{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "MS Sans Serif" 1 8 128 0 128 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Roter Text" -1 256 "Tahoma" 0 0 255 0 0 1 2 1 1 0 0 2 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 9 128 0 128 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE " " -1 -1 "Courier" 1 8 0 128 128 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Seitenumbruch" -1 258 1 {CSTYLE "" -1 -1 "Time s" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 1 2 0 1 } {PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 266 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }2 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 266 "" 0 "" {TEXT -1 21 "Modul Lineare Algebra" } }{PARA 266 "" 0 "" {TEXT -1 12 "WS 2005/2006" }}{PARA 266 "" 0 "" {TEXT -1 7 "Schmale" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 51 " I : Be ispiel zum Berechnen einer inversen Matrix" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "restart:with(linalg):" }{TEXT -1 165 "(dient dem La den des Paketes \"linalg\" zur lineaen Algebra, in Maple gibt es auch \+ noch das Paket \"LinearAlgebra\", das st\344rker auf numerische Fragen hin angelegt ist.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Gegeben se i die folgende Matrix mit rationalen Eintr\344gen:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "n:=4;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=matri x(n,n,[seq(op([seq(k+l^k,k=1..n-1),a[l]" }{TEXT -1 0 "" }{MPLTEXT 1 0 13 "]),l=1..n)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Aufgabe: bes timme alle " }{XPPEDIT 18 0 "a[1], `...`, a[n]" "6%&%\"aG6#\"\"\"%$... G&F$6#%\"nG" }{TEXT -1 33 " , f\374r die A invertierbar ist. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Berechnun g einer ZSF," }}{PARA 0 "" 0 "" {TEXT -1 13 "schrittweise:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B:=addrow(A,1,2,-3/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B:=addrow(B,1,3,-4/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B:=addrow(B,1,4,-5/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "etwas automatisiert:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 81 "B:=evalm(A):\nfor k from 2 to n do \nB:=addr ow(B,1,k,-B[k,1]/B[1,1]) \nod:\nevalm(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "und jetzt gleich f\374r alle Spalten (unter der Annahme, dass stets der neue Diagonaleintrag ungleich 0 ist. Wenn dies nicht z utrifft tritt Division durch 0 auf ! ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "B:=evalm(A):\nfor l to n-1 do \nfor k from l+1 to n do \nB:=addrow(B,l,k,-B[k,l]/B[l,l]) \nod:\nod:\nB:=evalm(B);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "An der Matrix B erkennt man, dass die reduzierte Zeilenstufenform genau dann zur Einheitsmatrix wird, \+ " }}{PARA 0 "" 0 "" {TEXT 258 4 "wenn" }{TEXT -1 11 " (hier " } {TEXT 257 8 "f\374r n=4 " }{TEXT -1 17 "angegeben) " }{XPPEDIT 18 0 "-35*a[1]+54*a[2]-33*a[3]+8*a[4];" "6#,**&\"#N\"\"\"&%\"aG6#F&F&! \"\"*&\"#aF&&F(6#\"\"#F&F&*&\"#LF&&F(6#\"\"$F&F**&\"\")F&&F(6#\"\"%F&F &" }{TEXT -1 9 " " }{TEXT 259 8 "nicht 0" }{TEXT -1 8 " ist. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Sei \+ " }{XPPEDIT 18 0 "a = matrix([[a[1]], [a[2]], [a[3]], [a[4]]]);" "6# /%\"aG-%'matrixG6#7&7#&F$6#\"\"\"7#&F$6#\"\"#7#&F$6#\"\"$7#&F$6#\"\"% " }{TEXT -1 121 " . Wenn wir L:=L\366s([-35,54,-33,8],0) bestimmen, d ann wissen wir, A ist genau dann invertierbar, wenn a nicht aus L ist ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Man \+ sieht direkt, dass L=" }{TEXT 268 2 "\{ " }{XPPEDIT 18 0 "54/35*a[2]-3 3/35*a[3]+8/35*a[4];" "6#,(*(\"#a\"\"\"\"#N!\"\"&%\"aG6#\"\"#F&F&*(\"# LF&F'F(&F*6#\"\"$F&F(*(\"\")F&F'F(&F*6#\"\"%F&F&" }{TEXT -1 9 " : \+ " }{XPPEDIT 18 0 "a[2],a[3],a[4];" "6%&%\"aG6#\"\"#&F$6#\"\"$&F$6#\" \"%" }{TEXT -1 29 " beliebige rationale Zahlen " }{TEXT 269 1 "\}" } {TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Nun wissen wir zwa r, wann A invertierbar ist, aber kennen noch nicht die inverse Matrix ." }}{PARA 0 "" 0 "" {TEXT -1 105 "Um diese zu erhalten m\374ssen wir bei der Rechnung die Zeilenumformungen in einem Speicherblock speiche rn:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "AE:=concat(A,array(1..n,1..n,identity));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Wir erhalten zun\344chst bei Wiederholung der obigen Rechnung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "B: =evalm(AE):\nfor l to n-1 do \nfor k from l+1 to n do \nB:=addrow(B,l ,k,-B[k,l]/B[l,l]) \nod:\nod:\nB:=evalm(B);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 111 "Wir nehmen nun an, dass der nxn-Eintrag nicht null ist und rechnen weiter bis zur reduzierten Zeilenstufenform:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "z.B. indem wir zu erst normieren:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for k to n do B: =mulrow(B,k,1/B[k,k]); od:evalm(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "und dann \374ber den 1-en ausr\344umen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "for k from 2 to n do for l to k-1 do B:=addrow(B,k,l, -B[l,k]/B[k,k]);od;od:evalm(map(simplify,B));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Nat\374rlich kann man auch einfach Maple fragen" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(A^(-1));" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "und dann erkennt man, \+ dass Maple's Ergebnis nur sinnvoll ist, wenn die Nenner nicht verschwi nden und das f\374hrt genau auf unser obiges Ergebnis." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 "II : Beispiel zum L \366sen linearer Gleichungssysteme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 " (a): zum Fa ll (a) in Satz 4.6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restar t:with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "A:=matri x(4,7,[1,2,3,4,5,6,7,2,3,4,5,6,7,1,3,4,5,6,7,1,2,4,5,6,7,1,2,3]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 7 "Fr age: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 99 "F\374r welche b ist das Gleichungssytem Ax=b l\366sbar und was ist da nn jeweils die genaue L\366sungsmenge ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Wir bilden die erweiterte Ma trix mit einer unbestimmten rechten Seite b:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Ab:=concat(A,matrix(4,1,[b[1],b[2],b[3],b[4]])); " }}{PARA 0 "" 0 "" {TEXT -1 41 "schrittweise und diesmal mit Abk\374r zungen:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Z3:=addrow; Z1:=mulrow;Z4:=swaprow;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bd:=Z3(Ab,1,2,-2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "Bd:=Z3(Bd,1,3,-3);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "Bd:=Z3(Bd,1,4,-4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Bd:=Z1(Bd,2,-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Bd:=Z3(Bd,2,3,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Bd:=Z3(Bd,2,4,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Bd:=Z4(Bd,3,4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bd:=Z1(Bd,3,-1/7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bd:=Z1(Bd,4,-1/7);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Bd ist nun immerhin schon in Zeilenstufenform und wird nun noch reduziert:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bd:=Z3(Bd,2, 1,-2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bd:=Z3(Bd,3,2,-4) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Bd:=Z3(Bd,3,1,3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bd:=Z3(Bd,4,3,-1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bd:=Z3(Bd,4,2,-1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Bd:=Z3(Bd,4,1,1);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Nun ist Bd in reduzierter Zeilenst ufenform und es sind" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "B=submatrix (Bd,1..4,1..7),d=submatrix(Bd,1..4,8..8);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 36 "Wegen Satz 4.3 ist L\366 s(A,b)=L\366s(B,d)" }{TEXT -1 1 " " }{TEXT 262 52 ", denn es wurden nu r Zeilenumformmungen vorgenommen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "Nach Sat z 4.6. (a) ist das LGS \"Bx=d\" unabh\344ngig von d l\366sbar, bzw. is t L\366s(Bd,d) nicht leer f\374r beliebige d und somit ist auch L\366 s(A,b) nicht leer f\374r beliebige b." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Wir k\366 nnen nun die L\366sungsmenge nach Satz 4.6 direkt angeben:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 18 "L\366s(A,b)=L \366s(B,d)=" }{TEXT -1 3 " " }{TEXT 265 1 "\{" }{TEXT -1 3 " " } {XPPEDIT 18 0 "matrix([[-4*b[1]+25/7*b[2]-1/7*b[3]-3/7*b[4]], [23/7*b[ 1]-3*b[2]+1/7*b[3]+4/7*b[4]], [-1/7*b[1]+1/7*b[2]+1/7*b[3]-1/7*b[4]], \+ [-1/7*b[1]+2/7*b[2]-1/7*b[3]]]);" "6#-%'matrixG6#7&7#,**&\"\"%\"\"\"&% \"bG6#F+F+!\"\"*(\"#DF+\"\"(F/&F-6#\"\"#F+F+*(F+F+F2F/&F-6#\"\"$F+F/*( F9F+F2F/&F-6#F*F+F/7#,**(\"#BF+F2F/&F-6#F+F+F+*&F9F+&F-6#F5F+F/*(F+F+F 2F/&F-6#F9F+F+*(F*F+F2F/&F-6#F*F+F+7#,**(F+F+F2F/&F-6#F+F+F/*(F+F+F2F/ &F-6#F5F+F+*(F+F+F2F/&F-6#F9F+F+*(F+F+F2F/&F-6#F*F+F/7#,(*(F+F+F2F/&F- 6#F+F+F/*(F5F+F2F/&F-6#F5F+F+*(F+F+F2F/&F-6#F9F+F/" }{TEXT -1 12 " \+ + " }{XPPEDIT 18 0 "matrix([[v[3]+2*v[4]+26*v[7]], [-2*v[3]-3*v[ 4]-22*v[7]], [v[7]], [v[7]]]);" "6#-%'matrixG6#7&7#,(&%\"vG6#\"\"$\"\" \"*&\"\"#F-&F*6#\"\"%F-F-*&\"#EF-&F*6#\"\"(F-F-7#,(*&F/F-&F*6#F,F-!\" \"*&F,F-&F*6#F2F-F=*&\"#AF-&F*6#F7F-F=7#&F*6#F77#&F*6#F7" }{TEXT -1 16 " : " }{XPPEDIT 18 0 "v[3],v[4],v[7];" "6%&%\"vG6#\"\" $&F$6#\"\"%&F$6#\"\"(" }{TEXT -1 42 " aus dem K\366rper der rational en Zahlen " }{TEXT 264 2 "\} " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 30 " (b): zum Fall (b) in Satz 4.6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 51 "Wir betrachten die Transponierte der Matrix aus (a)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "A:=transpose(matrix(4,7,[1, 2,3,4,5,6,7,2,3,4,5,6,7,1,3,4,5,6,7,1,2,4,5,6,7,1,2,3]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 23 "und ste llen die Frage: " }}{PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 " " {TEXT -1 51 "F\374r welche b ist das Gleichungssytem Ax=b l\366sbar \+ ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Wir bilden wieder die erweiterte Matrix mit einer unbestimmten \+ rechten Seite b" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Ab:=concat(A,mat rix(7,1,[seq(b[k],k=1..7)]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 " und benutzen wieder Abk\374rzungen" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Z3:=addrow; Z1:=mulrow;Z4:=swaprow;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 " rechnen aber jetzt teilautomatisiert:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Bd:=evalm(Ab):\nfor k from 2 to 7 do \nBd :=addrow(Bd,1,k,-Bd[k,1]/Bd[1,1]) \nod:\nevalm(Bd);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "for k from 3 to 7 do \nBd:=addrow(Bd,2,k, -Bd[k,2]/Bd[2,2]) \nod:\nevalm(Bd);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Bd:=Z4(Bd,3,7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Bd:=Z4(Bd,6,4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Bd:=Z3(Bd,3,4,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Bd:=Z3(Bd,4,5,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Bd:=Z1(Z1(Bd,3,1/7),4,1/7);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 54 "Nun ist Bd in reduzierter Zeilenstufenform und es sind " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "B=submatrix(Bd,1..7,1..4),d=sub matrix(Bd,1..7,5..5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Da wiede r nur Zeilenumformungen vorgenommen wurden, ist L\366s(A,b)=L\366s(B,d )," }}{PARA 0 "" 0 "" {TEXT -1 78 "und nach Satz 4.6 ist L\366s(A ,b)=L\366s(B,d) nicht leer genau dann wenn " }{XPPEDIT 18 0 "d[5 ] = 0,d[6] = 0,d[7] = 0;" "6%/&%\"dG6#\"\"&\"\"!/&F%6#\"\"'F(/&F%6#\" \"(F(" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 55 "Dies f\374hrt auf ein lineares Gleichungssystem f\374r die " }{XPPEDIT 18 0 "b[i] ;" "6#&%\"bG6#%\"iG" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 33 " Seine Koeffizientenmatrix lautet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "C:=matrix(3,7,[26,-22,0,0,1,1,1,26,-3,0,1,0,0,0,1,-2, 1,0,0,0,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "und es ist L\366 s(C,0) zu bestimmen mit denselben Methoden. Ist L\366s(C,0) bestimmt, \+ so hat man das Ergebnis: " }}{PARA 262 "" 0 "" {TEXT 267 116 "Das Glei chungssystem \"Ax=b\" ist genau dann l\366sbar bzw. L\366s(A,b) ist ge nau dann nicht leer, wenn b aus L\366s(C,0) ist. " }}{PARA 0 "" 0 "" {TEXT -1 100 "In den F\344llen, wo L\366sbarkeit vorliegt, erfolgt die Bestimmung der L\366sungsmenge direkt nach Satz 4.6." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 279 "Bei der Bestimmung vo n L\366s(C,0) machen wir es uns jetzt bequem und lassen Maple die redu zierte ZSF berechnen (das k\366nnen Sie sp\344ter auch so machen, aber f\374r die \334bungsaufgaben m\374ssen Sie die Matrizen schon mal sel bst anfassen, um zu lernen, wie Zeilenumformungen funktionieren.) " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Die erwei terte Matrix ist hier" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "C0 :=concat(C,matrix(3,1,0));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rref(C0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Zeilenumformunge n k\366nnen die letzte 0-Spalte nicht \344ndern Man kann sich bei homo genen Gleichungssystemen, die Erweiterung schenken." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Nun kann mit Satz 4.6 \+ direkt die L\366sungsmenge L\366s(C,0) angegeben werden. Es liegt der \+ Fall (a) vor und es sind alle " }{XPPEDIT 18 0 "d[i] = 0;" "6#/&%\"dG 6#%\"iG\"\"!" }{TEXT -1 4 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }