DERIVE for Windows version 5.02 DfW file saved on 24 Nov 2002 A:=[2, 0; 0, -2] Bild:=[2, -2; 4, -2; 4, -6; 6, -6; 6, -8; 4, -8; 4, -10; 8, -10; 8, -12; 2, -12; 2, -2] Orig:=[1, 1; 2, 1; 2, 3; 3, 3; 3, 4; 2, 4; 2, 5; 4, 5; 4, 6; 1, 6; 1, 1] hCross:=APPROX(5892857142857143/1250000000000000) o:=[1, 1; 2, 1; 2, 3; 3, 3; 3, 4; 2, 4; 2, 5; 4, 5; 4, 6; 1, 6; 1, 1] vCross:=APPROX(12210526315789473/1000000000000000)   CTextObj фV Є{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs28\cf0\b Affine Abbildungen mit Hilfe von Matrizen\plain\f6\fs26\cf0 \par \plain\f6\fs22\cf0 \par \plain\f6\fs22\cf0\b Eingabe der Originalkoordinaten: \plain\f6\fs22\cf0 Die abzubildende Figur - ein gro\'dfes F - geben wir als nx2 - Matrix ein. Die Punktkoordinaten werden also \plain\f6\fs22\cf0\i zeilenweise\plain\f6\fs22\cf0 eingegeben. Nur in dieser Darstellung kann DERIVE die Punkte direkt, also ohne Transponieren, plotten. \par }   CExpnObj8bx^BenutzerЁ┐C[[1,1],[2,1],[2,3],[3,3],[3,4],[2,4],[2,5],[4,5],[4,6],[1,6],[1,1]]АjфР ╜{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs22\cf0\b Benennung der Figur: \plain\f6\fs22\cf0 Es ist hilfreich, die Figur mit einem Namen zu versehen. Gib in der Eingabezeile \plain\f6\fs26\cf0\b Orig:=\plain\f6\fs22\cf0\b \plain\f6\fs22\cf0 ein und f\'fcge die #-Nummer des zu benennenden Ausdrucks an. Derive gibt als neue Ausgabe die mit \plain\f6\fs26\cf0\b Orig\plain\f6\fs22\cf0 benannte Matrix wieder. \plain\f4\fs22\cf0 \par } А8Ь╕ШBenutzerЁ┐IOrig:=[[1,1],[2,1],[2,3],[3,3],[3,4],[2,4],[2,5],[4,5],[4,6],[1,6],[1,1]]Адф  O{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs22\cf0\b Plotten der Originalfigur:\plain\f6\fs22\cf0 Markiere zun\'e4chst die Originalmatrix. \'d6ffne anschlie\'dfend mit dem Schlangen-Icon in der Symbolleiste das 2DGraphikFenster. Im \plain\f6\fs22\cf0\i Graphikfenster\plain\f6\fs22\cf0 bedeutet das Schlangen-Icon "Ausdruck zeichenen" und es bewirkt genau dieses. Unter dem Menuepunkt "Einstellen - Zeichenbereich" k\'f6nnen x- und y-Achsenabschnitte so gew\'e4hlt werden, dass sowohl das Original als auch die sp\'e4teren Bilder gut sichtbar sind. \par \par Der Menuepunkt "Datei - Einbetten" f\'fcgt das nachfolgende Bild direkt in das Algebra-Blatt ein. Sinnvoll ist u.U. die Fenstereinstellung "Fenster - Vertikal anordnen". Dies bringt den ausgw\'e4hlten Ausdruck und das zugeh\'f6rige Bild gemeinsam auf den BIldschirm. \par }   CPlotObj ▀ў   C2DPlotView  CPointListPlot   Ё? Ё? @ Ё? @ @ @ @ @ @ @ @ @ @ @ @ @ @ Ё? @ Ё? Ё?xy    @@ДB!Да┐|Ё┴Я?11Ё?Ё?BM╢,6(╙рАт                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           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MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs22\cf0\b Editieren der Originalfigur (Originalkoordinaten):\plain\f6\fs22\cf0 Eingabefehler oder ver\'e4nderte Fragestellungen machen es bisweilen n\'f6tig, die Originalmatrix zu bearbeiten. Die folgenden Situationen treten \'f6fter auf: \par \par \plain\f6\fs22\cf0\i 1) \'c4ndern eines Matrixelements [\'c4ndern einer Punktkoordinate]:\plain\f6\fs22\cf0 Klicke die Matrix an, klicke die Zeile an, klicke das Element an. Rechts-Klick auf das markierte Element f\'fcgt den aktuellen Wert in die Eingabezeile ein. \'c4ndere es dort und best\'e4tige. Dann wird das alte Elemet mit dem neuen \'fcberschrieben. \par \par \plain\f6\fs22\cf0\i 2) Einf\'fcgen einer Matrixzeile [Einf\'fcgen eine weiteren Punktes "mitten" im Polygonzug]:\plain\f6\fs22\cf0 Markiere die Matrix. Der Menuepunkt "Bearbeiten - Einf\'fcgen" bringt die Matrix in die Zwischenablage, von dort kann man mittels "Bearbeiten - Einf\'fcgen" die Werte in die Eingabezeile einf\'fcgen. Nun kann an der passenden Stelle eine zus\'e4tzliche Zeile eingegeben werden. Leider schreibt DERIVE den korrigierten Ausdruck ganz ans \plain\f6\fs22\cf0\i Ende\plain\f6\fs22\cf0 des Blattes. Mittels Drag&Drop kann man ihn allerdings bequem nach oben an die passende Stelle ziehen und anschlie\'dfend den alten Ausdruck l\'f6schen. \par \par \plain\f6\fs22\cf0\i 3) Anh\'e4ngen einer Matrixzeile:\plain\f6\fs22\cf0 Entweder wie in 1) oder wie in 2). Bei 1) wird ein Zwischenmenue ge\'f6ffnet. Dort kann die Zeilenzahl erh\'f6ht werden. Die neuen Zeilen werden unten angeh\'e4ngt. Die korrigierte Matrix \'fcberschreibt die alte. Bei 2) wird die korrigierte Matrix am Ende des Blattes ausgegeben. \par \par \plain\f6\fs22\cf0\b Eingabe der Abbildungsmatrix A:\plain\f6\fs22\cf0 Unten das zweischrittige Verfahren bei der Namensgebung. Wer sich an die Syntax schnell gew\'f6hnt, kann auch direkt in die Eingabezeile schreiben:\plain\f6\fs22\cf0\b A:=[2,0;0,-2] \par } А8А5BenutzerЁ┐[[2,0],[0,-2]]А8AиeBenutzerЁ┐A:=[[2,0],[0,-2]]Аqфи ┌{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs22\cf0\b Berechnung des Bildes:\plain\f6\fs22\cf0 Da \plain\f6\fs26\cf0\b A*Orig\plain\f6\fs22\cf0 nicht definiert ist, kann das Ergebnis nicht das gew\'fcnschte Bild sein. Koordinatenweise Neuberechnung gelingt nur, wenn \plain\f6\fs26\cf0\b Orig\plain\f6\fs22\cf0 vorher transponiert wird. Der Befehl dazu ist das \plain\f6\fs22\cf0\b Strg+Accent - Zeichen\plain\f6\fs22\cf0 . Sinnigerweise muss erst die Leertaste gedr\'fcckt werden, bevor der Befehl vollst\'e4ndig ist. DERIVE gibt nur den neuen Namen aus, also \plain\f6\fs26\cf0\b Orig`\plain\f6\fs22\cf0 . Das Transponieren erfolgt erst auf Befehl: "Vereinfachen - Algebraisch" oder "=" neben der Eingabezeile. \par } А8┤`└BenutzerЁ┐Orig`А╠ЁSimp(#5)1[[1,2,2,3,3,2,2,4,4,1,1],[1,1,3,3,4,4,5,5,6,6,1]]А№ф {\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs22\cf0 Man kann auch sofort den kompletten Rechenbefehl eingeben: \plain\f6\fs26\cf0\b Bild:=A*Orig`\plain\f6\fs22\cf0 . Dann erscheint wieder zun\'e4chst der Text. Die Rechnung wird mit "=" befohlen. \par } А8░'BenutzerЁ┐ Bild:=A*Orig`А└3PWSimp(Benutzer)№йё╥MbP?@[[2,4,4,6,6,4,4,8,8,2,2],[-2,-2,-6,-6,-8,-8,-10,-10,-12,-12,-2]]Аcфt я{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs22\cf0 Nun muss - als Vorbereitung f\'fcr das Plotten - erneut transponiert werden. Das geht auch in einem Komplettbefehl. Der macht die obigen Zeilen #6 bis #8 \'fcberfl\'fcssig. \par } А8А╚МBenutzerЁ┐ Bild:=(A*Orig`)`А`Ш░ФSimp(Benutzer)№йё╥MbP? R[[2,-2],[4,-2],[4,-6],[6,-6],[6,-8],[4,-8],[4,-10],[8,-10],[8,-12],[2,-12],[2,-2]]Ааф g{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs22\cf0\b Plotten der Bildfigur:\plain\f6\fs22\cf0 Ordne das Algebra- und das 2DGraphikfenster vertikal an, markiere die Bildmatrix, wechsle ins Graphikfenster und dr\'fccke das Schlangen - Icon. Zu dem schon vorhandenen Original wird das Bild gezeichnet. \par \par Ist das Original verschwunden, kann man im Algebra - Fenster erst das Original und dann unter Dr\'fccken der STRG - Taste zus\'e4tzlich das Bild markieren. Im Graphikfenster wird mittels Schlangen - Icon beides auf einmal gezeichnet. \par \par "Einbetten" liefert das folgenden Bild: \par } Аф$─ А А   @ └ @ └ @ └ @ └ @  └ @  └ @ $└  @ $└  @ (└ @ (└ @ └  А   Ё? Ё? @ Ё? @ @ @ @ @ @ @ @ @ @ @ @ @ @ Ё? @ Ё? Ё?xy    @@AA└┐)\П┬ї(└ 11Ё?Ё?BM╢#6($а┌                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   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 f{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f6\fs22\cf0\b Variieren der Abbildungsmatrix A:\plain\f6\fs22\cf0 Mittels "Kopieren" und "Einf\'fcgen" (im Menue Bearbeiten \plain\f6\fs22\cf0\i oder\plain\f6\fs22\cf0 durch \'fcber rechte Maustaste \plain\f6\fs22\cf0\i oder\plain\f6\fs22\cf0 \'fcber strg+c und strg+v) kann man \plain\f6\fs26\cf0\b A\plain\f6\fs22\cf0 mit ihren Elementen und \plain\f6\fs26\cf0\b Bild\plain\f6\fs22\cf0 samt Befehl gemeinsam ans Ende der Datei holen. Editiere \plain\f6\fs22\cf0\b A\plain\f6\fs22\cf0 wie oben beschrieben, markiere \plain\f6\fs22\cf0\b Bild\plain\f6\fs22\cf0 -Gleichung, dr\'fccke F3 (der markierte Ausdruck wandert in die Eingabezeile) und dr\'fccke "=": Die neuen Koordinaten werden berechnet und k\'f6nnen gezeichnet werden. F\'fcr weiter Ver\'e4nderungen l\'f6sche zun\'e4chst den eben berechneten Wert. \par } А8 и7 BenutzerЁ┐ A:=[[2,1],[0,-4]]А8C ╚O BenutzerЁ┐ Bild:=(A*Orig`)`А`[ ░W Simp(Benutzer) R[[2,-2],[4,-2],[4,-6],[6,-6],[6,-8],[4,-8],[4,-10],[8,-10],[8,-12],[2,-12],[2,-2]]А c ▀G  А А   Ё? Ё? @ Ё? @ @ @ @ @ @ @ @ @ @ @ @ @ @ Ё? @ Ё? Ё? А   @ └ @ └ @ └ @ └ @  └ @  └ @ $└  @ $└  @ (└ @ (└ @ └ xy   @@@└╖m█╢m█@╝ЖЄ╩k(@11Ё?Ё?BM36(╥фv                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   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фv  j{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 News Gothic MT;}{\f5\fswiss\fcharset1 Arial;}{\f6\fswiss\fprq2\fcharset1 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1031\pard\plain\f5\fs22\b Erste Anwendungen: Spiegelungs- und Drehmatrizen.\plain\f5\fs22 Elementare Achsenspiegelungen sowie Drehungen um den Ursprung lassen sich direkt als Matrizenmultiplikation beschreiben. 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