# Linear algebra (Osnabrück 2024-2025)/Part I/Lecture 6/latex

\setcounter{section}{6}
\subtitle {Vector spaces}
\image{ \begin{center} \includegraphics[width=5.5cm]{\imageinclude {Vector Addition.svg} } \end{center} \imagetext {The addition of two arrows $a$ and $b$, a typical example for vectors.} } \imagelicense { Vector Addition.svg } {Booyabazooka} {Commons} {PD}
The central concept of linear algebra is a vector space. \inputdefinition { } {
Let $K$ denote a field, and $V$ a set with a distinguished element \mathrelationchain {\relationchain {0 } { \in }{V } { }{ } { }{ } { }{ } }{,} and with two mappings \mathdisp {+ \colon V \times V \longrightarrow V , (u,v) \longmapsto u+v} { , } and \mathdisp {\cdot \colon K \times V \longrightarrow V , (s,v) \longmapsto s v = s \cdot v} { . } Then $V$ is called a \definitionwordpremath {K}{ vector space } \extrabracket {or a vector space over $K$} {,} if the following axioms hold \extrabracket {where \mathcor {r,s \in K} {and} {u,v,w \in V} are arbitrary} {.}
\enumerationeight { \mathrelationchain {\relationchain { u+v } { = }{v+u } { }{ } { }{ } { }{ } }{,} } { \mathrelationchain {\relationchain {(u+v)+w } { = }{ u +(v+w) } { }{ } { }{ } { }{ } }{,} } { \mathrelationchain {\relationchain { v+0 } { = }{v } { }{ } { }{ } { }{ } }{,} } {For every $v$, there exists a $z$ such that \mathrelationchain {\relationchain {v+z } { = }{ 0 } { }{ } { }{ } { }{ } }{,} } { \mathrelationchain {\relationchain {1 \cdot u } { = }{ u } { }{ } { }{ } { }{ } }{,} } { \mathrelationchain {\relationchain { r(su) } { = }{ (rs) u } { }{ } { }{ } { }{ } }{,} } { \mathrelationchain {\relationchain { r(u+v)
} { = }{ru + rv } { }{ } { }{ } { }{ } }{,} } { \mathrelationchain {\relationchain { (r+s) u } { = }{ru + su } { }{ } { }{ } { }{ } }{.}
The binary operation in $V$ is called (vector-)addition, and the operation $K \times V \rightarrow V$ is called \keyword {scalar multiplication} {.} The elements in a vector space are called \keyword {vectors} {,} and the elements \mathrelationchain {\relationchain {r } { \in }{K } { }{ } { }{ } { }{ } } are called \keyword {scalars} {.} The null element \mathrelationchain {\relationchain { 0 } { \in }{V } { }{ } { }{ } { }{ } } is called \keyword {null vector} {,} and for \mathrelationchain {\relationchain {v } { \in }{V } { }{ } { }{ } { }{ } }{,} the inverse element is called the \keyword {negative} of $v$, denoted by $-v$.
The field which occurs in the definition of a vector space is called the \keyword {base field} {.} All the concepts of linear algebra refer to such a base field. In case \mathrelationchain {\relationchain {K } { = }{\R } { }{ } { }{ } { }{ } }{,} we talk about a \keyword {real vector space} {,} and in case \mathrelationchain {\relationchain {K } { = }{ \Complex } { }{ } { }{ } { }{ } }{,} we talk about a \keyword {complex vector space} {.} For real and complex vector spaces there exist further structures like length, angle, inner product. But first we develop the algebraic theory of vector spaces over an arbitrary field.
\image{ \begin{center} \includegraphics[width=5.5cm]{\imageinclude {Vector_space_illust.svg} } \end{center} \imagetext } \imagelicense { Vector space illust.svg } {Oleg Alexandrov} {Commons} {PD}
\inputexample {
Let $K$ denote a field, and let \mathrelationchain {\relationchain { n } { \in }{ \N_+ } { }{ } { }{ } { }{ } }{.} Then the product set \mathrelationchaindisplay {\relationchain { K^n } { =} { \underbrace{K \times \cdots \times K }_{n\text{-times} } } { =} { { \left\{ (x_1 , \ldots , x_{ n }) \mid x_i \in K \right\} } } { } { } }{,} with componentwise addition and with scalar multiplication given by \mathrelationchaindisplay {\relationchain { s (x_1 , \ldots , x_{ n }) } { =} { (s x_1 , \ldots , s x_{ n }) } { } { } { } { } { } { } }{,} is a vector space. This space is called the $n$-dimensional \keyword {standard space} {.} In particular, \mathrelationchain {\relationchain {K^1 } { = }{K } { }{ } { }{ } { }{ } } is a vector space.
The null space $0$, consisting of just one element $0$, is a vector space. It might be considered as \mathrelationchain {\relationchain {K^0 } { = }{0 } { }{ } { }{ } { }{ } }{.}
The vectors in the standard space $K^n$ can be written as row vectors \mathdisp {\left( a_1 , \, a_2 , \, \ldots , \, a_n \right)} { } or as column vectors \mathdisp {\begin{pmatrix} a_1 \\a_2\\ \vdots\\a_n \end{pmatrix}} { . } The vector \mathrelationchaindisplay {\relationchain { e_i } { \defeq} { \begin{pmatrix} 0 \\ \vdots\\ 0\\1\\ 0\\ \vdots\\ 0 \end{pmatrix} } { } { } { } { } { } { } }{,} where the $1$ is at the $i$-th position, is called $i$-th \keyword {standard vector} {.} \inputexample {
Let $E$ be a \quotationshort{plane} with a fixed \quotationshort{origin} \mathrelationchain {\relationchain { Q } { \in }{ E } { }{ } { }{ } { }{ } }{.} We identify a point \mathrelationchain {\relationchain { P } { \in }{E } { }{ } { }{ } { }{ } } with the connecting vector \mathl{\overrightarrow{ Q P }} \extrabracket {the arrow from $Q$ to $P$} In this situation, we can introduce an intuitive coordinate-free vector addition and a coordinate-free scalar multiplication. Two vectors \mathcor {\overrightarrow{ Q P }} {and} {\overrightarrow{ Q R }} are added together by constructing the parallelogram of these vectors. The result of the addition is the corner of the parallelogram which lies in opposition to $Q$. In this construction, we have to draw a line parallel to \mathl{\overrightarrow{ Q P }} through $R$ and a line parallel to \mathl{\overrightarrow{ Q R }} through $P$. The intersection point is the point sought after. An accompanying idea is that we move the vector \mathl{\overrightarrow{ Q P }} in a parallel way so that the starting point of \mathl{\overrightarrow{ Q P }} becomes the ending point of \mathl{\overrightarrow{ Q R }}{.}
In order to describe the multiplication of a vector \mathl{\overrightarrow{ Q P }} with a scalar $s$, this number has to be given on a line $G$, which is also marked with a zero point \mathrelationchain {\relationchain { 0 } { \in }{ G } { }{ } { }{ } { }{ } } and a one point \mathrelationchain {\relationchain { 1 } { \in }{ G } { }{ } { }{ } { }{ } }{.} This line lies somehow in the plane. We move this line
\extrabracket {by translating and rotating} in such a way that $0$ becomes $Q$, and we avoid that the line is identical with the line given by \mathl{\overrightarrow{ Q P }} \extrabracket {which we call $H$} {.}
Now we connect $1$ and $P$ with a line $L$, and we draw the line $L'$ parallel to $L$ through $s$. The intersecting point of \mathcor {L'} {and} {H} is \mathl{s \overrightarrow{ Q P }}{.}
These considerations can also be done in higher dimensions, but everything takes place already in the plane spanned by these vectors.
\inputexample{ } {
The complex numbers $\Complex$ form a field, and therefore they form also a vector space over the field itself. However, the set of complex numbers equals $\R^2$ as an additive group. The multiplication of a complex number \mathl{a+b { \mathrm i}} with a real number \mathrelationchain {\relationchain { s } { = }{ (s,0) } { }{ } { }{ } { } } is componentwise, so this multiplication coincides with the scalar multiplication on $\R^2$. Hence, the complex numbers are also a real vector space.
\inputexample {
For a field $K$, and given natural numbers \mathl{m,n}{,} the set \mathdisp {\operatorname{Mat}_{ m \times n } (K)} { } of all \mathl{m \times n}{-}matrices with componentwise addition and componentwise scalar multiplication, is a $K$-vector space. The null element in this vector space is the \keyword {null matrix} \mathrelationchaindisplay {\relationchain {0 } { =} { \begin{pmatrix} 0 & \ldots & 0 \\ \vdots & \ddots & \vdots \\0 & \ldots & 0 \end{pmatrix} } { } { } { } { } { } { } }{.}
We will introduce polynomials later, they are probably known from school. \inputexample {
Let \mathrelationchain {\relationchain { R } { = }{ K[X] } { }{ } { }{ } { }{ } } be the polynomial ring in one variable over the field $K$, consisting of all polynomials, that is expressions of the form \mathdisp {a_nX^n+a_{n-1}X^{n-1} + \cdots + a_2X^2+a_1X+a_0} { , } with \mathrelationchain {\relationchain { a_i } { \in }{ K } { }{ } { }{ } { }{ } }{.} Using componentwise addition and componentwise multiplication with a scalar \mathrelationchain {\relationchain {s
} { \in }{ K } { }{ } { }{ } { }{ } } \extrabracket {this is also multiplication with the constant polynomial $s$} {,} the polynomial ring is a $K$-vector space.
\inputexample {
We consider the inclusion \mathrelationchain {\relationchain { \Q } { \subseteq }{ \R } { }{ } { }{ } { }{ } } of the rational numbers inside the real numbers. Using the real addition and the multiplication of rational numbers with real numbers, we see that $\R$ is a vector space, as follows directly from the field axioms. This is a quite crazy vector space.
\factsituation {Let $K$ be a field, and let $V$ be a $K$-vector space.} \factsegue {Then the following properties hold \extrabracket {for \mathcor {v \in V} {and} {s \in K}} {.}} \factconclusion {\enumerationfour {We have \mathrelationchain {\relationchain { 0v } { = }{ 0 } { }{ } { }{ } { }{ } }{.}
} {We have \mathrelationchain {\relationchain { s 0 } { = }{ 0 } { }{ } { }{ } { }{ } }{.} } {We have \mathrelationchain {\relationchain { (-1) v } { = }{ -v } { }{ } { }{ } { }{ } }{.} } {If \mathrelationchain {\relationchain { s } { \neq }{ 0 } { }{ } { }{ } { }{ } } and \mathrelationchain {\relationchain { v } { \neq }{ 0 } { }{ } { }{ } { }{ } }{,} then \mathrelationchain {\relationchain { s v } { \neq }{ 0 } { }{ } { }{ } { }{ } }{.} }} \factextra
\subtitle {Linear subspaces}
\inputdefinition { } {
Let $K$ be a field, and $V$ be a $K$-vector space. A subset \mathrelationchain {\relationchain { U } { \subseteq }{ V } { }{ } { }{ } { }{ } } is called a \definitionword {linear subspace}{,} if the following properties hold. \enumerationthree { \mathrelationchain {\relationchain { 0 } { \in }{ U } { }{ } { }{ } { }{ } }{.} } {If \mathrelationchain {\relationchain { u,v } { \in }{U } { }{ } { }{ } { }{ } }{,} then also \mathrelationchain {\relationchain { u+v } { \in }{U } { }{ } { }{ } { }{ } }{.} } {If \mathrelationchain {\relationchain { u } { \in }{ U } { }{ } { }{ } { }{ } } and \mathrelationchain {\relationchain { s } { \in }{ K } { }{ } { }{ } { }{ } }{,} then also \mathrelationchain {\relationchain { s u } { \in }{ U } { }{ } { }{ } { }{ } } holds.
Addition and scalar multiplication can be restricted to such a linear subspace. Hence, the linear subspace is itself a vector space, see Exercise 6.10 . The simplest linear subspaces in a vector space $V$ are the null space $0$ and the whole vector space $V$.
\factsituation {Let $K$ be a field, and let \mathdisp {\begin{matrix} a _{ 1 1 } x _1 + a _{ 1 2 } x _2 + \cdots + a _{ 1 n } x _{ n } & = & 0 \\ a _{ 2 1 } x _1 + a _{ 2 2 } x _2 + \cdots + a _{ 2 n } x _{ n } & = & 0 \\ \vdots & \vdots & \vdots \\ a _{ m 1 } x _1 + a _{ m 2 } x _2 + \cdots + a _{ m n } x _{ n } & = & 0 \end{matrix}} { } be a homogeneous system of linear equations over $K$.} \factconclusion {Then the set of all solutions to the system is a linear subspace of the standard space $K^n$.} \factextra
Therefore, we talk about the \keyword {solution space} of the linear system. In particular, the sum of two solutions of a system of linear equations is again a solution. The solution set of an inhomogeneous linear system is not a vector space. However, one can add, to a solution of an inhomogeneous system, a solution of the corresponding homogeneous system, and get a solution to the inhomogeneous system again.
\inputexample {
We have a look at the homogeneous version of Example 5.1 , so we consider the homogeneous linear system \mathdisp {\begin{matrix} 2x & +5y & +2z & & -v & = & 0 \\ \, 3x & -4y & & +u & +2v & = & 0 \\ \, 4x & & -2z & +2u & & = & 0 \, . \end{matrix}} { } over $\R$. Due to Lemma 6.11 , the solution set $L$ is a linear subspace of $\R^5$. We have described it explicitly in Example 5.1
as \mathdisp {{ \left\{ u { \left(- { \frac{ 1 }{ 3 } }, 0 , { \frac{ 1 }{ 3 } } ,1,0\right) } + v { \left(- { \frac{ 2 }{ 13 } }, { \frac{ 5 }{ 13 } }, -{ \frac{ 4 }{ 13 } },0,1\right) } \mid u,v \in \R \right\} }} { , } which also shows that the solution set is a vector space. With this description, it is clear that $L$ is in bijection with $\R^2$, and this bijection respects the addition and also the scalar multiplication \extrabracket {the solution set $L'$ of the inhomogeneous system is also in bijection with $\R^2$, but there is no reasonable addition nor scalar multiplication on $L'$} {.} However, this bijection depends heavily on the chosen \quotationshort{basic solutions} \mathcor {{ \left(- { \frac{ 1 }{ 3 } }, 0 , { \frac{ 1 }{ 3 } } ,1,0\right) }} {and} {{ \left(- { \frac{ 2 }{ 13 } }, { \frac{ 5 }{ 13 } }, -{ \frac{ 4 }{ 13 } },0,1\right) }} {,} which depends on the order of elimination. There are several equally good basic solutions for $L$.
This example shows also the following: the solution space of a linear system over $K$ is \quotationshort{in natural way}{,} that means independent on any choice, a linear subspace of $K^n$ \extrabracket {where $n$ is the number of variables} {.} For this solution space, there always exists a \quotationshort{linear bijection} (an \quotationshort{isomorphism}) to some \mathl{K^{d}} \extrabracket {\mathrelationchainb {\relationchainb {d } { \leq }{n } { }{ } { }{ } { }{ } }} {,} but for is no natural choice for such a bijection. This is one of the main reasons to work with abstract vector spaces, instead of just $K^n$.
\subtitle {Generating systems}
The solution set of a homogeneous linear system in $n$ variables over a field $K$ is a linear subspace of $K^n$. This solution space is often described as the set of all \quotationshort{linear combinations} of finitely many
\extrabracket {simple} solutions. In this and the next lecture, we will develop the concepts to make this precise.
\image{ \begin{center} \includegraphics[width=5.5cm]{\imageinclude {VectorGenerado.gif} } \end{center} \imagetext {The plane generated by two vectors $v_1$ and $v_2$ consists of all linear combinations \mathrelationchain {\relationchain { u } { = }{ xv_1+yv_2 } { }{ } { }{ } { }{ } }{.}} } \imagelicense { VectorGenerado.gif } {Marianov} {Commons} {PD}
\inputdefinition { } {
Let $K$ be a field, and let $V$ be a $K$-vector space. Let \mathl{v_1 , \ldots , v_n} denote a family of vectors in $V$. Then the vector \mathdisp {s_1v_1+s_2v_2 + \cdots + s_nv_n \text{ with } s_i \in K} { } is called a \definitionword {linear combination} of this vectors
Two different coefficient tuples can define the same vector.
\inputdefinition { } {
Let $K$ be a field, and let $V$ be a $K$-vector space. A family \mathcond {v_i \in V} {i \in I} {,} is called a \definitionword {generating system} \extrabracket {or \definitionword {spanning system}} of $V$, if every vector \mathrelationchain {\relationchain {v } { \in }{V } { }{ } { }{ } { }{ } } can be written as \mathrelationchaindisplay {\relationchain {v } { =} {\sum_{j \in J} s_j v_j } { } { } { } { } { } { } }{,} with a finite subfamily \mathrelationchain {\relationchain {J } { \subseteq }{I } { }{ } { }{ } { }{ } }{,} and with \mathrelationchain {\relationchain { s_j } { \in }{ K } { }{ } { }{ } { }{ } }
In $K^n$, the standard vectors \mathcond {e_i} {1 \leq i \leq n} {,} form a generating system. In the polynomial ring \mathl{K[X]}{,} the powers \mathcond {X^n} {n \in \N} {,} form an \extrabracket {infinite} generating system.
\inputdefinition { } {
Let $K$ be a field, and let $V$ be a $K$-vector space. For a family \mathcond {v_i} {i \in I} {,} we set \mathrelationchaindisplay {\relationchain { \langle v_i ,\, i \in I \rangle } { =} { { \left\{ \sum_{i \in J} s_i v_i \mid s_i \in K , \, J \subseteq I \text{ finite subset} \right\} } } { } { } { } { } { } { } }{,}
The empty set generates the null space\extrafootnote {This follows from the definition, if we use the convention that the empty sum equals $0$} {.} {.} The null space is also generated by the element $0$. A single vector $v$ spans the space \mathrelationchain {\relationchain { Kv } { = }{ { \left\{ s v \mid s \in K \right\} } } { }{ } { }{ } { }{ } }{.} For \mathrelationchain {\relationchain {v } { \neq }{ 0 } { }{ } { }{ } { }{ } }{,} this is a \keyword {line} {,} a term we will make more precise in the framework of dimension theory. For two vectors \mathcor {v} {and} {w} {,} the \quotationshort{form} of the spanned space depends on how the two vectors are related to each other. If they both lie on a line, say \mathrelationchain {\relationchain {w } { = }{ s v } { }{ } { }{ } { }{ } }{,} then $w$ is superfluous, and the linear subspace generated by the two vectors equals the linear subspace generated by $v$. If this is not the case \extrabracket {and \mathcor {v} {and} {w} are not $0$} {,} then the two vectors span a \quotationshort{plane}{.}
We list some simple properties for generating systems and linear subspaces.
\factsituation {Let $K$ be a field and let $V$ be a $K$-vector space.} \factsegue {Then the following hold.} \factconclusion {\enumerationthree {Let \mathcond {U_j} {j \in J} {,} be a family of subspaces of $V$. Then the intersection \mathrelationchaindisplay {\relationchain { U } { =} { \bigcap_{j \in J} U_j } { } { } { } { } { } { } } is a linear subspace. } {Let \mathcond {v_i} {i \in I} {,} be a family of elements of $V$ and consider the subset $W$ of $V$ which is given by all linear combinations of these elements. Then $W$ is a linear subspace of $V$. } {The family \mathcond {v_i} {i \in I} {,} is a system of generators of $V$ if and only if \mathrelationchaindisplay {\relationchain { \langle v_i ,\, i\in I \rangle } { =} { V } { } { } { } { } { } { } }{.} }} \factextra