{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "Script" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 258 "Script" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -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 "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 "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 1 2 0 1 }{PSTYLE "Normal" -1 259 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" -1 260 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 2 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" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 9 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 "Normal" -1 264 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 "Normal" -1 265 1 {CSTYLE "" -1 -1 "T imes" 1 9 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 18 "Zu Aufgabe (27/28)" }}{PARA 0 "" 0 "" {TEXT -1 16 "Bitte \+ beachten: " }}{PARA 0 "" 0 "" {TEXT -1 98 " -- die griechisch en Buchstaben sehen hier zum Teil anders aus als auf dem Aufgabenblatt ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 " \+ -- die zugeh\366rige mws-Datei ist " }{URLLINK 17 "hier" 4 "Aufg abe_27_28.mws" "" }{TEXT -1 12 " hinterlegt." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with (LinearAlgebra):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "phi:=2*x1^2+x2^2-x3^2-x4^2+c ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Die Matrix dieses quadratisc hen Polynoms ist" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "A:=Matrix([[co eff(phi,x1,2),0,0,0],[0,coeff(phi,x2,2),0,0],[0,0,coeff(phi,x3,2),0],[ 0,0,0,coeff(phi,x4,2)]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lambda:=x1-x2-x4-1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "mu:=x1+x 2-x3-1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Entsprechende Normalen vektoren sind" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B:=u->:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B(lambda),B(m u);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Nicht gefragt ist die folg ende Bestimmung eines Polynoms, das sich bei Elimination von " } {XPPEDIT 18 0 "x3,x4" "6$%#x3G%#x4G" }{TEXT -1 24 " mit Hilfe der Poly nome " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 5 " und " } {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 8 " ergibt:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "sx4:=solve(lambda,x4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sx3:=solve(subs(x4=sx4,mu),x3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(x4=sx4,x3=sx3,phi);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "xi:=expand(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Die Nullstellenmenge von " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 5 " in " }{XPPEDIT 18 0 "R^2" "6#*$%\"RG\"\"#" }{TEXT -1 14 " sieh t so aus:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plots[implicit plot](xi,x1=-100..100,x2=-100..100,numpoints=10000,scaling=constrained ,axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Die Nullstellenm enge von " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 5 " in " } {XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 23 " sieht z.B. so aus (in " }{XPPEDIT 18 0 "R^4" "6#*$%\"RG\"\"%" }{TEXT -1 50 " kann man e s sich leider nicht mit Paple ansehen):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "plots[implicitplot3d](xi,x1=-100..100,x2=-100..100,x 3=-100..100,numpoints=10000,scaling=constrained,axes=boxed);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Das Polynom " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 46 " ist aber nicht das in der Aufgabe gesuchte! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Warum?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Orthonormale Basi serg\344nzung zum Orthogonalraum zu " }{XPPEDIT 18 0 "" "6#-%$<,>G 6#%#BBG" }{TEXT -1 80 ". Dadurch erh\344lt man eine orthogonale Matrix f\374r die gesuchte lineare Abbildung " }{TEXT 257 1 "l" }{TEXT 258 2 " " }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "BB:=;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "N:=NullSpace(Transp ose(BB));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "NB:=[N[1],N[2],B(lambd a),B(mu)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "P:=convert(GramSchmid t(NB,conjugate=false,normalized),Matrix);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "P1:=Transpose(P):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sqrt(6)*P;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Pro ben:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Multiply(P1,P),Determinant( P);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "gemeinsamer St\374tzvektor f\374r die durch " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 5 " und " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 23 " gegebenen Hypereb enen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "v:=<1,-1,-1,1>;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "#v:=<2,0,2,0>;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fl:=u->subs (seq(x||k=u[k],k=1..4),lambda):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " fm:=u->subs(seq(x||k=u[k],k=1..4),mu):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fl(v),fm(v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Bestimmun g der Transformation " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 16 "1 des Polynoms " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 2 " :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y:=:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "AA:=P1.A.P;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "phi||1:=sort(expand(Transpose(P.y+v).A.(P.y+v)+c),[y1 ,y2,y3,y4],plex);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "bb:=2* P1.A.v;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "cc:=Transpose(v) .A.v+c;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "expand(phi||1-(Transpose(y).AA.y+Transpose(y).bb +cc));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "lambda||1:=collec t(Transpose(B(lambda)).(P.y+v)+tcoeff(lambda),[y1,y2,y3,y4]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "mu||1:=collect(Transpose(B(mu)).(P. y+v)+tcoeff(mu),[y1,y2,y3,y4]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sy3:=solve(lambda||1,y3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sy4:=solve(mu||1,y4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "psi:=sort(simplify(subs(y3=sy3,y4=sy4,phi||1)),[y1,y2,y3,y4],plex) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "So sieht die Kurve aus:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "plots[implicitplot](psi,y1=-100..10 0,y2=-100..100,numpoints=10000,scaling=constrained,axes=boxed);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "So sieht die urspr\374ngliche Fl \344che (\"" }{XPPEDIT 18 0 "2*x1^2+x2^2-x3^2-x4^2=0" "6#/,**&\"\"#\" \"\"*$%#x1GF&F'F'*$%#x2GF&F'*$%#x3GF&!\"\"*$%#x4GF&F.\"\"!" }{TEXT -1 30 "\") in der Hyperebene x4=1 aus:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "plots[implicitplot3d](subs(x4=1,phi),x1=-10..10,x2=-10..10,x3=-10 ..10,numpoints=10000,axes=boxed,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Sieht sie wirklich so aus, oder hat impl icitplot zu ungenau gerechnet in der N\344he des Nullpunktes, was durc haus vorkommen kann ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Wie sieht der Schnitt mit der Ebene x3=0 aus ?" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(x4=1,x3=0,phi);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 99 "plots[implicitplot](subs(x4=1,x3=0,phi),x1=-10 ..10,x2=-10..10,numpoints=50000,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "5 1" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }