From b87c960654ecda9b85a1f80a0008c7bb30291a23 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 19 Nov 2019 21:00:43 +0100 Subject: [PATCH 1/7] finish la5, improve lecture class --- lecture.cls | 8 ++ ws2019/la/uebungen/la5.pdf | Bin 104227 -> 104558 bytes ws2019/la/uebungen/la5.tex | 174 ++++++++++++++++++++++++++----------- 3 files changed, 133 insertions(+), 49 deletions(-) diff --git a/lecture.cls b/lecture.cls index f663181..4fde8c7 100644 --- a/lecture.cls +++ b/lecture.cls @@ -14,6 +14,14 @@ \RequirePackage{transparent} \RequirePackage{xcolor} +\DeclareOption{uebung}{ + \makeatletter + \lhead{\@title} + \rhead{\@author} + \makeatother +} +\ProcessOptions\relax + % PAGE GEOMETRY \geometry{ left=15mm, diff --git a/ws2019/la/uebungen/la5.pdf b/ws2019/la/uebungen/la5.pdf index 63d413a6164c0265c54eaa7d5f187ff7c1f59b71..7b8a62ad0b732f8c213b9128650d7aef9a0f1c97 100644 GIT binary patch delta 83533 zcmZttWmFwq6Sj-4g=?^d1$TFc;I6^lEy3M_EZn^b!7UI7Zo%E%-JReBx4_Qx?r)zx 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zxB?R2L6VoLKVWDBq~C$zH(+EDj3&VKB)Aa)H#U&?HIV@0e36Gy&7{1mxuFi}kYfyn`o50U?%z+*fwn62u_ zS>icRc`Iq b^i1{vBF0bKmqnNX9||=yF$yImMNdWw-&ojD diff --git a/ws2019/la/uebungen/la5.tex b/ws2019/la/uebungen/la5.tex index 2a74975..2c6f887 100644 --- a/ws2019/la/uebungen/la5.tex +++ b/ws2019/la/uebungen/la5.tex @@ -1,9 +1,21 @@ -\documentclass{../../../lecture} +\documentclass[uebung]{../../../lecture} \usepackage{enumerate} +\usepackage{array} + +\title{Übungsblatt Nr. 5} +\author{Christian Merten, Mert Biyikli} \begin{document} +\begin{tabular}{|c|m{1cm}|m{1cm}|m{1cm}|m{1cm}|m{1cm}|@{}m{0cm}@{}} + \hline + Aufgabe & \centering A1 & \centering A2 & \centering A3 & \centering A4 & \centering $\sum$ & \\[5mm] \hline + Punkte & & & & & & \\[5mm] \hline +\end{tabular} + +\vspace{5mm} + \begin{aufgabe} Es sei $K$ Körper, $M$ eine Menge und $m_0 \in M$ ein fest gewähltes @@ -11,11 +23,23 @@ Element. In \\$V = \text{Abb}(M, K)$ betrachten wir die Teilmengen $U = \{f \in V \mid f(m_0) = 0\} $ und \\$W = \{f \in V \mid \forall x, y \in M \colon f(x) = f(y)\} $ -Zunächst: $K$ ist K-Vektorraum mit $(K, +, 0)$. Damit wird -$V = \text{Abb}(M, K)$ zum Vektorraum. - \begin{enumerate}[a)] - \item Beh.: $U \subset V$ ist Untervektorraum. + \item Zunächst: $K$ ist K-Vektorraum. Damit wird + $V = \text{Abb}(M, K)$ mit $0_V(m) = 0 \text{ } \forall m \in M$ zum K-Vektorraum. + + Damit eine Teilmenge $M \subset V$ zum Untervektorraum von $V$ wird, muss gelten: + \[ + m_1 + m_2 \in M \text{ } \forall m_1,m_2 \in M + .\] und + \[ + a m_1 \in M \text{ } \forall m_1 \in M, a \in K + .\] Die Inversen der zugehörigen Untergruppe sind gegeben durch + \[ + m^{-1} = (-1)_K m \in M \text{ } \forall m \in M + .\] + + + Beh.: $U \subset V$ ist Untervektorraum. \begin{proof} Seien $f_1, f_2 \in U$, $a \in K$ beliebig. Zu zeigen: @@ -48,11 +72,15 @@ $V = \text{Abb}(M, K)$ zum Vektorraum. \item Beh.: $U \cap W = \{0\} $ \begin{proof} + Zunächst: $0_V(m_0) = 0 \implies 0_V \in U$ und + $0_V(x) = 0 = 0_V(y) \text{ } \forall x,y \in M \implies 0_V \in W$. Daraus folgt + $0_V \in U \cap W \implies U \cap W \neq \emptyset$. + Sei $f \in U \cap W$ beliebig: \begin{align*} &\forall m \in M \colon f(m) = f(m_0) \land f(m_0) = 0 \\ \implies &\forall m \in M \colon f(m) = 0 \\ - \implies &f = 0 + \implies &f = 0_V .\end{align*} \end{proof} \item Beh.: $V = U + W$ @@ -123,13 +151,13 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. \begin{align*} \partial(v_1+v_2)(i) &= (i+1) \cdot (v_1 + v_2)(i+1) \\ &= (i+1) \cdot (v_1(i+1) + v_2(i+1)) \\ - &= (i+1)v_1(i+1) + (i+1)v_2(i+1) \\ + &= (i+1) \cdot v_1(i+1) + (i+1) \cdot v_2(i+1) \\ &= \partial(v_1)(i) + \partial(v_2)(i) .\end{align*} \begin{align*} - \partial(a v_1)(i) &= (i + 1)(a v_1)(i+1) \\ - &= a (i+1) v_1 (i+1) \\ - &= a \partial(v_1)(i) + \partial(a v_1)(i) &= (i + 1) \cdot (a v_1)(i+1) \\ + &= a (i+1) \cdot v_1 (i+1) \\ + &= a \cdot \partial(v_1)(i) .\end{align*} \end{proof} @@ -139,17 +167,17 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Seien $v_1, v_2 \in V$ mit $\psi(v_1) = \psi(v_2)$. Dann \begin{align*} - &\psi(v_1) = \left( f_1(0), f_1(1), \ldots, f_1(n+1) \right) - = \left( f_2(0), f_2(1), \ldots, f_2(n+1) \right) = \psi(v_2)\\ - \implies& f_1(k) = f_2(k) \text{ }\forall k \in \{0, \ldots, n+1\} \\ - \implies& f_1 = f_2 + &\psi(v_1) = \left( v_1(0), v_1(1), \ldots, v_1(n+1) \right) + = \left( v_2(0), v_2(1), \ldots, v_2(n+1) \right) = \psi(v_2)\\ + \implies& v_1(k) = v_2(k) \text{ }\forall k \in \{0, \ldots, n+1\} \\ + \implies& v_1 = v_2 .\end{align*} $\implies \psi$ ist injektiv. - Sei $c = (c_0, \ldots, c_{n+1}) \in K^{n+2}$, dann ex. ein $f \in V$, s.d. + Sei $c = (c_0, \ldots, c_{n+1}) \in K^{n+2}$, dann ex. ein $v \in V$, s.d. \begin{align*} - &f(k) = c_k \text{ } \forall k \in \{0, \ldots, n+1\} \\ - \implies &\psi(f) = c + &v(k) = c_k \text{ } \forall k \in \{0, \ldots, n+1\} \\ + \implies &\psi(v) = c .\end{align*} $\implies \psi$ ist surjektiv. \end{proof} @@ -175,32 +203,63 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. k+1 \neq 0$ $\forall k \in \{0, \ldots, n\} $. \begin{align*} &k + 1 \neq 0 \\ - \stackrel{k > 0}{\iff} & k + 1 \neq \text{char}K \\ - \stackrel{0 \le k \le n}\iff & \text{char}K = 0 \lor \text{char}K > n + 1 \\ + \stackrel{k \ge 0}{\iff} & k + 1 \neq \text{char}K \\ + \stackrel{1 \le k + 1 \le n + 1}\iff & \text{char}K = 0 \lor \text{char}K > n + 1 \\ \iff & \text{char}K \not\in \{2, \ldots, n+1\} .\end{align*} \end{proof} - \item Beh.: $\psi(\text{ker }\partial) = - \left\{ (c, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid c \in K\right\} $ - - \begin{proof} - Zunächst: $\text{ker }\partial$. - - Damit $r \in V$ im Kern von $\partial$ liegt, muss gelten: - $\partial(r)(k) = 0$ $\forall k \in \{0, \ldots, n\}$ + \item Bestimmen Sie $\psi(\text{ker }K) \subset K^{n+2}$. + \begin{proof}[Lösung] \begin{align*} - &\partial(r)(k) = (k+1) \cdot r(k+1) \\ - \stackrel{k+1 \neq 0}{\implies} &r(k+1) = 0 + &\ker \partial = + \{f \in V \mid \left( \partial(f) \right)(k) = 0 \text{ } \forall k \in \{0, 1, \ldots n\} \} .\end{align*} - Damit: $r(k) = 0$ $\forall k \in \{1, \ldots, n+1\} $. + Damit $f \in \text{ker } \partial$, muss folglich gelten: \begin{align*} - \psi(r) &= \left( r(0), r(1), \ldots, r(n+1) \right) \\ - &= (c, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \text{ } \forall c \in K + &(\partial(f))k = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} .\end{align*} - Das heißt: - \[ - \psi(\text{ker }\partial) = \left\{ (c, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid c \in K\right\} - .\] + $\iff$ + \begin{align*} + (k+1) \cdot f(k+1) = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} + .\end{align*} + $\stackrel{K \text{ Körper}}{\iff}$ + \begin{align*} + k+1 = 0 \lor f(k+1) = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} + .\end{align*} + + Aus (c) folgt: $k+1 \neq 0 \iff \text{char K} \not\in \{2, \ldots, n+1\} $. + + \begin{enumerate}[(i)] + \item $\text{char }K \not\in \{2, \ldots, n+1\} $. Dann ist $k + 1 \neq 0$, d.h. + \begin{align*} + &f(k+1) = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} \\ + \implies &f(k) = 0 \text{ } \forall k \in \{1, \ldots, n+1\} \\ + \implies & \text{ker } \partial = \{f \in V \mid f(k) = 0 \text{ } \forall k \in \{1, \ldots, n+1\} \} + .\end{align*} + Damit folgt: + \[ + \psi(\text{ker }\partial) = + \{(a, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid c \in K\} + .\] + \item $\text{char }K \in \{2, \ldots, n+1\} $. Dann gilt für $k = \text{char }K-1$: + \[ + k + 1 = \text{char } K - 1 + 1 = \text{char } K = 0_K + .\] + Für alle $k \in \{0, 1, \ldots, n\}, k \neq \text{char } K - 1$, folgt analog zu (i): + \[ + f(k + 1) = 0 + .\] + Damit folgt: + \[ + \text{ker } \partial + = \left\{ f \in V \mid f(k) = 0 \text{ } \forall k \in \{1, \ldots, n+1\} + \setminus \{\text{char } K - 1\} \right\} + .\] Damit ergibt sich: + \[ + \psi(\text{ker } \partial) = \{ (a_0, a_1, \ldots, a_{n+1}) \in K^{n+2} + \mid a_k = 0 \text{ } \forall k \in \{1, \ldots, n+1\} \setminus \{\text{char }K - 1\} \} + .\] + \end{enumerate} \end{proof} \end{enumerate} @@ -221,8 +280,8 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. = f^{*}(\varphi_1) + f^{*}(\varphi_2) \\ f^{*}(a \varphi_1) &= (a \varphi_1) \circ f - \stackrel{\varphi_1 \text{ linear}}{=} - a ((\varphi_1) \circ f) + = + a (\varphi_1 \circ f) = a f^{*}(\varphi_1) .\end{align*} \end{proof} @@ -234,14 +293,14 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. \begin{proof} Seien $u_1, u_2 \in U$, $a \in K$ und $f \in U^{*}$ beliebig. \begin{align*} - \text{ev}(u_1 + u_2)(f) &= + (\text{ev}(u_1 + u_2))(f) &= f(u_1 + u_2) \stackrel{f \text{ linear}} {=} f(u_1) + f(u_2) - = \text{ev}(u_1)(f) + \text{ev}(u_2)(f) \\ - \text{ev}(a u_1)(f) &= + = (\text{ev}(u_1))(f) + (\text{ev}(u_2))(f) \\ + \left(\text{ev}(a u_1)\right)(f) &= f(a u_1) \stackrel{f \text{ linear}} {=} a f(u_1) - = a \cdot \text{ev}(u_1)(f) + = a \cdot (\text{ev}(u_1))(f) .\end{align*} \end{proof} \end{enumerate} @@ -251,7 +310,7 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Es sei $K$ ein Körper und $U, V$ zwei $K$-Vektorräume. \begin{enumerate}[a)] - \item Die Abbildung $*$: + \item Beh.: Die Abbildung $*$: $\text{Hom}_K(U,V) \to \text{Hom}_K(V^{*}, U^{*})$ ist linear. @@ -260,19 +319,36 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. $\varphi \in V^{*}$ und $a \in K$ beliebig. \begin{align*} - *(f_1 + f_2)(\varphi) &= (f_1 + f_2)^{*}(\varphi) + (*(f_1 + f_2))(\varphi) &= ((f_1 + f_2)^{*})(\varphi) = \varphi \circ (f_1 + f_2) = \varphi \circ f_1 + \varphi \circ f_2 - = *(f_1)(\varphi) + *(f_2)(\varphi) \\ - *(a f_1)(\varphi) - &= (a f_1)*(\varphi) + = (*(f_1))(\varphi) + (*(f_2))(\varphi) \\ + (*(a f_1))(\varphi) + &= ((a f_1)^{*})(\varphi) = \varphi \circ (a f_1) = a (\varphi \circ f_1) - = a*(f_1)(\varphi) + = a\cdot (*(f_1))(\varphi) .\end{align*} \end{proof} - \item Ist $f\colon U \to V$ linear und surjektiv, so ist + \item Beh.: Ist $f\colon U \to V$ linear und surjektiv, so ist $f^{*}\colon V^{*} \to U^{*}$ injektiv. + + \begin{proof} + Seien $\varphi_1, \varphi_2 \in \text{Hom}_K(V,K) = V^{*}$ mit + $f^{*}(\varphi_1) = f^{*}(\varphi_2)$. Dann folgt: + \begin{align*} + \varphi_1 \circ f = \varphi_2 \circ f + .\end{align*} + das heißt: + \begin{align*} + \forall u \in U\colon \varphi_1(f(u)) = \varphi_2(f(u)) + .\end{align*} + Wegen $f$ surjektiv gilt: $V = f(U)$ und damit: + \begin{align*} + \forall v \in V\colon \varphi_1(v) = \varphi_2(v) + .\end{align*} + $\implies \varphi_1 = \varphi_2$ + \end{proof} \end{enumerate} \end{aufgabe} From 6f25011a8401f625ab4aaf5367eb29fc79939513 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 20 Nov 2019 13:38:34 +0100 Subject: [PATCH 2/7] add ipi4 --- ws2019/ipi/uebungen/ipi4.pdf | Bin 0 -> 96148 bytes ws2019/ipi/uebungen/ipi4.tex | 120 ++++++++++++++++++++++++++++++++ ws2019/ipi/uebungen/uebung4.cpp | 70 +++++++++++++++++++ 3 files changed, 190 insertions(+) create mode 100644 ws2019/ipi/uebungen/ipi4.pdf create mode 100644 ws2019/ipi/uebungen/ipi4.tex create mode 100644 ws2019/ipi/uebungen/uebung4.cpp diff --git a/ws2019/ipi/uebungen/ipi4.pdf b/ws2019/ipi/uebungen/ipi4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..15028fae4c4462cd767d2913889c0ba695938ef1 GIT binary patch literal 96148 zcmb?@by(E-_wNitcXyXGGjua_BMlNtBM8z+ceiv%rwS5+f=ZW!G=hM%DAFlCcieUN z`}?kQ@9MqRefD{FpWQz`=e$q6&g;C+uxQCEazl6oFB02QmJXN_5}3T| zHuvp3?$YxMiSp6^@duMv(caO+#*Lm=(b2-gM&8EC+1dtEN($56!_CIR3DbLa+r-43 z>Lp3wbj^}Ue{Z;JTlYIT9r2yg)p)Ac{qd{nb}Y{-tNYVmna(~IJKL2+YhRH0Sjiml zC5{x!G9=9XY8ZHQwEdvwgJX+$(evZ@;oJHC$0utW)9UZ2U2`@+%pGR0$?s>|go3SX zoeiJUecp{ouDEb(5&vlI*bNAQ=SS?4qHI@x$upkdoQOX`uHh1W5Pz9vmSD4mTs0)t6OB{QrN@@ z32VsMnG*FnJ!0svQmdNaFQHl1L1qnVmcC5GFWP(^^4?aDfW7&SX+zxe zjTChkSvq?SjJBKa*J1>54o+R)H{{Ip=UBdCMbpnrdd~Veva&yiudX~LT+;vDbxNfi zal?4KR|t9A+GcmGX6G)Rk*lRFI7DDega~QMvc15{ySiB6Z6U;B@!~)(AnbVF??bpC zHNR5HN2Q@LhfRpP!0dw|>I5koh>{*#M|A3e6sBPG?a-UWWZ-O_0V%;=*>J5XxHkQ5G~wS2h5Er zA?4x@3>tI0HwnJf9m!^ZSrU!kQcU;$PZZ3Ri_;{XPMflxLar<)g4&WPJp5^rs7ivBg{K1}NlQ!m4)tHc4U;$xYaY8|f z1aWMVYm8gZORhIl7<}Q3ktb7$+EIKRN*`Mwv6dY#oxF*cwIh|@w{*PS5g%S-AQD0? zw7J*ujg`s`-+tG!EtwVFubZ#@a7*}B4O7}78gB`@dCBV;9(iR??1?Jyx96s~?7_t+ zunfP_PpnKWZulDv@rz93NzV`E$JgoOLXyk4SST$^E0)0y#+!5i-TRLqap4E@ZUW)<0wIkiVCmmnz87^L6-=}@yM)JkPX090bT|N5HQG6$l z7!Rr?bD7t+feoL} zJnP;G(ZZrM9=F%l_U>?eRatVl-4hUG#Aa=J#_V4=FtCNaX+^etrK?@dh{xUydd4b& zJnPO&yjiSOl~({_`lO&AwNs4U@|1D$)7ui*k^;eJcaw!R#H@U?z2SMB}U+|LvSXKz|N3WDyG zb@LL;HtO6m+qo(1BZm8hr|hxui%bT6B+YQtTD%9zgwpi53%=Ur6Nwpf@2wXbQlfmG zU-4w{J$$}^Q{-4%;!Y=JAIA|YZ!>i)^%aleQ__&CERHvGbf;;zSnoy=^1oCnb#SdjwT}*xLqTPI1xs%JgLCXMSz6n)y zncVlZTlTB^WDAy`GsA|>ykBEGPcLhSE0G31Ma!d;vm48u2!H4*xfpZooJ7LVS&Nzh zm{;v2?2YLqU*nd4aDjGqJ5ni~kibRl0e~S!4Hz{_8f7lDIg2eM%CHILiq|bvxxr3a$=hj|DBviv!0P}k`gYS|o=wlyY;UO(K zkuA7n%KE&oHNYd!m9y4rN=AJ%x}<*@gLd5kT4+iGWM-n0$J55;W;mmMtH|p0@eoha zmfXODFlQW#yhf2X9><``bX7(MT}yS4@j8|(Fe7Et^#jj*-G*FDa70B2I3#f<8Zq7l z%R(RcY9*->RSNDtW~soz!4ALTDpN(0t+YN*=FyIE!=-Y0Xd1>sRr9P+->xH>s^-G) zE1|QeedpIoss{a6b*TM?GG3K3IU7Zq*DqcP=zqx-tdv%f4Xc!fHr1dCa#^2;`7mL9y$En?X@FRd;?pc?NA&SXkY|+3liS~??w@;@h>}t1 zMK19n#F)J88(#Q*zA-d!NNADdm^xVj4?Qp0<5UDU>FOX* zrIxif>ZP{H`msVHVcV)U>l`HRb=NbR0;=8SS5AT23aZAXpGQ9Uyv%0h4PyliViU!W zE^MRDag=+wRiQsP3mQ4OxIodQ6y$g?6XJ7>OCqHP!Img@EfG@)LRD^8=fy76biXm- zE_34OM-s<=lYzk|{AEPkUF7KtLZI7{3rAi?Jow&g53yl&yBrbFV{bnJMnLsO$QLCg zE}f9{V@1!~_TIzeYRJ(F$Xd{9{Db`a*vDq;Na_uX#Vmfm%~6;Vf3}TQ2&rh5G4->ZX`xNxOy5n1HMM<*!|l z-5NY;O_lXOd%8p+O$17EpfV=4w>HUK$S#oYdhXF2JV-e6^FH)p5ymDwq*-vsh;pCV ze=RVfT)cAUd#q{znL1f1x>-3A(Mk|_Uq$pKa;X{i?HI!+O+PN!V`nD zJ@J9Qc`iDaLeQB%cJW#GGdiD$h(tE#c-f)&aN@`se+t+_lb{zF~V$ z{tG?gE^dzTCxGoiIlsGorwXFi?U7$^LJi3er<8RARM4M3Y{f)L=sMYmYV>4aE^$=N zzU6(*UsJO==I>3#Fo_p6gA#=G{eGiFSN9uh zjCXm#F9g!}CM_aKyG{d@i9RyjEk+#@doLddCCWaGc&{`nPle0t$8U3Ss7F}_RLkyL zLcZy;LYk68>BePvuDr9EZc}Bm@(_wNs9O9XjM<$k&Y-YQpJuRP;c3*keL*=bE`gK= zo95=eXWrPE1gq$hb=E`_&*h$<>8a0tKIX$9z*5WIWJ3G{Tu++x)+5{c!LOT3XtNj6 z#QFQs4NuSJXZ71c{;xO>;v_;)GNLz`oPmF6sAL1qY#vDBm@an%`Y^DtPqb|`LU%@Xw@KNYd$=Exo1$=9W}UyTIOrl{n?*%qQY(RChl0p- zFW}OD>6XhH9@SQRhFyb# z0n6;2T=_Lb8@O^RE}^DFzE(ThG*o2aTQrX8A9d&^O3ulx>F$(nTB&Du^l!_1HlFDV z!Cu$;ir`I)J(R`Uj+ym~Ssi_LWVNHZpmf@jS0~qe`FRyzy5Wu$2GQ)IPKANs^b@I1 zJv+UXJn?vX?llt!HJr~12z|$6^0;Y+ON6ZcP~Bj8St9b4KYb3 z+GOQL(6t zoj#Wn45#xh>M@=xDWf@D+3pVl=K9pwV%prd{xNpK|LOb4$_M#*c7;F$g?@QYIQ-}>3slyf`n3!)C%EOY6bt+^WcFyu&|o$3b16S9dEs(yT8w9hkJ`7H_8TB^KK zju}k|2}q^Q z4w7>)A>&SAl7AeRmhoDz{86(21p%qOGIp&WSZqx0E$yn}bLL;sazxJ)SLKQ^u9$uLI z`S0t<(VtMyY3TDI>2Mens_fy-KNb;d$=sE4$zBfjN}F3}kQ$#Y>Bm7fuQN3#bF|4d zuqCc0)tAaqGaTob;>y4+aRzOn2NvOu-{}Mnp%-~L0CYne4SZavv2p2>9FuRvY;}E; 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$\sum$ & \\[5mm] \hline + Punkte & & & & & & \\[5mm] \hline +\end{tabular} + +\begin{aufgabe} + Zahlendarstellung + + \begin{enumerate}[a)] + \item Berechnung der Determinante einer $2\times 2 $ Matrix. + \begin{lstlisting}[language=C++, title=Berechnung der Determinante, captionpos=b] +double determinante_double(double a, double b, double c, double d) { + return a*d - b*c; +} + +float determinante(float a, float b, float c, float d) { + return a*d - b*c; +} + \end{lstlisting} + + Das Ergebnis bei Verwendung von \verb+float+ ist $10000$ und damit nicht exakt. + Das liegt an der zu geringen Größe eines \verb+float+'s, der nur rund $7$ Dezimalstellen exakt + speichern kann, danach wird gerundet. Das führt in diesem Fall zum Verlust aller + Nachkommastellen. Bei Verwendung des Datentyps \verb+double+ reichen die rund $16$ Stellen aus, um + das Ergebnis exakt darzustellen. + \item Assoziativität bei \verb+float+s + \begin{lstlisting}[language=C++, title=Vergleich der zwei Versionen, captionpos=b] +float testAssoziativitaet() { + for (int n = 6; n <= 14; n++) { + // (a+b)+c + float vers1 = (pow(10, n) + pow(-10, n)) + pow(10,-n); + // a+(b+c) + float vers2 = pow(10, n) + (pow(-10, n) + pow(10,-n)); + print(n, vers1, vers2, 0); + } +} + \end{lstlisting} + + Das Ergebnis ist bereits ab $n = 5$ nicht mehr assoziativ. Das liegt daran, dass + bei $(a+b)+c$ zunächst $10^{n} - 10^{n}$ berechnet wird, das aber immer null ist und danach + einfach $10^{-n}$ ausgewertet wird. Dabei kann die gesamte Präzision des \verb+float+'s + für die Nachkommastellen von $10^{-n}$ verwendet werden. Bei $a + (b+c)$ wird erst + $-10^{n} + 10^{-n}$ berechnet. Hierbei reicht die Präzision nicht aus, um die Nachkommastellen + darzustellen, da jetzt die Zahl bereits vor dem Komma sehr groß ist. Dadurch wird + $-10^{n} + 10^{-n}$ auf $-10^{n}$ gerundet und dann mit $10^{n}$ addiert, was dann null ergibt. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} Effektiver Zinssatz + + Programm siehe \verb+uebung4.cpp+ + + Die Ergebnisse erleiden durch Runden und den begrenzten Speicherplatz des \verb+float+'s einige + Ungenauigkeiten. Die Differenz zum Grenzwert wird wie erwartet immer geringer. + + Die Genauigkeit der \verb+double+ Werte ist höher. +\end{aufgabe} + +\begin{aufgabe} Multiplikation im Zweierkomplement + + Zeigen Sie, dass für $n \in \N$ mit + \[ + a, b, a \cdot b \in \left[ -2^{n-1}, 2^{n-1}- 1 \right] + .\] für die Multiplikation im Zweierkomplement folgendes gilt: + \[ + d_n(a\cdot b) = s_n(d_n(a)\cdot d_n(b)) + .\] + + \begin{proof} + Seien $a, b \in \left[ -2^{n-1}, 2^{n-1}-1 \right] $ mit $-2^{n-1} \le a\cdot b \le 2^{n-1}-1$ beliebig. + + Fallunterscheidung: + \begin{enumerate}[(i)] + \item $a, b \ge 0$. Dann ist $a \cdot b \ge 0$. Damit + \[ + s_n(d_n(a)\cdot d_n(b)) = s_n(a\cdot b) = a\cdot b = d_n(a\cdot b) + .\] + \item $a < 0, b > 0$ (analog $a > 0, b <0)$. Dann ist $a \cdot b < 0$. + \begin{align*} + s_n(d_n(a)\cdot d_n(b)) &= s_n((2^{n} - |a|) \cdot b) \\ + &= s_n((2^{n} + a) \cdot b) \\ + &= s_n(2^{n} \cdot b + a\cdot b) \\ + &= s_n(2^{n} \cdot b - |a+b|) \\ + &= 2^{n} - |a+b| \\ + &= d_n(a\cdot b) + .\end{align*} + \item $a < 0, b < 0$. Dann ist $a \cdot b > 0$. + \begin{align*} + s_n(d_n(a)\cdot d_n(b)) &= s_n((2^{n} - |a|)(2^{n} - |b|) \\ + &= s_n((2^{n} \cdot 2^{n} + 2^{n} \cdot b + a\cdot 2^{n} + a\cdot b) \\ + &= a \cdot b \\ + &= d_n(a\cdot b) + .\end{align*} + \item $a < 0, b = 0$ (analog $a = 0, b < 0$). Dann ist $a \cdot b = 0$. + \begin{align*} + s_n(d_n(a) \cdot d_n(b)) = s_n(\left( 2^{n} - |a| \right) \cdot 0) = s_n(0) = 0 = d_n(0) = d_n(a\cdot b) + .\end{align*} + \end{enumerate} + \end{proof} + +\end{aufgabe} + +\end{document} diff --git a/ws2019/ipi/uebungen/uebung4.cpp b/ws2019/ipi/uebungen/uebung4.cpp new file mode 100644 index 0000000..c4699b8 --- /dev/null +++ b/ws2019/ipi/uebungen/uebung4.cpp @@ -0,0 +1,70 @@ +#include "cpp_headers/fcpp.hh" + +// Berechne die Determinante einer 2x2 Matrix mit doubles +double determinante_double(double a, double b, double c, double d) { + return a*d - b*c; +} + +// Berechne die Determinante einer 2x2 Matrix mit floats +float determinante(float a, float b, float c, float d) { + return a*d - b*c; +} + +// Teste die Assoziativitaet von floating point Zahlen +float testAssoziativitaet() { + for (int n = 1; n <= 14; n++) { + // (a+b)+c + float vers1 = (pow(10, n) + -pow(10, n)) + pow(10,-n); + // a+(b+c) + float vers2 = pow(10, n) + (-pow(10, n) + pow(10,-n)); + print(n, vers1, vers2, 0); + } +} + +// Berechnet den effektiven Zinssatz mit floats +// z: Zinssatz, n: Abrechnungsvorgaenge +float zins(float z, int n) { + return pow(1 + z/n, n) - 1; +} + +// Berechnet den effektiven Zinssatz mit doubles +// z: Zinssatz, n: Abrechnungsvorgaenge +double zins(double z, int n) { + return pow(1 + z/n, n) - 1; +} + +// Vergleiche die Berechnung des effektiven Zinssatzes bei n Abrechnungsvorgaengen +// mit floats und doubles und vergleiche mit dem Grenzwert für n -> unendlich. +void compare(int n) { + // Berechne effektiven Zins mit float + float zf = zins(0.06, n); + // Berechne die Differenz zu exp(z) - 1 + float difff = exp(0.06) - 1 - zf; + // Berechne effektiven Zins mit double + double zd = zins(0.06, n); + // Berechne die Differenz zu exp(z) - 1 + double diffd = exp(0.06) - 1 - zd; + // Gebe alles aus + print("Abrechnungsvorgaenge:", n, 0); + print("Zinssatz (float):", zf, 0); + print("Abweichung von exp(z) - 1:", difff, 0); + print("Zinssatz (double):", zd, 0); + print("Abweichung von exp(z) - 1:", diffd, 0); + print(""); +} + +int main() { + print("det", determinante(100, 0.01, -0.01, 100), 0); + print("double det", determinante_double(100, 0.01, -0.01, 100), 0); + testAssoziativitaet(); + + // vergleiche fuer verschiedene Werte + compare(1); + compare(4); + compare(12); + compare(365); + compare(365*24); + compare(365*24*60); + compare(365*24*60*2); + compare(365*24*60*60); +} From 952bb0985fba93fcc4659a0dd5c36c0138265ce5 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 20 Nov 2019 13:39:34 +0100 Subject: [PATCH 3/7] add comments --- ws2019/ipi/uebungen/uebung4.cpp | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/ws2019/ipi/uebungen/uebung4.cpp b/ws2019/ipi/uebungen/uebung4.cpp index c4699b8..a5bf06b 100644 --- a/ws2019/ipi/uebungen/uebung4.cpp +++ b/ws2019/ipi/uebungen/uebung4.cpp @@ -54,11 +54,13 @@ void compare(int n) { } int main() { + // Determinantenberechnung (Aufgabe 1) print("det", determinante(100, 0.01, -0.01, 100), 0); print("double det", determinante_double(100, 0.01, -0.01, 100), 0); + // Teste die Assoziativitaet (Aufgabe 1) testAssoziativitaet(); - // vergleiche fuer verschiedene Werte + // vergleiche fuer verschiedene Werte (Aufgabe 2) compare(1); compare(4); compare(12); From 902c7802faaecff463fbde29241cdc96217f57cb Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 20 Nov 2019 20:22:12 +0100 Subject: [PATCH 4/7] update la5 --- ws2019/la/uebungen/la5.pdf | Bin 104558 -> 104841 bytes ws2019/la/uebungen/la5.tex | 15 ++++++++------- 2 files changed, 8 insertions(+), 7 deletions(-) diff --git a/ws2019/la/uebungen/la5.pdf b/ws2019/la/uebungen/la5.pdf index 7b8a62ad0b732f8c213b9128650d7aef9a0f1c97..cf98acfc9a3213a13afae5d86064bf3c2c57ab35 100644 GIT binary patch delta 18459 zcmV)6K*+!Dum*{<2C%vm0x~g^feR>q-CJ98+r|}s*RQ}MH8wVTzsQqLs=DnoZ8}Y@ z_Muj$BSjF6n4(isv>mto?>!5E02VyBQk1AZi6R$!IoI!eXYtj=U;o0Q`TB%L^N2>j 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z%-=cZ1|pn`GvUm+c&9X8OcSF;y2ubS#SAfBWQyTt9ew^-Pz(|g=3>HrZLoicG#9wz zKZlAaF~nSQH!U-r_zS_?ko6X3y@J^fU`{>E&476c;u~at^reQ0Xc1#>{w+3gdSO8` zNE87v%3Q%ujuqa7<-cG>BormW z%2X(hg;o0?6oQffl$OBiPTzkhNsJL=#W-_ode|*{2IZ=^E(+GGXvHGfPz)O@U{kvI z1e-6x7QJI@Hf*bd?N?w&A5{Ky0!KbT!*@9P9FD2SaZTgILukAPC-1|l8aQ1GXVzrG*-ALq08Qo4 zTm>zq(3%71^WlQ-*LEG+-$2KE=v2Ln@1QFhx|96;CWwh*lDW$vzm{z7UYSoR=Dzm$ zG}&DLW1mvZ(t3TGVpj0~r%5xb{_N9Kk!9BY#hG30a{mDQ*l9zTV3`3P3o|$|ISM5u HMNdWw6W8{o diff --git a/ws2019/la/uebungen/la5.tex b/ws2019/la/uebungen/la5.tex index 2c6f887..e7ccc00 100644 --- a/ws2019/la/uebungen/la5.tex +++ b/ws2019/la/uebungen/la5.tex @@ -2,6 +2,7 @@ \usepackage{enumerate} \usepackage{array} +\usepackage{mathtools} \title{Übungsblatt Nr. 5} \author{Christian Merten, Mert Biyikli} @@ -202,10 +203,10 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Bleibt zu zeigen: char$K \not\in \{2, \ldots, n+1\} \iff k+1 \neq 0$ $\forall k \in \{0, \ldots, n\} $. \begin{align*} - &k + 1 \neq 0 \\ - \stackrel{k \ge 0}{\iff} & k + 1 \neq \text{char}K \\ - \stackrel{1 \le k + 1 \le n + 1}\iff & \text{char}K = 0 \lor \text{char}K > n + 1 \\ - \iff & \text{char}K \not\in \{2, \ldots, n+1\} + &k + 1 \neq 0 \\[-2mm] + \stackrel{\mathclap{\strut k \ge 0}}{\qquad \iff \qquad} &k + 1 \neq \text{char}K \\[-2mm] + \stackrel{\mathclap{\strut 1 \le k + 1 \le n + 1}}{\qquad \iff \qquad} &\text{char}K = 0 \lor \text{char}K > n + 1 \\[1mm] + \qquad \iff \qquad & \text{char}K \not\in \{2, \ldots, n+1\} .\end{align*} \end{proof} \item Bestimmen Sie $\psi(\text{ker }K) \subset K^{n+2}$. @@ -239,7 +240,7 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Damit folgt: \[ \psi(\text{ker }\partial) = - \{(a, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid c \in K\} + \{(a, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid a \in K\} .\] \item $\text{char }K \in \{2, \ldots, n+1\} $. Dann gilt für $k = \text{char }K-1$: \[ @@ -321,12 +322,12 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. \begin{align*} (*(f_1 + f_2))(\varphi) &= ((f_1 + f_2)^{*})(\varphi) = \varphi \circ (f_1 + f_2) - = \varphi \circ f_1 + \varphi \circ f_2 + \stackrel{\varphi \text{ linear}}{=} \varphi \circ f_1 + \varphi \circ f_2 = (*(f_1))(\varphi) + (*(f_2))(\varphi) \\ (*(a f_1))(\varphi) &= ((a f_1)^{*})(\varphi) = \varphi \circ (a f_1) - = a (\varphi \circ f_1) + \stackrel{\varphi \text{ linear}}{=} a \cdot (\varphi \circ f_1) = a\cdot (*(f_1))(\varphi) .\end{align*} \end{proof} From 482802ee2a15d9c7b98a72e2935dc216c1221888 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 20 Nov 2019 20:29:11 +0100 Subject: [PATCH 5/7] update la5 --- ws2019/la/uebungen/la5.pdf | Bin 104841 -> 104849 bytes ws2019/la/uebungen/la5.tex | 9 +++++---- 2 files changed, 5 insertions(+), 4 deletions(-) diff --git a/ws2019/la/uebungen/la5.pdf b/ws2019/la/uebungen/la5.pdf index 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zTH$mZoT-QAB%EBWg_cG*7l+nTI9~y6`o9bJp#2kce1)X?E`EVaVYr;<=a(hMiwWjB zV}31c?)Dy^CYt-$?b9T4eYbqdHj6y*X|h@LsZUeP5Z2{1ScB_%~qMhYUF;Zpzr diff --git a/ws2019/la/uebungen/la5.tex b/ws2019/la/uebungen/la5.tex index e7ccc00..0278f3e 100644 --- a/ws2019/la/uebungen/la5.tex +++ b/ws2019/la/uebungen/la5.tex @@ -231,7 +231,7 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. Aus (c) folgt: $k+1 \neq 0 \iff \text{char K} \not\in \{2, \ldots, n+1\} $. \begin{enumerate}[(i)] - \item $\text{char }K \not\in \{2, \ldots, n+1\} $. Dann ist $k + 1 \neq 0$, d.h. + \item $\text{char }K \not\in \{2, \ldots, n+1\} $: Dann ist $k + 1 \neq 0$, d.h. \begin{align*} &f(k+1) = 0 \text{ } \forall k \in \{0, 1, \ldots, n\} \\ \implies &f(k) = 0 \text{ } \forall k \in \{1, \ldots, n+1\} \\ @@ -242,7 +242,7 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. \psi(\text{ker }\partial) = \{(a, \underbrace{0, \ldots, 0}_{n+1\text{-mal}}) \mid a \in K\} .\] - \item $\text{char }K \in \{2, \ldots, n+1\} $. Dann gilt für $k = \text{char }K-1$: + \item $\text{char }K \in \{2, \ldots, n+1\} $: Dann gilt für $k = \text{char }K-1$: \[ k + 1 = \text{char } K - 1 + 1 = \text{char } K = 0_K .\] @@ -346,9 +346,10 @@ $V = \text{Abb}\left( \{0, 1, \ldots, n+1\}, K\right)$. .\end{align*} Wegen $f$ surjektiv gilt: $V = f(U)$ und damit: \begin{align*} - \forall v \in V\colon \varphi_1(v) = \varphi_2(v) + & \forall v \in V\colon \varphi_1(v) = \varphi_2(v) .\end{align*} - $\implies \varphi_1 = \varphi_2$ + $\implies \varphi_1 = \varphi_2$ \\ + $\implies f^{*}$ injektiv \end{proof} \end{enumerate} \end{aufgabe} From 02e93ac868687f82fe10aa5b1d83481555ee833f Mon Sep 17 00:00:00 2001 From: flavis Date: Thu, 21 Nov 2019 08:33:06 +0100 Subject: [PATCH 6/7] add wohldefiniertheits --- ws2019/la/uebungen/la5.pdf | Bin 104849 -> 105514 bytes ws2019/la/uebungen/la5.tex | 13 ++++++++++--- 2 files changed, 10 insertions(+), 3 deletions(-) diff --git a/ws2019/la/uebungen/la5.pdf b/ws2019/la/uebungen/la5.pdf index 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