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zZ^lYIw*r``R>|@MSL#PSM_v#sk6(wJw~@%W+{WuzB<0Ahujx!BLu#1$OfG{X@CxY~ z^ZV4u!+go%#Y`%|(0$n{;56_#z-ZUZ`>X(|VQg^~ViaP0h0R0`8ef$J819-MaNelS z(AByHscCPUUN;H{8cT;;B{}*j-c|}MZn<*m#wYMJiwAFMXkr(?yy0f1Mwps&}*>gV;Zq@Ga=}NC@Kg4Le5cn$2;r~i(Qx4!J z>AWM4|0@LGi1vJ9dMemi=b&&aIX-;Qxw;XQn&6^QPwZgSz`?S!H(F0au?U#d@FI5O0EhvIDF>6x+gh>b?O)pwVBF)@%XN5!~}a_>}w3 zfg0#1EDMu=Qey)kU9Xqe{}1H(YfJI8qHd`BYIfn4pm7Yt^1q2j^M<-enaHGNWy$#X J)Sju6{SOuHgO&gQ diff --git a/docs/loesungen.tex b/docs/loesungen.tex index 40a09dd..58aafc7 100644 --- a/docs/loesungen.tex +++ b/docs/loesungen.tex @@ -6414,27 +6414,30 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve Da $w_{i}\in\range(\phi)$ für alle ${i\in\{1,2,\ldots,d\}}$, reicht es aus wie oben zu zeigen, dass ${(w_{1},w_{2},\ldots,w_{d})}$ linear unabhängig sind. - Seien also $c_{1},c_{2},\ldots,c_{d}\in K$ beliebig - mit $\sum_{i=1}^{d}c_{i}w_{i}=\zerovector$. - \textbf{Zu zeigen:} $c_{1},c_{2},\ldots,c_{d}=0_{K}$. - Es gilt nun + Für $c_{1},c_{2},\ldots,c_{d}\in K$ gelten nun folgende Implikationen: - \begin{mathe}[mc]{rclql} - \sum_{i=1}^{d}c_{i}x_{i} - &= &\sum_{i=1}^{d}c_{i}\psi(w_{i}) - &\text{(siehe oben)}\\ - &= &\psi(\sum_{i=1}^{d}c_{i}w_{i}) - &\text{wegen Linearität}\\ - &= &\psi(\zerovector) - &\text{per Voraussetzung}\\ - &= &\zerovector - &\text{(siehe \cite[Lemma~6.1.2]{sinn2020})}.\\ - \end{mathe} + \begin{longmathe}[mc]{RCL} + \sum_{i=1}^{d}c_{i}w_{i}=\zerovector + &\Longrightarrow + &\psi(\sum_{i=1}^{d}c_{i}w_{i})=\psi(\zerovector)\\ + &\Longrightarrow + &\sum_{i=1}^{d}c_{i}\psi(w_{i})=\zerovector\\ + &&\text{wegen Linearität von $\psi$ und \cite[Lemma~6.1.2]{sinn2020}}\\ + &\Longrightarrow + &\sum_{i=1}^{d}c_{i}x_{i}=\zerovector\\ + &&\text{per Konstruktion von $w_{i}$}\\ + &\Longrightarrow + &c_{1},c_{2},\ldots,c_{d}=0_{K}\\ + &&\text{weil ${(x_{1},x_{2},\ldots,x_{d})}$ linear unabhängig ist.}\\ + \end{longmathe} - Wegen linearer Unabhängigkeit von ${(x_{1},x_{2},\ldots,x_{d})}$, - folgt hieraus, dass $c_{1},c_{2},\ldots,c_{d}=0_{K}$. - Damit haben wir die lineare Unabhängigkeit von ${(w_{1},w_{2},\ldots,w_{d})}$ bewiesen. - Da $w_{i}\in\range(\phi)$ für alle ${i\in\{1,2,\ldots,d\}}$, + Damit gilt + ${\sum_{i=1}^{d}c_{i}w_{i}=\zerovector + \Rightarrow + c_{1},c_{2},\ldots,c_{d}=0_{K}}$ + für alle $c_{1},c_{2},\ldots,c_{d}\in K$. + Also ist ${(w_{1},w_{2},\ldots,w_{d})}$ linear unabhängig. + Da ${w_{i}\in\range(\phi)}$ für alle ${i\in\{1,2,\ldots,d\}}$, erschließt sich aus der linearen Unabhängigkeit \fbox{$d\leq\dim(\range(\phi))$} (siehe \cite[Satz~5.3.4]{sinn2020}). @@ -6468,30 +6471,35 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve Wir zeigen nun, dass $(x_{1},x_{2},\ldots,x_{d})$ linear unabhängig ist. - Seien also $c_{1},c_{2},\ldots,c_{d}\in K$ beliebig - mit $\sum_{i=1}^{d}c_{i}x_{i}=\zerovector$. - \textbf{Zu zeigen:} $c_{1},c_{2},\ldots,c_{d}=0_{K}$. - Es gilt nun + Für $c_{1},c_{2},\ldots,c_{d}\in K$ gelten nun folgende Implikationen: - \begin{mathe}[mc]{rclql} - \psi(\sum_{i=1}^{d}c_{i}w_{i}) - &= &\sum_{i=1}^{d}c_{i}\psi(w_{i}) - &\text{wegen Linearität}\\ - &= &\sum_{i=1}^{d}c_{i}x_{i} - &\text{per Konstruktion}\\ - &= &\zerovector - &\text{per Voraussetzung}.\\ - \end{mathe} + \begin{longmathe}[mc]{RCL} + \sum_{i=1}^{d}c_{i}x_{i}=\zerovector + &\Longrightarrow + &\sum_{i=1}^{d}c_{i}\psi(w_{i})=\zerovector\\ + &&\text{per Konstruktion von $x_{i}$}\\ + &\Longrightarrow + &\psi(\sum_{i=1}^{d}c_{i}w_{i})=\zerovector\\ + &&\text{wegen Linearität von $\psi$}\\ + &\Longrightarrow + &\sum_{i=1}^{d}c_{i}w_{i}\in\ker(\psi)=\{\zerovector\}\\ + &&\text{wegen \uline{Injektivität} von $\psi$ + \cite[Lemma~6.1.4]{sinn2020}}\\ + &\Longrightarrow + &\sum_{i=1}^{d}c_{i}w_{i}=\zerovector\\ + &\Longrightarrow + &c_{1},c_{2},\ldots,c_{d}=0_{K}\\ + &&\text{weil ${(x_{1},x_{2},\ldots,x_{d})}$ linear unabhängig ist.}\\ + \end{longmathe} - Darum $\sum_{i=1}^{d}c_{i}w_{i}\in\ker(\psi)=\{\zerovector\}$ - wegen \uline{Injektivität} von $\psi$ (siehe \cite[Lemma~6.1.4]{sinn2020}). - Also $\sum_{i=1}^{d}c_{i}w_{i}=\zerovector$, - woraus sich ergibt, dass $c_{1},c_{2},\ldots,c_{d}=0_{K}$, - weil ${(w_{1},w_{2},\ldots,w_{d})}$ linear unabhängig ist. - Darum haben wir die lineare Unabhängigkeit von $(x_{1},x_{2},\ldots,x_{d})$ bewiesen. + Damit gilt + ${\sum_{i=1}^{d}c_{i}x_{i}=\zerovector + \Rightarrow + c_{1},c_{2},\ldots,c_{d}=0_{K}}$ + für alle $c_{1},c_{2},\ldots,c_{d}\in K$. + Also ist ${(x_{1},x_{2},\ldots,x_{d})}$ linear unabhängig. Da $(x_{1},x_{2},\ldots,x_{d})$ linear unabhängig ist - und $x_{i}\in\range(\psi\circ\phi)$ für alle ${i\in\{1,2,\ldots,d\}}$, + und ${x_{i}\in\range(\psi\circ\phi)}$ für alle ${i\in\{1,2,\ldots,d\}}$, folgt \fbox{$% d\leq\dim(\range(\psi\circ\phi)) \textoverset{Defn}{=}\rank(\psi\circ\phi)%