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$0(n) + 0(n+1) + 0(n+2) = 0$ + + Seien $w_1, w_2 \in W, a \in K$ und $n \in \N$ beliebig. + \begin{align*} + &(aw_1 + w_2)(n) + (aw_1 + w_2)(n+1) + (aw_1 + w_2)(n+2) \\ + &= a (\underbrace{w_1(n) + w_1(n+1) + w_1(n+2)}_{= 0}) + + \underbrace{w_2(n) + w_2(n+1) + w_2(n+2)}_{= 0} \\ + &= 0 + .\end{align*} + $\implies (aw_1 + w_2) \in W$ + \end{proof} + \item Beh.: Sind $f, g \in W$ derart, dass $f(1) = g(1)$ und $f(2) = g(2)$ gelten, so ist $f = g$. + \begin{proof} + Seien $f, g \in W$ mit $f(1) = g(1)$ und $f(2) = g(2)$. + + Zz.: $\forall n \in \N\colon f(n) = g(n)$\\ + Beweis durch vollständige Induktion + + I.A.: Nach Voraussetzung gilt $f(1) = g(1)$ und $f(2) = g(2)$. + + I.S.: Es existiere ein festes aber beliebiges $n \in \N, n \ge 2$ mit + $f(n) = g(n)$ und $f(n-1) = g(n-1)$. + + $n \to n+1$: Wegen $f, g \in W$ gilt: + \begin{align*} + f(n-1) + f(n) + f(n+1) = 0 = g(n-1) + g(n) + g(n+1) + .\end{align*} + $\implies f(n+1) = g(n+1)$. + \end{proof} + \item Beh.: $W$ ist endlich erzeugt. + \begin{proof} + Definiere: + \begin{align*} + &w_1 := \begin{cases} + 1 & \exists k \in \N\colon n = 3k+1 \\ + 0 & \exists k \in \N\colon n = 3k+2 \\ + -1 & \exists k \in \N\colon n = 3k + \end{cases}, \qquad + w_2 \colon= \begin{cases} + 0 & \exists k \in \N\colon n = 3k+1 \\ + 1 & \exists k \in \N\colon n = 3k+2 \\ + -1 & \exists k \in \N\colon n = 3k + \end{cases} + .\end{align*} + + Zz.: $w_1, w_2 \in W$. Sei $n \in \N$ beliebig. + + Falls $\exists k \in \N\colon n = 3k+1$, dann $n + 1 = 3k+2$ und $n + 2 = 3(k+1)$. + \begin{align*} + w_1(n) + w_1(n+1) + w_1(n+2) &= 1 + 0 - 1 = 0 + \intertext{und} + w_2(n) + w_2(n+1) + w_2(n+2) &= 0 + 1 - 1 = 0 + .\end{align*} + + Fälle $\exists k \in \N\colon n = 3k+2$ bzw. $n = 3k$ folgen analog. + + Zz.: $\{w_1, w_2\} $ ist Erzeugendensystem. Sei $f \in W$ beliebig. + Wähle $a_1 := f(1)$ und $a_2 := f(2)$. Damit gilt wegen (b): + \begin{align*} + a_1 w_1(1) + a_2 w_2(1) = a_1 = f(1) + \intertext{und} + a_1 w_1(2) + a_2 w_2(2) = a_2 = f(2) + .\end{align*} Wegen (b) $\implies f = a_1 w_1 + a_2 w_2$. \\ + $\implies \{w_1, w_2\} $ ist endliches Erzeugendensystem von $W$. + \end{proof} + \item Beh.: $\text{dim}(W) = 2$ + \begin{proof} + Zz.: $\{w_1, w_2\}$ aus (c) ist Basis, also linear unabhängig. + + Sei $a_1 w_1(n) + a_2 w_2(n) = 0 \quad \forall n \in \N$. + + Für $n = 1$ folgt $a_1 w_1(1) + a_2 w_2(1) = a_1 = 0$.\\ + Für $n = 2$ folgt $a_1 w_1(2) + a_2 w_2(2) = a_2 = 0$.\\ + + $\implies \{w_1, w_2\}$ linear unabhängig und wegen (c) Basis von $W$ + $\implies \text{dim}(W) = 2$ + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(c)] + \item Beh.: $(u, v, x)$ ist Basis des $\R^{3}$. + \begin{proof} + Seien $a, b, c \in \R$ mit $a \cdot (0,1,2) + b \cdot (2, 1, 0) + c \cdot (1, 0, 0) = 0$. + \begin{align*} + 2a = 0 \quad \implies a = 0 \\ + a + b = 0 \quad \implies b = 0 \\ + 2b + c = 0 \quad \implies c = 0 + .\end{align*} + $\implies (u,v,x)$ linear unabhängig. + + Sei $z \in \R^{3}$ mit $(z_1, z_2, z_3)$ beliebig. Dann wähle $a = \frac{z_3}{2}$, + $b = z_2 - \frac{z_3}{2}$ und $c = z_1 - 2z_2 + z_3$. + + Dann gilt: + \begin{align*} + &\frac{z_3}{2} \cdot (0,1,2) + \left(z_2 - \frac{z_3}{2}\right) \cdot (2, 1, 0) + + (z_1 - 2z_2 + z_3) \cdot (1, 0, 0) \\ + &= (2z_2 - z_3 + z_1 - 2z_2 + z_3, \frac{z_3}{2} + z_2 - \frac{z_3}{2}, z_3)\\ + &= (z_1, z_2, z_3) + .\end{align*} + $\implies (u, v, x)$ ist Erzeugendensystem. + $\implies (u,v,x) $ ist Basis des $\R^{3}$. + \end{proof} + \end{enumerate} + \begin{enumerate}[(a)] + \item Beh.: $(u,v)$ ist linear unabhängig, aber keine Basis. + \begin{proof} + $(u,v)$ ist ein Teilsystem von $(u,v,x)$ und damit wegen (c) ebenfalls linear unabhängig, + + Da $(u,v,x)$ linear unabhängig ist, ist $(u,v)$ nicht maximal, also keine Basis + und damit kein Erzeugendensystem. + \end{proof} + \item Beh.: $(u,v,w)$ ist weder linear unabhängig, noch Erzeugendensystem. + \begin{proof} + Wähle $a := \frac{1}{2}$ und $b := \frac{1}{2}$. Damit folgt + \[ + \frac{1}{2} u + \frac{1}{2} v = \frac{1}{2}(0,1,2) + \frac{1}{2} (2,1,0) = (1,1,1) = w + .\] $\implies (u,v,w)$ nicht linear unabhängig. + + Da $w$ nicht zu $\text{Lin}((u,v,w))$ beiträgt, und $(u,v)$ wegen (a) kein Erzeugendensystem + sind, ist $(u,v,w)$ ebenfalls kein Erzeugendensystem und damit keine Basis. + \end{proof} + \end{enumerate} + \begin{enumerate}[(d)] + \item Beh.: $(u,v,w,x)$ ist Erzeugendensystem, aber nicht linear unabhängig und damit keine Basis. + \begin{proof} + Da die Basis $(u,v,x)$ ein Teilsystem von $(u,v,w,x)$ ist, folgt, dass $(u,v,w,x)$ + Erzeugendensystem ist. + + Allerdings ist dieses nicht minimal, da $(u,v,x)$ Basis ist. Also ist $(u,v,w,x)$ keine + Basis und damit nicht linear unabhängig. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $\text{Rg}(g \circ f) \le \text{min}(\text{Rg}(f), \text{Rg}(g))$ + \begin{proof} + Zz.: $\text{Rg}(g \circ f) \le \text{Rg}(f)$. + + Wegen $g \circ f$ folgt, + $\text{Rg}(g \circ f) = \text{dim}(\text{Bild}(g \circ f)) \le \text{dim}(\text{Bild}(f)) = \text{Rg}(f)$. + + Zz.: $\text{Rg}(g \circ f) \le \text{Rg}(g)$. + + Wegen $\text{Bild}(g \circ f) \subset \text{Bild}(g)$ und $f$ und $g$ linear, folgt + $\text{Bild}(g \circ f)$ ist UVR von $\text{Bild}(g)$. \\ + $\implies \text{Rg}(g \circ f) = \text{dim}(\text{Bild}(g\circ f)) + \le \text{dim}(\text{Bild}(g)) = \text{Rg}(g)$ + \end{proof} + \item Für $U = V = W = \R^{2}$ und + \begin{align*} + f\colon (x, y) \mapsto (x, 0), \qquad + g\colon (x, y) \mapsto (0, y) + .\end{align*} + $\implies g \circ f\colon (x, y) \mapsto (0, 0)$. + + Wegen $\text{Rg}(f) = 1 = \text{Rg}(g)$, aber $\text{Rg}(g \circ f) = 0$ folgt: + \[ + \text{Rg}(g \circ f) = 0 < 1 = \text{min}\left( \text{Rg}(f), \text{Rg}(g) \right) + .\] + \item Für $U = V = W$ und $f = g = id$ folgt $g \circ f = id \circ id = id$, also + $f = g = g \circ f$, also $\text{Rg}(g \circ f) = \text{Rg}(g) = \text{Rg}(f)$. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Ist $f^{*}\colon V^{*} \to U^{*}$ injektiv, so ist $f$ surjektiv. + \begin{proof} + Kontraposition. Zz.: Ist $f$ nicht surjektiv, dann ist $f^{*}$ nicht injektiv. + + Sei $(v_i)_{i \in I}$ Basis von $V$. + Wegen $f$ nicht surjektiv und linear, ex. ein $J \subset I, J\neq \emptyset$, s.d. + $\forall j \in J\colon v_j \not\in \text{Bild}(f)$. + + Nun wähle $j_0 \in J$ und $\varphi_1, \varphi_2 \in V^{*}$ mit + $\varphi_1(v_{j_0}) \neq \varphi_2(v_{j_0})$ und + $\varphi_1(v_i) = \varphi_2(v_i) \quad \forall i \in I \setminus J$ + + $\implies \varphi_1(f(u)) = \varphi_2(f(u)) \quad \forall u \in U$. \\ + Aber $\varphi_1(v_{j_0}) \neq \varphi_2(v_{j_{0}}) \implies \varphi_1 \neq \varphi_2$. + + $\implies f^{*}$ nicht injektiv. + \end{proof} + \item Beh.: $f$ injektiv $\iff$ $f^{*}$ surjektiv + \begin{proof} + \begin{enumerate}[(i)] + \item Zz.: $f$ injektiv $\implies f^{*}$ surjektiv + + Sei $u^{*} \in U^{*}$ beliebig und $(u_i)_{i \in I}$ eine Basis von $U$. + Wegen $f$ injektiv folgt: + \[ + (f(u_i))_{i \in I} = (v_i)_{i \in I} \quad \text{linear unabhängig} + .\] Wegen Basisergänzungssatz, sei $J$ Indexmenge mit $I \cap J = \emptyset$ und + $(v_i)_{i \in I \cup J}$ Basis von $V$. + + Nun definiere $\varphi \in V^{*}$ für $(v_i)_{i \in I \cup J}$ mit: + \begin{align*} + \varphi(v_i) = \begin{cases} + u^{*}(u_i) & i \in I \\ + 0 & i \in J + \end{cases} + .\end{align*} + $\varphi$ ist damit durch Basisvektoren eindeutig bestimmt. + + Außerdem gilt $\forall u_i \in (u_i)_{i \in I}\colon u^{*}(u_i) = \varphi(f(u_i))$. + Wegen $(u_i)_{i\in I}$ Basis von $U$, folgt: $\forall u \in U$ + $u^{*}(u) = \varphi(f(u)$. $\implies f^{*}(\varphi) = u^{*}$. + + $\implies f^{*}$ surjektiv. + \item Zz.: $f^{*}$ surjektiv $\implies f$ injektiv. + + Angenommen: $f$ nicht injektiv. Dann ex. $u_1, u_2 \in U$ + mit $f(u_1) = f(u_2)$, aber $u_1 \neq u_2$. + + Wähle $u^{*} \in U^{*}$, s.d. $u^{*}(u_1) \neq u^{*}(u_2)$. Wegen $f^{*}$ + surjektiv, ex. $\varphi \in V^{*}\colon \varphi(f(u)) = u^{*}(u)$ $\forall u \in U$. + Damit: + \[ + u^{*}(u_1) = \varphi(f(u_1)) = \varphi(f(u_2)) = u^{*}(u_2) + .\] Widerspruch zu $u^{*}(u_1) \neq u^{*}(u_2)$. + + $\implies f$ injektiv. + \end{enumerate} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}