# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 24/latex

\setcounter{section}{24}
\zwischenueberschrift{Base change}
We know, due to Theorem 23.15 , that in a finite-dimensional vector space, any two bases have the same length, the same number of vectors. Every vector has, with respect to every basis, unique coordinates \zusatzklammer {the coefficient tuple} {.} How do these coordinates behave when we change the bases? This is answered by the following statement.
\faktsituation {Let $K$ be a field, and let $V$ be a $K$-vector space of dimension $n$. Let \mathkor {\mathfrak{ v } = v_1 , \ldots , v_n} {and} {\mathfrak{ w } = w_1 , \ldots , w_n} denote bases of $V$.} \faktvoraussetzung {Suppose that \mavergleichskettedisp {\vergleichskette {v_j } { =} { \sum_{ i = 1 }^{ n } c_{ij} w_i } { } { } { } { } { } { } } with coefficients \mavergleichskette {\vergleichskette { c_{ij} } { \in }{ K } { }{ } { }{ } { }{ } }{,} which we collect into the $n \times n$-matrix \mavergleichskettedisp {\vergleichskette { M^{ \mathfrak{ v } }_{ \mathfrak{ w } } } { =} { { \left( c_{ij} \right) }_{ij} } { } { } { } { } { } { } }{.}} \faktfolgerung {Then a vector $u$, which has the coordinates $\begin{pmatrix} s_{1 } \\ \vdots\\ s_{ n } \end{pmatrix}$ with respect to the basis $\mathfrak{ v }$, has the coordinates \mavergleichskettedisp {\vergleichskette {\begin{pmatrix} t _{1 } \\ \vdots\\ t _{ n } \end{pmatrix} } { =} { M^{ \mathfrak{ v } }_{ \mathfrak{ w } } \begin{pmatrix} s_{1 } \\ \vdots\\ s_{ n } \end{pmatrix} } { =} { \begin{pmatrix} c_{11 } & c_{1 2} & \ldots & c_{1 n } \\ c_{21 } & c_{2 2} & \ldots & c_{2 n } \\ \vdots & \vdots & \ddots & \vdots \\ c_{ n 1 } & c_{ n 2 } & \ldots & c_{ n n } \end{pmatrix} \begin{pmatrix} s_{1 } \\ \vdots\\ s_{ n } \end{pmatrix} } { } { } { } { } } with respect to the basis $\mathfrak{ w }$.} \faktzusatz } {
This follows directly from \mavergleichskettedisp {\vergleichskette { u } { =} { \sum_{ j = 1 }^{ n } s_j v_j } { =} { \sum_{ j = 1 }^{ n } s_j { \left( \sum_{ i = 1 }^{ n } c_{ij} w_i \right) } } { =} { \sum_{ i = 1 }^{ n } { \left( \sum_{ j = 1 }^{ n } s_j c_{ij} \right) } w_i } { } { } }{,} and the definition of matrix multiplication.
The matrix \mathl{M^{ \mathfrak{ v } }_{ \mathfrak{ w } }}{,} which describes the base change from $\mathfrak{ v }$ to $\mathfrak{ w }$, is called the \stichwort {transformation matrix} {.} In the $j$-th column of the transformation matrix, there are the coordinates of $v_j$ with respect to the basis $\mathfrak{ w }$. When we denote, for a vector \mavergleichskette {\vergleichskette {u } { \in }{V } { }{ } { }{ } { }{ } } and a basis $\mathfrak{ v }$, the corresponding coordinate tuple by $\Psi_{ \mathfrak{ v } } (u)$, then the transformation can be quickly written as \mavergleichskettedisp {\vergleichskette { \Psi_{ \mathfrak{ w } } (u) } { =} { M^{ \mathfrak{ v } }_{ \mathfrak{ w } } ( \Psi_{ \mathfrak{ v } } (u)) } { } { } { } { } { } { } }{.}
\inputbeispiel {
We consider in $\R^2$ the standard basis, \mavergleichskettedisp {\vergleichskette { \mathfrak{ u } } { =} { \begin{pmatrix} 1 \\0 \end{pmatrix} , \, \begin{pmatrix} 0 \\1 \end{pmatrix} } { } { } { } { } { } { } }{,} and the basis \mavergleichskettedisp {\vergleichskette { \mathfrak{ v } } { =} { \begin{pmatrix} 1 \\2 \end{pmatrix} , \, \begin{pmatrix} -2 \\3 \end{pmatrix} } { } { } { } { } { } { } }{.} The basis vectors of $\mathfrak{ v }$ can be expressed directly with the standard basis, namely \mathdisp {v_1= \begin{pmatrix} 1 \\2 \end{pmatrix} = 1 \begin{pmatrix} 1 \\0 \end{pmatrix} + 2 \begin{pmatrix} 0 \\1 \end{pmatrix} \text{ and } v_2= \begin{pmatrix} -2 \\3 \end{pmatrix} = -2 \begin{pmatrix} 1 \\0 \end{pmatrix} + 3 \begin{pmatrix} 0 \\1 \end{pmatrix}} { . } Therefore, we get immediately \mavergleichskettedisp {\vergleichskette { M^{ \mathfrak{ v } }_{ \mathfrak{ u } } } { =} { \begin{pmatrix} 1 & -2 \\ 2 & 3 \end{pmatrix} } { } { } { } { } { } { } }{.} For example, the vector which has with respect to $\mathfrak{ v }$ the coordinates \mathl{(4,-3)}{,} has the coordinates \mavergleichskettedisp {\vergleichskette { M^{ \mathfrak{ v } }_{ \mathfrak{ u } } \begin{pmatrix} 4 \\-3 \end{pmatrix} } { =} { \begin{pmatrix} 1 & -2 \\ 2 & 3 \end{pmatrix} \begin{pmatrix} 4 \\-3 \end{pmatrix} } { =} { \begin{pmatrix} 10 \\-1 \end{pmatrix} } { } { } { } { } } with respect to the standard basis $\mathfrak{ u }$. The transformation matrix \mathl{M^{ \mathfrak{ u } }_{ \mathfrak{ v } }} is more difficult to compute: We have to write the standard vectors as linear combinations of \mathkor {v_1} {and} {v_2} {.} A direct computation \zusatzklammer {solving two linear systems} yields \mavergleichskettedisp {\vergleichskette { \begin{pmatrix} 1 \\0 \end{pmatrix} } { =} { { \frac{ 3 }{ 7 } } \begin{pmatrix} 1 \\2 \end{pmatrix} - { \frac{ 2 }{ 7 } } \begin{pmatrix} -2 \\3 \end{pmatrix} } { } { } { } { } { } { } } and \mavergleichskettedisp {\vergleichskette { \begin{pmatrix} 0 \\1 \end{pmatrix} } { =} { { \frac{ 2 }{ 7 } } \begin{pmatrix} 1 \\2 \end{pmatrix} + { \frac{ 1 }{ 7 } } \begin{pmatrix} -2 \\3 \end{pmatrix} } { } { } { } { } { } { } }{.} Hence, \mavergleichskettedisp {\vergleichskette { M^{ \mathfrak{ u } }_{ \mathfrak{ v } } } { =} { \begin{pmatrix} { \frac{ 3 }{ 7 } } & { \frac{ 2 }{ 7 } } \\ - { \frac{ 2 }{ 7 } } & { \frac{ 1 }{ 7 } } \end{pmatrix} } { } { } { } { } { } { } }{.}
\zwischenueberschrift{Linear mappings}
\inputdefinition { } {
Let $K$ be a field, and let \mathkor {V} {and} {W} be $K$-vector spaces. A mapping \mathdisp {\varphi \colon V \longrightarrow W} { } is called a \definitionswort {linear mapping}{,} if the following two properties are fulfilled. \aufzaehlungzwei { \mavergleichskette {\vergleichskette { \varphi(u+v) } { = }{ \varphi(u) + \varphi(v) } { }{ } { }{ } { }{ } } for all \mavergleichskette {\vergleichskette { u,v } { \in }{V } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \varphi( s v) } { = }{ s \varphi(v) } { }{ } { }{ } { }{ } } for all \mathkor {s \in K} {and} {v \in V} {.}
Here, the first property is called \stichwort {additivity} and the second property is called \stichwort {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 \mavergleichskette {\vergleichskette {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, \mavergleichskettedisp {\vergleichskette { \varphi { \left( \sum_{i = 1}^n s_iv_i \right) } } { =} { \sum_{i = 1}^n s_i \varphi(v_i) } { } { } { } { } { } { } }{,} see exercise *****.
\inputbeispiel {
Let $K$ denote a field, and let $K^n$ be the $n$-dimensional standard space. Then the $i$-th \stichwort {projection} {,} this is the mapping \mathdisp {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 \stichwort {coordinate function} {.}
\faktsituation {Let $K$ denote a field, and let \mathl{U,V,W} denote vector spaces over $K$. Suppose that \mathdisp {\varphi : U \longrightarrow V \text{ and } \psi : V \longrightarrow W} { } are linear mappings.} \faktfolgerung {Then also the composition \mathdisp {\psi \circ \varphi \colon U \longrightarrow W} { } is a linear mapping.} \faktzusatz
\faktsituation {Let $K$ be a field, and let \mathkor {V} {and} {W} be $K$-vector spaces. Let \mathdisp {\varphi \colon V \longrightarrow W} { } be a bijective linear map.} \faktfolgerung {Then also the inverse mapping \mathdisp {\varphi^{-1} \colon W \longrightarrow V} { } is linear.} \faktzusatz
\zwischenueberschrift{Determination on a basis}
Behind the following statement \zusatzklammer {the \stichwort {determination theorem}} {,} there is the important principle that in linear algebra \zusatzklammer {of finite-dimensional vector spaces} {,} the objects are determined by finitely many data.
\faktsituation {Let $K$ be a field, and let \mathkor {V} {and} {W} be $K$-vector spaces. Let \mathbed {v_i} {i \in I} {,} denote a basis of $V$, and let \mathbed {w_i} {i \in I} {,} denote elements in $W$.} \faktfolgerung {Then there exists a unique linear mapping \mathdisp {f \colon V \longrightarrow W} { } with \mathdisp {f(v_i)= w_i \text { for all } i \in I} { . } } \faktzusatz } {
Since we want \mavergleichskette {\vergleichskette {f(v_i) } { = }{ w_i } { }{ } { }{ } { }{ } } and since a linear mapping respects all linear combinations, that is
\mavergleichskettedisp {\vergleichskette { f { \left( \sum_{i \in I} s_i v_i \right) } } { =} { \sum_{i \in I} s_i f { \left( v_i \right) }
} { } { } { } { } { } { } } holds, and since every vector \mavergleichskette {\vergleichskette { v } { \in }{ V } { }{ } { }{ } { }{ } } is such a linear combination, there can at most exist one such a linear mapping. We define now a mapping \mathdisp {f \colon V \longrightarrow W} { } in the following way: we write every vector \mavergleichskette {\vergleichskette { v } { \in }{ V } { }{ } { }{ } { }{ } } with the given basis as \mavergleichskettedisp {\vergleichskette {v } { =} {\sum_{i \in I} s_i v_i } { } { } { } { } { } { } } \zusatzklammer {where \mavergleichskette {\vergleichskette { s_i } { = }{ 0 } { }{ } { }{ } { }{ } } for almost all \mavergleichskette {\vergleichskette { i } { \in }{ I } { }{ } { }{ } { }{ } }} and define \mavergleichskettedisp {\vergleichskette { f(v) } { \defeq} { \sum_{i \in I} s_i w_i } { } { } { } { } { } { } }{.} Since the representation of $v$ as such a linear combination is unique, this mapping is well-defined. Linearity. For two vectors \mathkor {u= \sum_{i \in I} s_iv_i} {and} {v= \sum_{i \in I} t_iv_i} {,} we have \mavergleichskettealign {\vergleichskettealign { f { \left( u+v \right) } } { =} { f { \left( { \left( \sum_{i \in I} s_iv_i \right) } + { \left( \sum_{i \in I} t_iv_i \right) } \right) }
} { =} { f { \left( \sum_{i \in I} { \left( s_i + t_i \right) } v_i \right) } } { =} { \sum_{i \in I} (s_i + t_i) f { \left( v_i \right) } } { =} { \sum_{i \in I} s_i f { \left( v_i \right) } + \sum_{i \in I} t_i f(v_i) } } { \vergleichskettefortsetzungalign { =} { f { \left( \sum_{i \in I} s_iv_i \right) } + f { \left( \sum_{i \in I} t_iv_i \right) }
} { =} { f(u) +f(v) } { } { } }{.} The compatibility with scalar multiplication is shown in a similar way, see exercise *****.
\inputbeispiel {
The easiest
linear mappings are \zusatzklammer {beside the null mapping} the linear maps from $K$ to $K$. Such a linear mapping \mathdisp {\varphi \colon K \longrightarrow K , x \longmapsto \varphi(x)} { , } is determined \zusatzklammer {by Theorem 24.7 , but this is also directly clear} by \mathl{\varphi(1)}{,} or by the value \mathl{\varphi(t)} for an single element \mathbed {t \in K} {t \neq 0} {.} In particular, \mavergleichskette {\vergleichskette { \varphi(x) } { = }{ ax } { }{ } { }{ } { }{ } }{,} with a uniquely determined \mavergleichskette {\vergleichskette {a } { \in }{K } { }{ } { }{ } { }{ } }{.} In the context of physics, for \mavergleichskette {\vergleichskette {K } { = }{\R } { }{ } { }{ } { }{ } }{,} and if there is a linear relation between two measurable quantities, we talk about \stichwort {proportionality} {,} and $a$ is called the \stichwort {proportionality factor} {.} In school, such a linear relation occurs as \anfuehrung{rule of three}{.}
\zwischenueberschrift{Linear mappings and matrices}
Due to Theorem 24.7 , a linear mapping \mathdisp {\varphi \colon K^n \longrightarrow K^m} { } is determined by the images \mathbed {\varphi(e_j)} {j = 1 , \ldots , n} {,} of the standard vectors. Every \mathl{\varphi(e_j)} is a linear combination \mavergleichskettedisp {\vergleichskette { \varphi(e_j) } { =} { \sum_{i = 1}^m a_{ij} e_i } { } { } { } { } { } { } }{,} and therefore the linear mapping is determined by the elements \mathl{a_{ij}}{.} So, such a linear map is determined by the $mn$ elements \mathbed {a_{ij}} {1 \leq i \leq m} {1 \leq j \leq n} {,} from the field. We can write such a data set as a matrix. Because of the determination theorem, this holds for linear maps in general, as soon as in both vector spaces bases are fixed.
\inputdefinition { } {
Let $K$ denote a field, and let $V$ be an $n$-dimensional vector space with a basis \mavergleichskette {\vergleichskette { \mathfrak{ v } } { = }{ v_1 , \ldots , v_n } { }{ } { }{ } { }{ } }{,} and let $W$ be an $m$-dimensional vector space with a basis \mavergleichskette {\vergleichskette { \mathfrak{ w } } { = }{ w_1 , \ldots , w_m } { }{ } { }{ } { }{ } }{.}
For a linear mapping \mathdisp {\varphi \colon V \longrightarrow W} { , } the matrix \mavergleichskettedisp {\vergleichskette { M } { =} { M^{ \mathfrak{ v } }_{ \mathfrak{ w } } ( \varphi) } { =} { (a_{ij})_{ij} } { } { } { } { } }{,} where \mathl{a_{ij}} is the $i$-th coordinate of \mathl{\varphi(v_j )} with respect to the basis $\mathfrak{ w }$, is called the \definitionswort {describing matrix for} $\varphi$ with respect to the bases.
For a matrix \mavergleichskette {\vergleichskette {M } { = }{ (a_{ij})_{ij} } { \in }{ \operatorname{Mat}_{ m \times n } (K) } { }{ } { }{ } }{,} the linear mapping \mathl{\varphi^{ \mathfrak{ v } }_{ \mathfrak{ w } } (M)} determined by \mathdisp {v_j \longmapsto \sum_{ i = 1 }^{ m } a_{ij} w_i} { } in the sense of Theorem 24.7 ,
For a linear mapping $\varphi \colon K^n \rightarrow K^m$, we always assume that everything is with respect to the standard bases, unless otherwise stated. For a linear mapping $\varphi \colon V \rightarrow V$ from a vector space in itself \zusatzklammer {what is called an \stichwort {endomorphism}} {,} one usually takes the same bases on both sides. The identity on a vector space of dimension $n$ is described by the identity matrix, with respect to every basis.
\faktsituation {Let $K$ be a field, and let $V$ be an $n$-dimensional vector space with a basis \mavergleichskette {\vergleichskette { \mathfrak{ v } } { = }{ v_1 , \ldots , v_n
} { }{ } { }{ } { }{ } }{,} and let $W$ be an $m$-dimensional vector space with a basis \mavergleichskette {\vergleichskette { \mathfrak{ w }
} { = }{ w_1 , \ldots , w_m } { }{ } { }{ } { }{ } }{.}} \faktfolgerung {Then the mappings \mathdisp {\varphi \longmapsto M^{ \mathfrak{ v } }_{ \mathfrak{ w } } ( \varphi) \text{ and } M \longmapsto \varphi^{ \mathfrak{ v } }_{ \mathfrak{ w } } (M)} { , } defined in definition, are inverse to each other.} \faktzusatz } {
We show that both compositions are the identity. We start with a matrix \mavergleichskette {\vergleichskette { M } { = }{ { \left( a_{ij} \right) }_{ij} } { }{ } { }{ } { }{ } } and consider the matrix \mathdisp {M^{ \mathfrak{ v } }_{ \mathfrak{ w } }( \varphi^{ \mathfrak{ v } }_{ \mathfrak{ w } }(M) )} { . } Two matrices are equal, when the entries coincide for every index pair \mathl{(i,j)}{.} We have \mavergleichskettealign {\vergleichskettealign {(M^{ \mathfrak{ v } }_{ \mathfrak{ w } }( \varphi^{ \mathfrak{ v } }_{ \mathfrak{ w } }(M) ))_{ij} } { =} { i-\text{th coordinate of } ( \varphi^{ \mathfrak{ v } }_{ \mathfrak{ w } }(M)) (v_j) } { =} { i-\text{th coordinate of } \sum_{ i = 1 }^{ m } a_{ij} w_i } { =} { a_{ij} } { } { } }{.} Now, let $\varphi$ be a linear mapping, we consider \mathdisp {\varphi^{ \mathfrak{ v } }_{ \mathfrak{ w } }( M^{ \mathfrak{ v } }_{ \mathfrak{ w } }(\varphi) )} { . }
Two linear mappings coincide, due to Theorem 24.7 , when they have the same values on the basis \mathl{v_1 , \ldots , v_n}{.} We have \mavergleichskettedisp {\vergleichskette { (\varphi^{ \mathfrak{ v } }_{ \mathfrak{ w } }( M^{ \mathfrak{ v } }_{ \mathfrak{ w } }(\varphi) ))(v_j) } { =} { \sum_{ i = 1 }^{ m } (M^{ \mathfrak{ v } }_{ \mathfrak{ w } } (\varphi))_{ij} \, w_i } { } { } { } { } { } { } }{.} Due to the definition, the coefficient \mathl{(M^{ \mathfrak{ v } }_{ \mathfrak{ w } } (\varphi))_{ij}} is the $i$-th coordinate of \mathl{\varphi(v_j)} with respect to the basis \mathl{w_1 , \ldots , w_m}{.} Hence, this sume equals \mathl{\varphi(v_j)}{.}
\inputbeispiel {
A linear mapping \mathdisp {\varphi \colon K^n \longrightarrow K^m} { } is usually described by the matrix $M$ with respect to the standard bases on the left and on the right. The result of the matrix multiplication \mavergleichskettedisp {\vergleichskette { \begin{pmatrix} y_{1 } \\ \vdots\\ y_{ m } \end{pmatrix} } { =} { M \begin{pmatrix} x_{1 } \\ \vdots\\ x_{ n } \end{pmatrix} } { } { } { } { } { } { } } can be interpreted directly as a point in $K^m$. The $j$-th column of $M$ is the image of the $j$-th standard vector $e_j$.
\zwischenueberschrift{Rotations}
A rotation of the real plane $\R^2$ around the origin, given the angle $\alpha$ counterclockwise, maps \mathl{\begin{pmatrix} 1 \\0 \end{pmatrix}} to \mathl{\begin{pmatrix} \cos \alpha \\ \sin \alpha \end{pmatrix}} and \mathl{\begin{pmatrix} 0 \\1 \end{pmatrix}} to \mathl{\begin{pmatrix} - \sin \alpha \\ \cos \alpha \end{pmatrix}}{.} Therefore, plane rotations are described in the following way. \inputdefinition { {{{2}}} } {
A linear mapping \mathdisp {D(\alpha) \colon \R^2 \longrightarrow \R^2} { , } which is given by a \definitionswort {rotation matrix} \mathl{\begin{pmatrix} \operatorname{cos} \, \alpha & - \operatorname{sin} \, \alpha \\ \operatorname{sin} \, \alpha & \operatorname{cos} \,\alpha
\end{pmatrix}} \zusatzklammer {with some \mavergleichskette {\vergleichskette { \alpha } { \in }{\R } { }{ } { }{ } { }{ } }} is called
A \stichwort {space rotation} is a linear mapping of the space $\R^3$ in itself around a rotation axis \zusatzklammer {a line through the origin} with an certain angle $\alpha$. If the vector \mavergleichskette {\vergleichskette { v_1 } { \neq }{ 0 } { }{ } { }{ } { }{ } } defines the axis, and $u_2$ and $u_3$ are orthogonal to $v_1$ and to each other, and all have length $1$, then the rotation is described by the matrix \mathdisp {\begin{pmatrix} 1 & 0 & 0 \\ 0 & \operatorname{cos} \, \alpha & - \operatorname{sin} \, \alpha \\ 0 &\operatorname{sin} \, \alpha & \operatorname{cos} \,\alpha
\end{pmatrix}} { } with respect to the basis \mathl{v_1,u_2,u_3}{.}
\zwischenueberschrift{The kernel of a linear mapping}
\inputdefinition { } {
Let $K$ denote a field, let \mathkor {V} {and} {W} denote $K$-vector spaces, and let \mathdisp {\varphi \colon V \longrightarrow W} { } denote a $K$-linear mapping. Then \mavergleichskettedisp {\vergleichskette { \operatorname{kern} \varphi } { \defeq} { \varphi^{-1}(0) } { =} { { \left\{ v \in V \mid \varphi(v) = 0 \right\} } } { } { } { } { } }
The kernel is a linear subspace of $V$.
The following \stichwort {criterion for injectivity} is important.
\faktsituation {Let $K$ denote a field, let \mathkor {V} {and} {W} denote $K$-vector spaces, and let \mathdisp {\varphi \colon V \longrightarrow W} { } denote a $K$-linear mapping.} \faktfolgerung {Then $\varphi$ is injective if and only if \mavergleichskette {\vergleichskette { \operatorname{kern} \varphi } { = }{ 0 } { }{ } { }{ } { }{ } } holds.} \faktzusatz } {
If the mapping is injective, then there can exist, apart from \mavergleichskette {\vergleichskette {0 } { \in }{V } { }{ } { }{ } { }{ } }{,} no other vector \mavergleichskette {\vergleichskette {v } { \in }{V } { }{ } { }{ } { }{ } } with \mavergleichskette {\vergleichskette { \varphi(v) } { = }{ 0 } { }{ } { }{ } { }{ } }{.} Hence, \mavergleichskette {\vergleichskette { \varphi^{-1}(0) } { = }{ \{ 0 \} } { }{ } { }{ } { }{ } }{.} So suppose that \mavergleichskette {\vergleichskette { \operatorname{kern} \varphi } { = }{ 0 } { }{ } { }{ } { }{ } }{,} and let \mavergleichskette {\vergleichskette { v_1,v_2 } { \in }{ V } { }{ } { }{ } { }{ } } be given with \mavergleichskette {\vergleichskette { \varphi(v_1) } { = }{ \varphi(v_2) } { }{ } { }{ } { }{ } }{.} Then, due to linearity, \mavergleichskettedisp {\vergleichskette {\varphi(v_1 - v_2) } { =} {\varphi(v_1) - \varphi(v_2) } { =} { 0 } { } { } { } { } }{.} Therefore, \mavergleichskette {\vergleichskette { v_1-v_2 } { \in }{ \operatorname{kern} \varphi } { }{ } { }{ } { }{ } }{,} and so \mavergleichskette {\vergleichskette {v_1 } { = }{v_2 } { }{ } { }{ } { }{ } }{.}