {x } { \in }{L } { }{ } { }{ } { }{ } }{.} So let \marelationchain {\relationchain {x } { \in }{L } { }{ } { }{ } { }{ } }{.} Then \marelationchainalign {\relationchainalign { ( 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) } }{.}
\subtitle {Graph, image and preimage of a mapping}
\inputdefinition { } {
Let \mathkor {L} {and} {M} be sets and let \mathdisp {F \colon L \longrightarrow M} { } be a mapping. Then the set \marelationchaindisplay {\relationchain { \Gamma } { =} { \Gamma_F } { =} { { \left\{ (x,F(x)) \mid x \in L \right\} } } { \subseteq} { L \times M } { } { } }
The graph is a concept of set theory. Whether it is possible to \quotationshort{visualize} it in a picture, depends on whether we can visualize the product set \mathl{L \times M}{.}
\inputdefinition { } {
Let \mathkor {L} {and} {M} be sets and let \mathdisp {F \colon L \longrightarrow M} { } be a mapping. For a subset \marelationchain {\relationchain {S } { \subseteq }{L } { }{ } { }{ } { }{ } }{,} we call \marelationchaindisplay {\relationchain { F(S) } { =} { { \left\{ y \in M \mid \text{there exists an } x \in S \text{ such that } F(x)= y \right\} } } { } { } { } { } { } { } } the \definitionword {image of} $S$ under $F$. For \marelationchain {\relationchain {S } { = }{L } { }{ } { }{ } { }{ } }{,} \marelationchaindisplay {\relationchain { F(L) } { =} { \operatorname{Im} F } { } { } { } { } { } { } }
\inputdefinition { } {
Let \mathkor {L} {and} {M} be sets and let \mathdisp {F \colon L \longrightarrow M} { } be a mapping. For a subset \marelationchain {\relationchain {T } { \subseteq }{M } { }{ } { }{ } { }{ } }{,} we call \marelationchaindisplay {\relationchain { F^{-1}(T) } { =} { { \left\{ x \in L \mid F(x) \in T \right\} } } { } { } { } { } { } { } } the \definitionword {preimage of} $T$ under $F$. For an subset \marelationchain {\relationchain {T } { = }{ \{y\} } { }{ } { }{ } { }{ } } with one element, we call \mathdisp {F^{-1}(\{y\})} { }
For the mapping \mathdisp {\R \longrightarrow \R , x \longmapsto x^2} { , } the image of \mathl{[1,2]} is the set of all squares of real numbers between \mathkor {1} {and} {2} {,} this is thus \mathl{[1,4]}{.} The preimage of \mathl{[1,4]} consists of all real numbers, whose square is between \mathkor {1} {and} {4} {.} This is \mathl{[-2,-1] \cup [1,2]}{.}
For two given sets \mathkor {L} {and} {M} {,} we denote the \keyword {set of mappings} from $L$ to $M$ by \marelationchaindisplay {\relationchain { \operatorname{Map} \, { \left( L , M \right) } } { =} { { \left\{ f: L \rightarrow M \mid f \text{ mapping} \right\} } } { } { } { } { } { } { } }{.}
\subtitle {Binary operations}
The natural addition assigns to two real numbers another real number, its structure is \mathdisp {+ \colon \R \times \R \longrightarrow \R , (x,y) \longmapsto x+y} { . } Such binary operations play an important role in mathematics.
\inputdefinition { } {
An \definitionword {operation} \extrabracket {or \definitionword {binary operation}} $\circ$ on a set $M$ is a mapping \mathdisp {\circ \colon M\times M \longrightarrow M
A binary operation assigns to a pair \marelationchaindisplay {\relationchain { (x,y) } { \in} { M \times M } { } { } { } { } { } { } } another element \marelationchaindisplay {\relationchain { 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 \keyword {multiplication} or \keyword {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.
\inputdefinition { } {
A binary operation \mathdisp {\circ \colon M \times M \longrightarrow M , (x,y) \longmapsto x \circ y} { , } on a set $M$ is called \definitionword {commutative}{,} if for all \marelationchain {\relationchain {x,y } { \in }{M } { }{ } { }{ } { }{ } } the equality \marelationchaindisplay {\relationchain { x \circ y } { =} { y \circ x } { } { } { } { } { } { } }
\inputdefinition { } {
A binary operation \mathdisp {\circ \colon M \times M \longrightarrow M , (x,y) \longmapsto x \circ y} { , } on a set $M$ is called \definitionword {associative}{,} if for all \marelationchain {\relationchain {x,y,z } { \in }{M } { }{ } { }{ } { }{ } } the equality \marelationchaindisplay {\relationchain {( x \circ y ) \circ z } { =} { y \circ ( x \circ z) } { } { } { } { } { } { } }
\inputdefinition { } {
Let a set $M$ and a binary operation \mathdisp {\circ \colon M \times M \longrightarrow M , (x,y) \longmapsto x \circ y} { , } be given. An element \marelationchain {\relationchain {e } { \in }{M } { }{ } { }{ } { }{ } } is called \definitionword {neutral element} of the operation, if for all \marelationchain {\relationchain {x } { \in }{M } { }{ } { }{ } { }{ } } the equalities \marelationchain {\relationchain { x \circ e } { = }{ x } { = }{ e \circ x } { }{ } { }{ } }
In the commutative case, it is enough to check only one property of the neutral element.
\inputdefinition { } {
Let a set $M$ with a binary operation \mathdisp {\circ \colon M \times M \longrightarrow M , (x,y) \longmapsto x \circ y} { , } and a neutral element \marelationchain {\relationchain {e } { \in }{M } { }{ } { }{ } { }{ } } be given. For an element \marelationchain {\relationchain {x } { \in }{M } { }{ } { }{ } { }{ } }{,} an element \marelationchain {\relationchain {y } { \in }{M } { }{ } { }{ } { }{ } } is called \definitionword {inverse element} \extrabracket {for $x$} {,} if the equalities \marelationchaindisplay {\relationchain { x \circ y } { =} { e } { =} { y \circ x } { } { } { } { } }
\inputexample {
Let $L$ be a set and let \marelationchaindisplay {\relationchain {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 Fact *****. 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.