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Line \303\241ris algebra" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 39 "1.1. \+ M\303\241trixok \303\251s vektorok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" } }{SECT 0 {PARA 5 "" 0 "" {TEXT 207 12 "1.1.1. Test." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "1,2+2,3^2,3.45/3,Pi,3+4*I,x^2=-1,Alma,infinity, -infinity;" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"%\"\"*$\"++++]6!\"*I#PiG%*protectedG,&\"\"$F# *&F$F#^#F#F#F#/*$)I\"xG6\"\"\"#F#!\"\"I%AlmaGF3I)infinityGF*,$F7F5" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6,$\"\"\"\"\"!$\"\"%F%$\"\"*F%$\"++++]6!\"*$\"+aEfTJF,, &$\"\"$F%F$*&F&F$^#F$F$F$/*$)I\"xG6\"\"\"#F$$!\"\"F%I%AlmaGF8$F$I)infi nityG%*protectedG$F;F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "% %[5],%%[6];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$I#PiG%*protectedG,&\"\" $\"\"\"*&\"\"%F'^#F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 " ?Pi" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=50;" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S^P*Rp r>%)G]zKQVEYQKz*e`EfTJ!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=Alma; Alma:=1; A; Alma:='Alma'; \+ A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%AlmaG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%AlmaG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I% AlmaG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "whattype(1); wh attype(1/2); whattype(0.5); whattype(Alma);\n" }{MPLTEXT 1 0 47 "whatt ype(infinity);whattype(Pi);whattype(A); \n" }{MPLTEXT 1 0 64 "whattype (2); whattype(krikszkraksz); whattype(\"krikszkraksz\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "I(integerG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I)fractionG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 " I&floatG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I'symbolG%*prote ctedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I1extended_numericG%*protectedG " }}{PARA 11 "" 1 "" {XPPMATH 20 "I'symbolG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I'symbolG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I(integerG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I'symbolG% *protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I'stringG%*protectedG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "type(1,integer);type(1,floa t);type(1+3*I,complexcons);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG %*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 28 "z:=3+4*I;Re(z);Im(z);abs(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"$\"\"\"*&\"\"%F$^#F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(x^2=-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^#\"\"\",$F#!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(x^5+x+1=0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6',&#\"\"\"\"\"#!\"\"*&*&F$F%^#F%F%F%)\" \"$F$F%F%,&F$F'*&,$F)F%F%F+F%F',(*&#F%\"\"'F%),&\"$+\"F%*&\"#7F%)\"#pF $F%F%#F%F,F%F'*&#F&F,F%F4F'F'F;F%,**&#F%F8F%F4F%F%*&F;F%F4F'F%F;F%*(F) F%F+F%,&F1F'F " 0 "" {MPLTEXT 1 0 19 "solve(x^5+x^2+1=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'-I'RootOfG6$%*protectedGI(_syslibG6\"6$,(*$)I#_ZGF%\"\" &\"\"\"F/*$)F-\"\"#F/F/F/F//I&indexGF%F/-F$6$F*/F4F2-F$6$F*/F4\"\"$-F$ 6$F*/F4\"\"%-F$6$F*/F4F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve(x^7+x^6+x^5+x^4+x^3+2*x^2+x+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6),&#\"\"\"\"\"#!\"\"*&*&F$F%^#F%F%F%)\"\"$F$F%F%,&F$F'*&,$F)F%F%F+F %F'-I'RootOfG6$%*protectedGI(_syslibG6\"6$,(*$)I#_ZGF2\"\"&F%F%*$)F:F& F%F%F%F%/I&indexGF2F%-F16$F7/F?F&-F16$F7/F?F,-F16$F7/F?\"\"%-F16$F7/F? 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eGF$6&-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'- F56#-F86$Q\"2F'F;-F56#-F86$Q\"0F'F;-F26%F>-F56#-F86$Q\"3F'F;FC-F26%FJ- F56#-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/F " 0 "" {MPLTEXT 1 0 40 "transpose(c);d:=transpose(%);equal( c,d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I ,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6$-I$mtrGF$6%-I$mtd GF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"2F'F;-F56#- F86$Q\"3F'F;-F26%F>FC-F56#-F86$Q\"4F'F;/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6%Q\"dF'/%'italicGQ%trueF'/%,mathvariantGQ'it alicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/% )stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF =/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ \"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF $6$-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fao6$Q\"2F'F9-F[o6$Fco-F^o6#-Fao6$Q \"3F'F9-F[o6$Fjo-F^o6#-Fao6$Q\"4F'F9/%%openGQ\"[F'/%&closeGQ\"]F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "a:=matrix(3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$;F&\"\"#E\\[l!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "entermatrix(a);" }}{PARA 7 "" 1 "" {TEXT 208 33 "Warning, computation interrupted" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 43 "1.1. 6. M\305\261veletek m\303\241trixokkal." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a:=matrix([[1,0],[0,2 ]]); b:=matrix([[0,1],[0,0]]);\n" }{MPLTEXT 1 0 31 "c:=matrix([[1,0],[ 0,1],[0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"aF'/%'italicGQ%trueF'/%,m athvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/% *separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimit sGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspace GFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtab leGF$6$-I$mtrGF$6$-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fao6$Q\"0F'F9-F[o6$F co-F^o6#-Fao6$Q\"2F'F9/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6% -I#miGF$6%Q\"bF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q #:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*sy mmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF '/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ )infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6$-I$mtrGF$6$-I$mtdGF$6#-I# mnGF$6$Q\"0F'F9-F^o6#-Fao6$FTF9-F[o6$F]oF]o/%%openGQ\"[F'/%&closeGQ\"] F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Types ettingGI(_syslibGF'6%-I#miGF$6%Q\"cF'/%'italicGQ%trueF'/%,mathvariantG Q'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorG F=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'acce ntGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsi zeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$m trGF$6$-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fao6$Q\"0F'F9-F[o6$FcoF]o-F[o6$ FcoFco/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "d:=evalm(2*a+3*b); evalm(a&*b); evalm(b&*a); evalm(si n(d));\n" }{MPLTEXT 1 0 37 "evalm(d^2); evalm(c&*a); evalm(a&*c);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6%Q\"dF'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)s tretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\" 1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6$-I$mtrGF$6 $-I$mtdGF$6#-I#mnGF$6$Q\"2F'F9-F^o6#-Fao6$Q\"3F'F9-F[o6$-F^o6#-Fao6$Q \"0F'F9-F^o6#-Fao6$Q\"4F'F9/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sys libGF'6%-I%mrowGF$6#-I'mtableGF$6$-I$mtrGF$6$-I$mtdGF$6#-I#mnGF$6$Q\"0 F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"1F'F;-F26$F4F4/%%openGQ\"[F'/% &closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6$-I$mtrGF$ 6$-I$mtdGF$6#-I#mnGF$6$Q\"0F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"2F' F;-F26$F4F4/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowG F$6#-I'mtableGF$6$-I$mtrGF$6$-I$mtdGF$6#-F,6%-I#miGF$6%Q$sinF'/%'itali cGQ&falseF'/%,mathvariantGQ'normalF'-I#moGF$60Q0⁡F'F@/%& fenceGF?/%*separatorGF?/%)stretchyGF?/%*symmetricGF?/%(largeopGF?/%.mo vablelimitsGF?/%'accentGF?/%%formGQ&infixF'/%'lspaceGQ$0emF'/%'rspaceG FZ/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F#6#-F,6#-I#mnGF$6$Q\"2F'F@ -F56#-F,6%F9FC-F#6#-F,6#-Fbo6$Q\"3F'F@-F26$-F56#-Fbo6$Q\"0F'F@-F56#-F, 6%F9FC-F#6#-F,6#-Fbo6$Q\"4F'F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI( _syslibGF'6%-I%mrowGF$6#-I'mtableGF$6$-I$mtrGF$6$-I$mtdGF$6#-I#mnGF$6$ Q\"4F'/%,mathvariantGQ'normalF'-F56#-F86$Q#18F'F;-F26$-F56#-F86$Q\"0F' F;-F56#-F86$Q#16F'F;/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6$-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,ma thvariantGQ'normalF'-F56#-F86$Q\"0F'F;-F26$F>-F56#-F86$Q\"2F'F;-F26$F> F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 8 "" 1 "" {TEXT 209 78 "Error , (in linalg:-multiply) non matching dimensions for vector/matrix prod uct" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 36 "1.1.7. \303\201ll\303\255t\303\241s." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 75 "1.1.8. Nullm\303\241trix, egys\303\251gm\303\241tri x, inverz m\303\241trix." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a:=matrix(3,2,0); iszero(a); b:=mat rix(2,2,[1,0,0,0]); iszero(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mro wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"aF'/%'it alicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%& fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopG F=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmat hspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfenced GF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6$-I$mtdGF$6#-I#mnGF$6$Q\"0F'F9F]oFjn Fjn/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%tr ueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulena meG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"bF'/%'italicGQ%trueF'/% ,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF' /%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelim itsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspa ceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mt ableGF$6$-I$mtrGF$6$-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fao6$Q\"0F'F9-F[o6 $FcoFco/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "row dim(a); coldim(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a:=diag(3,4,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modu lenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"aF'/%'italicGQ%true F'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&fal seF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movabl elimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%' rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#- I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"3F'F9-F^o6#-Fao6$Q\"0F 'F9Fdo-F[o6%Fdo-F^o6#-Fao6$Q\"4F'F9Fdo-F[o6%FdoFdo-F^o6#-Fao6$Q\"5F'F9 /%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "krondelta:=(i,j)-> if i=j then 1 else 0 fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"iG6\"I\"jGF%F%6$I)operatorGF%I&arrowGF%F%@%/F$F&\" \"\"\"\"!F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a:=matri x(3,3,krondelta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"aF'/%'italicGQ%trueF' /%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&false F'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablel imitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rs paceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I' mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fao6$Q\"0F'F9Fco -F[o6%FcoF]oFco-F[o6%FcoFcoF]o/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "b:=matrix(3,2,krondelta);" } }{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesettin gGI(_syslibGF'6%-I#miGF$6%Q\"bF'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%) stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF= /%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ \"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF $6$-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fao6$Q\"0F'F9-F[o6$FcoF]o-F[o6$FcoF co/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "c:=hilbert(3); d:=inverse(c); evalm(c&*d); evalm(d&*c);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6%Q\"cF'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)s tretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\" 1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6 %-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linet hicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F ^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcp-F[o6%FcoFep-F^o6#-Ffo6(F`o-Fao 6$Q\"4F'F9F[pF^pFapFcp-F[o6%FepF^q-F^o6#-Ffo6(F`o-Fao6$Q\"5F'F9F[pF^pF apFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"dF'/ %'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF '/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(larg eopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thic kmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfe ncedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"9F'F9-F ^o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/FNQ$0emF'/FQQ 2verythinmathspaceF'FRFU-Fao6$Q#36F'F9-F^o6#-Fao6$Q#30F'F9-F[o6%Fdo-F^ o6#-Fao6$Q$192F'F9-F^o6#-F#6$Fho-Fao6$Q$180F'F9-F[o6%FdpF`q-F^o6#Fdq/% %openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfence dG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtable GF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F 56#-F86$Q\"0F'F;F>-F26%F>F4F>-F26%F>F>F4/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Types ettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#- I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"0F'F;F>-F26%F>F4F >-F26%F>F>F4/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(b);" }}{PARA 8 "" 1 "" {TEXT 209 53 "Error, \+ (in linalg:-inverse) expecting a square matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a:=matrix([[2,4],[1,2]]); inverse(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sy slibGF'6%-I#miGF$6%Q\"aF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I #moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretch yGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%form GQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%( maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6$-I$mtrGF$6$-I$mt dGF$6#-I#mnGF$6$Q\"2F'F9-F^o6#-Fao6$Q\"4F'F9-F[o6$-F^o6#-Fao6$FTF9F]o/ %%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 8 "" 1 "" {TEXT 209 43 "Error, ( in linalg:-inverse) singular matrix" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 51 "1.1. 9. Elemi sor- \303\251s oszlopm\305\261veletek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a:=matrix([ [1,2,3],[4,5,6]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modul enameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"aF'/%'italicGQ%trueF '/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&fals eF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movable limitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'r spaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I 'mtableGF$6$-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fao6$Q\"2F'F9-F ^o6#-Fao6$Q\"3F'F9-F[o6%-F^o6#-Fao6$Q\"4F'F9-F^o6#-Fao6$Q\"5F'F9-F^o6# -Fao6$Q\"6F'F9/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "swaprow(a,1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "- I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#- I'mtableGF$6$-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"4F'/%,mathvariantGQ'no rmalF'-F56#-F86$Q\"5F'F;-F56#-F86$Q\"6F'F;-F26%-F56#-F86$Q\"1F'F;-F56# -F86$Q\"2F'F;-F56#-F86$Q\"3F'F;/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "swapcol(a,2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sy slibGF'6%-I%mrowGF$6#-I'mtableGF$6$-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\" 1F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"3F'F;-F56#-F86$Q\"2F'F;-F26%- F56#-F86$Q\"4F'F;-F56#-F86$Q\"6F'F;-F56#-F86$Q\"5F'F;/%%openGQ\"[F'/%& closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "addrow(a,1, 2,-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6$-I$mtrGF$6%-I$mt dGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"2F'F;-F56# -F86$Q\"3F'F;-F26%-F56#-F86$Q\"0F'F;-F56#-F,6$-I#moGF$60Q*&uminus0;F'F ;/%&fenceGQ&falseF'/%*separatorGFY/%)stretchyGFY/%*symmetricGFY/%(larg eopGFY/%.movablelimitsGFY/%'accentGFY/%%formGQ'prefixF'/%'lspaceGQ$0em F'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGF:/%(maxsizeGQ)infinityF'F E-F56#-F,6$FS-F86$Q\"6F'F;/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "addcol(a,2,3,-3/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sysli bGF'6%-I%mrowGF$6#-I'mtableGF$6$-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"1F' /%,mathvariantGQ'normalF'-F56#-F86$Q\"2F'F;-F56#-F86$Q\"0F'F;-F26%-F56 #-F86$Q\"4F'F;-F56#-F86$Q\"5F'F;-F56#-F,6$-I#moGF$60Q*&uminus0;F'F;/%& fenceGQ&falseF'/%*separatorGFhn/%)stretchyGFhn/%*symmetricGFhn/%(large opGFhn/%.movablelimitsGFhn/%'accentGFhn/%%formGQ'prefixF'/%'lspaceGQ$0 emF'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGF:/%(maxsizeGQ)infinityF '-I&mfracGF$6(-F86$Q\"3F'F;F@/%.linethicknessGQ\"1F'/%+denomalignGQ'ce nterF'/%)numalignGF^q/%)bevelledGFhn/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 67 "1.1.10. Gauss-f\303\251le kik\303\274sz\303\266b\30 3\266l\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x:='x'; y:='y'; z:='z';\n" }{MPLTEXT 1 0 57 " eqns:=\{x+y/2+z/3=1,x/2+y/3+z/4=7/12,x/3+y/4+z/5=13/30\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"yG6 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%/,(I\"xG6\"\"\"\"*&#F'\"\"#F'I\"yGF&F'F'*&#F'\"\"$F'I\" zGF&F'F'F'/,(*&F)F'F%F'F'*&F-F'F+F'F'*&#F'\"\"%F'F/F'F'#\"\"(\"#7/,(*& F-F'F%F'F'*&F5F'F+F'F'*&#F'\"\"&F'F/F'F'#\"#8\"#I" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "solve(eqns);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%/I\"zG6\"\"\"$/I\"xGF%\"\"\"/I\"yGF%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "a:=genmatrix(eqns,[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6%Q\"aF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#: =F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symm etricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/ %'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)i nfinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mn GF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/% +denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao 6$Q\"3F'F9F[pF^pFapFcp-F[o6%FcoFep-F^o6#-Ffo6(F`o-Fao6$Q\"4F'F9F[pF^pF apFcp-F[o6%FepF^q-F^o6#-Ffo6(F`o-Fao6$Q\"5F'F9F[pF^pFapFcp/%%openGQ\"[ F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a:=gen matrix(eqns,[x,y,z],'b'); print(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "- I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"aF' /%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normal F'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(lar geopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thi ckmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mf encedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6 #-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/%+denomalignGQ' centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF ^pFapFcp-F[o6%FcoFep-F^o6#-Ffo6(F`o-Fao6$Q\"4F'F9F[pF^pFapFcp-F[o6%Fep F^q-F^o6#-Ffo6(F`o-Fao6$Q\"5F'F9F[pF^pFapFcp/%%openGQ\"[F'/%&closeGQ\" ]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,T ypesettingGI(_syslibGF'6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6 $Q\"1F'/%,mathvariantGQ'normalF'-F26#-I&mfracGF$6(-F56$Q\"7F'F8-F56$Q# 12F'F8/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFK/%) bevelledGQ&falseF'-F26#-F>6(-F56$Q#13F'F8-F56$Q#30F'F8FFFIFLFN/%%openG Q\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ge neqns(a,[x,y,z],b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%/,(I\"xG6\"\"\" \"*&#F'\"\"#F'I\"yGF&F'F'*&#F'\"\"$F'I\"zGF&F'F'F'/,(*&F)F'F%F'F'*&F-F 'F+F'F'*&#F'\"\"%F'F/F'F'#\"\"(\"#7/,(*&F-F'F%F'F'*&F5F'F+F'F'*&#F'\" \"&F'F/F'F'#\"#8\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c:= augment(a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"cF'/%'italicGQ%trueF'/%,m athvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/% *separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimit sGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspace GFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtab leGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6 $Q\"2F'F9/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF` p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&FcoFe p-F^o6#-Ffo6(F`o-Fao6$Q\"4F'F9F[pF^pFapFcp-F^o6#-Ffo6(-Fao6$Q\"7F'F9-F ao6$Q#12F'F9F[pF^pFapFcp-F[o6&FepF^q-F^o6#-Ffo6(F`o-Fao6$Q\"5F'F9F[pF^ pFapFcp-F^o6#-Ffo6(-Fao6$Q#13F'F9-Fao6$Q#30F'F9F[pF^pFapFcp/%%openGQ\" [F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "genma trix(eqns,[x,y,z],flag);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$ 6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F56# -I&mfracGF$6(F7-F86$Q\"2F'F;/%.linethicknessGQ\"1F'/%+denomalignGQ'cen terF'/%)numalignGFK/%)bevelledGQ&falseF'-F56#-FA6(F7-F86$Q\"3F'F;FFFIF LFNF4-F26&F>FQ-F56#-FA6(F7-F86$Q\"4F'F;FFFIFLFN-F56#-FA6(-F86$Q\"7F'F; -F86$Q#12F'F;FFFIFLFN-F26&FQFZ-F56#-FA6(F7-F86$Q\"5F'F;FFFIFLFN-F56#-F A6(-F86$Q#13F'F;-F86$Q#30F'F;FFFIFLFN/%%openGQ\"[F'/%&closeGQ\"]F'" }} }{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "1.1.11. P\303\251ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "c1:=pivot(c ,1,1,2..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#c1F'/%'italicGQ%trueF'/%,mat hvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*s eparatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsG F=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGF O/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtable GF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q \"2F'F9/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/ %)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&-F^o6#- Fao6$Q\"0F'F9-F^o6#-Ffo6(F`o-Fao6$Q#12F'F9F[pF^pFapFcpFcqFcq-F[o6&F^qF cq-F^o6#-Ffo6(-Fao6$Q\"4F'F9-Fao6$Q#45F'F9F[pF^pFapFcp-F^o6#-Ffo6(F`o- Fao6$Q#10F'F9F[pF^pFapFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "c2:=pivot(c1,2,2,3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl ibGF'6%-I#miGF$6%Q#c2F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#m oGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyG F=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ &infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(ma xsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdG F$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethickness GQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo 6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&-F^o6#-Fao6$Q\"0F'F9-F^o6#-Ffo 6(F`o-Fao6$Q#12F'F9F[pF^pFapFcpFcqFcq-F[o6&F^qF^q-F^o6#-Ffo6(F`o-Fao6$ Q$180F'F9F[pF^pFapFcp-F^o6#-Ffo6(F`o-Fao6$Q#60F'F9F[pF^pFapFcp/%%openG Q\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ga usselim(c,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$ 6&-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F56#-I&mfracGF$ 6(F7-F86$Q\"2F'F;/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)num alignGFK/%)bevelledGQ&falseF'-F56#-FA6(F7-F86$Q\"3F'F;FFFIFLFNF4-F26&- F56#-F86$Q\"0F'F;-F56#-FA6(F7-F86$Q#12F'F;FFFIFLFNFinFin-F26&FZFZ-F56# -FA6(F7-F86$Q$180F'F;FFFIFLFN-F56#-FA6(F7-F86$Q#60F'F;FFFIFLFN/%%openG Q\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ba cksub(c2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I #mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F26#-I%mrowGF$6$-I#moGF$60Q*&u minus0;F'F8/%&fenceGQ&falseF'/%*separatorGFF/%)stretchyGFF/%*symmetric GFF/%(largeopGFF/%.movablelimitsGFF/%'accentGFF/%%formGQ'prefixF'/%'ls paceGQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGF7/%(maxsizeGQ)i nfinityF'-F56$Q\"2F'F8-F26#-F56$Q\"3F'F8/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "d:=copy(c); d[3,3]:=7/36 ; print(d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"dF'/%'italicGQ%trueF'/%,mat hvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*s eparatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsG F=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGF O/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtable GF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q \"2F'F9/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/ %)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&FcoFep- F^o6#-Ffo6(F`o-Fao6$Q\"4F'F9F[pF^pFapFcp-F^o6#-Ffo6(-Fao6$Q\"7F'F9-Fao 6$Q#12F'F9F[pF^pFapFcp-F[o6&FepF^q-F^o6#-Ffo6(F`o-Fao6$Q\"5F'F9F[pF^pF apFcp-F^o6#-Ffo6(-Fao6$Q#13F'F9-Fao6$Q#30F'F9F[pF^pFapFcp/%%openGQ\"[F '/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"(\"#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI( _syslibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$ Q\"1F'/%,mathvariantGQ'normalF'-F56#-I&mfracGF$6(F7-F86$Q\"2F'F;/%.lin ethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFK/%)bevelledGQ&f alseF'-F56#-FA6(F7-F86$Q\"3F'F;FFFIFLFNF4-F26&F>FQ-F56#-FA6(F7-F86$Q\" 4F'F;FFFIFLFN-F56#-FA6(-F86$Q\"7F'F;-F86$Q#12F'F;FFFIFLFN-F26&FQFZ-F56 #-FA6(F_o-F86$Q#36F'F;FFFIFLFN-F56#-FA6(-F86$Q#13F'F;-F86$Q#30F'F;FFFI FLFN/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d1:=pivot(d,1,1,2..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q #d1F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'n ormalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/ %(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceG Q/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF' -I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9 -F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/%+denomali gnGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F 9F[pF^pFapFcpF]o-F[o6&-F^o6#-Fao6$Q\"0F'F9-F^o6#-Ffo6(F`o-Fao6$Q#12F'F 9F[pF^pFapFcpFcqFcq-F[o6&F^qFcqFcq-F^o6#-Ffo6(F`o-Fao6$Q#10F'F9F[pF^pF apFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "d2:=pivot(d1,2,2,3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q #d2F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'n ormalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/ %(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceG Q/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF' -I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9 -F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/%+denomali gnGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F 9F[pF^pFapFcpF]o-F[o6&-F^o6#-Fao6$Q\"0F'F9-F^o6#-Ffo6(F`o-Fao6$Q#12F'F 9F[pF^pFapFcpFcqFcq-F[o6&F^qF^qF^q-F^o6#-Ffo6(F`o-Fao6$Q#60F'F9F[pF^pF apFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "backsub(d2);" }}{PARA 8 "" 1 "" {TEXT 209 47 "Error, \+ (in linalg:-backsub) inconsistent system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "e:=copy(d); e[3,4]:=5/12; print(e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I#miGF$6%Q\"eF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60 Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*s ymmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infix F'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeG Q)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I #mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F '/%+denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o- Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&FcoFep-F^o6#-Ffo6(F`o-Fao6$Q\"4F'F9F [pF^pFapFcp-F^o6#-Ffo6(-Fao6$Q\"7F'F9-Fao6$Q#12F'F9F[pF^pFapFcp-F[o6&F epF^q-F^o6#-Ffo6(Fiq-Fao6$Q#36F'F9F[pF^pFapFcp-F^o6#-Ffo6(-Fao6$Q#13F' F9-Fao6$Q#30F'F9F[pF^pFapFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"&\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mf encedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mt ableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normal F'-F56#-I&mfracGF$6(F7-F86$Q\"2F'F;/%.linethicknessGQ\"1F'/%+denomalig nGQ'centerF'/%)numalignGFK/%)bevelledGQ&falseF'-F56#-FA6(F7-F86$Q\"3F' F;FFFIFLFNF4-F26&F>FQ-F56#-FA6(F7-F86$Q\"4F'F;FFFIFLFN-F56#-FA6(-F86$Q \"7F'F;-F86$Q#12F'F;FFFIFLFN-F26&FQFZ-F56#-FA6(F_o-F86$Q#36F'F;FFFIFLF N-F56#-FA6(-F86$Q\"5F'F;FboFFFIFLFN/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "e1:=pivot(e,1,1,2..3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6%Q#e1F'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)s tretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\" 1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6 &-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linet hicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F ^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&-F^o6#-Fao6$Q\"0F'F9-F ^o6#-Ffo6(F`o-Fao6$Q#12F'F9F[pF^pFapFcpFcqFcqF\\q/%%openGQ\"[F'/%&clos eGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "e2:=pivot(e1,2, 2,3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I ,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#e2F'/%'italicGQ%trueF'/%,mathva riantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*sepa ratorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/ %'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/% (minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$ 6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2 F'F9/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/%)b evelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&-F^o6#-Fao 6$Q\"0F'F9-F^o6#-Ffo6(F`o-Fao6$Q#12F'F9F[pF^pFapFcpFcqFcq-F[o6&F^qF^qF ^qF^q/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "backsub(e2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfen cedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I'mtableGF$6#-I$mt rGF$6%-I$mtdGF$6#-I%mrowGF$6%-I&mfracGF$6(-I#mnGF$6$Q\"1F'/%,mathvaria ntGQ'normalF'-F;6$Q\"2F'F>/%.linethicknessGQ\"1F'/%+denomalignGQ'cente rF'/%)numalignGFI/%)bevelledGQ&falseF'-I#moGF$60Q\"+F'F>/%&fenceGFN/%* separatorGFN/%)stretchyGFN/%*symmetricGFN/%(largeopGFN/%.movablelimits GFN/%'accentGFN/%%formGQ&infixF'/%'lspaceGQ0mediummathspaceF'/%'rspace GF`o/%(minsizeGF=/%(maxsizeGQ)infinityF'-F56%-F86(F:-F;6$Q\"6F'F>FDFGF JFL-FP60Q1⁢F'F>FSFUFWFYFenFgnFinF[o/F_oQ$0emF'/FboFcpFc oFeo-F56#-I%msubGF$6%-I#miGF$6%Q#_tF'/%'italicGQ%trueF'/F?Q'italicF'-F 56#F:/%/subscriptshiftGQ\"0F'-F26#-F56%F:-FP60Q(−F'F>FSFUFWFYFen FgnFinF[oF^oFaoFcoFeoFgp-F2Ffp/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "evalm(randmatrix(3,4)+I*rand matrix(3,4)); 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M\303\263dos\303\255tott Gauss-f\303\25 1le kik\303\274sz\303\266b\303\266l\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(c);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#m nGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F56#-I&mfracGF$6(F7-F86$Q\"2F'F; /%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFK/%)bevell edGQ&falseF'-F56#-FA6(F7-F86$Q\"3F'F;FFFIFLFNF4-F26&F>FQ-F56#-FA6(F7-F 86$Q\"4F'F;FFFIFLFN-F56#-FA6(-F86$Q\"7F'F;-F86$Q#12F'F;FFFIFLFN-F26&FQ FZ-F56#-FA6(F7-F86$Q\"5F'F;FFFIFLFN-F56#-FA6(-F86$Q#13F'F;-F86$Q#30F'F ;FFFIFLFN/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "c1:=pivot(c,1,1,2..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q #c1F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'n ormalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/ %(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceG Q/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF' -I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9 -F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/%+denomali gnGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F 9F[pF^pFapFcpF]o-F[o6&-F^o6#-Fao6$Q\"0F'F9-F^o6#-Ffo6(F`o-Fao6$Q#12F'F 9F[pF^pFapFcpFcqFcq-F[o6&F^qFcq-F^o6#-Ffo6(-Fao6$Q\"4F'F9-Fao6$Q#45F'F 9F[pF^pFapFcp-F^o6#-Ffo6(F`o-Fao6$Q#10F'F9F[pF^pFapFcp/%%openGQ\"[F'/% &closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "mulrow(c1, 2,12); c2:=pivot(%,2,2,3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfen cedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtab leGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF' -F56#-I&mfracGF$6(F7-F86$Q\"2F'F;/%.linethicknessGQ\"1F'/%+denomalignG Q'centerF'/%)numalignGFK/%)bevelledGQ&falseF'-F56#-FA6(F7-F86$Q\"3F'F; FFFIFLFNF4-F26&-F56#-F86$Q\"0F'F;F4F4F4-F26&FZ-F56#-FA6(F7-F86$Q#12F'F ;FFFIFLFN-F56#-FA6(-F86$Q\"4F'F;-F86$Q#45F'F;FFFIFLFN-F56#-FA6(F7-F86$ Q#10F'F;FFFIFLFN/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6%Q#c2F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#: =F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symm etricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/ %'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)i nfinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mn GF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/% +denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao 6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&-F^o6#-Fao6$Q\"0F'F9F]oF]oF]o-F[o6&F^qF ^q-F^o6#-Ffo6(F`o-Fao6$Q$180F'F9F[pF^pFapFcp-F^o6#-Ffo6(F`o-Fao6$Q#60F 'F9F[pF^pFapFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c3:=mulrow(c2,3,180);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6%Q#c3F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#: =F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symm etricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/ %'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)i nfinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mn GF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/% +denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao 6$Q\"3F'F9F[pF^pFapFcpF]o-F[o6&-F^o6#-Fao6$Q\"0F'F9F]oF]oF]o-F[o6&F^qF ^qF]o-F^o6#Fip/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "backsub(c3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(m fencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I'mtableGF$6#-I $mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F26#-I%m rowGF$6$-I#moGF$60Q*&uminus0;F'F8/%&fenceGQ&falseF'/%*separatorGFF/%)s tretchyGFF/%*symmetricGFF/%(largeopGFF/%.movablelimitsGFF/%'accentGFF/ %%formGQ'prefixF'/%'lspaceGQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(mi nsizeGF7/%(maxsizeGQ)infinityF'-F56$Q\"2F'F8-F26#-F56$Q\"3F'F8/%%openG Q\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%? G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 99 "1.1.13. M\303\241trixinve rzi\303\263 Gauss-f\303\251le kik\303\274sz\303\266b\303\266l\303\251s sel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a:=hilbert(3); b:=diag(1,1,1); c:=augment(a,b);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6%Q\"aF'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)s tretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\" 1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6 %-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linet hicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F ^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcp-F[o6%FcoFep-F^o6#-Ffo6(F`o-Fao 6$Q\"4F'F9F[pF^pFapFcp-F[o6%FepF^q-F^o6#-Ffo6(F`o-Fao6$Q\"5F'F9F[pF^pF apFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"bF'/ %'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF '/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(larg eopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thic kmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfe ncedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6# -Fao6$Q\"0F'F9Fco-F[o6%FcoF]oFco-F[o6%FcoFcoF]o/%%openGQ\"[F'/%&closeG Q\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,T ypesettingGI(_syslibGF'6%-I#miGF$6%Q\"cF'/%'italicGQ%trueF'/%,mathvari antGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separa torGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%' accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(m insizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6% -I$mtrGF$6(-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F' F9/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/%)bev elledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F^o6#-Fao6$Q\"0F' F9F\\q-F[o6(FcoFep-F^o6#-Ffo6(F`o-Fao6$Q\"4F'F9F[pF^pFapFcpF\\qF]oF\\q -F[o6(FepFcq-F^o6#-Ffo6(F`o-Fao6$Q\"5F'F9F[pF^pFapFcpF\\qF\\qF]o/%%ope nGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " cc:=gausselim(c,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modu lenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#ccF'/%'italicGQ%true F'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&fal seF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movabl elimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%' rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#- I'mtableGF$6%-I$mtrGF$6(-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F` o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numal ignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcpF]o-F^o6 #-Fao6$Q\"0F'F9F\\q-F[o6(F\\q-F^o6#-Ffo6(F`o-Fao6$Q#12F'F9F[pF^pFapFcp Fcq-F^o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/FNQ$0emF '/FQQ2verythinmathspaceF'FRFUFeoF]oF\\q-F[o6(F\\qF\\q-F^o6#-Ffo6(F`o-F ao6$Q$180F'F9F[pF^pFapFcp-F^o6#-Ffo6(F`o-Fao6$Q\"6F'F9F[pF^pFapFcp-F^o 6#-F#6$F^rF`oF]o/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "aa:=submatrix(cc,1..3,1..3); bb:=submatrix(cc,1. .3,4..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#aaF'/%'italicGQ%trueF'/%,mathv ariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*sep aratorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF= /%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/ %(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF $6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\" 2F'F9/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`p/%) bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[pF^pFapFcp-F[o6%-F^o6#-Fao6$ Q\"0F'F9-F^o6#-Ffo6(F`o-Fao6$Q#12F'F9F[pF^pFapFcpFcq-F[o6%F^qF^q-F^o6# -Ffo6(F`o-Fao6$Q$180F'F9F[pF^pFapFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6%Q#bbF'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)s tretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\" 1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6 %-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fao6$Q\"0F'F9Fco-F[o6%-F^o6#-F#6$-F66 0Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/FNQ$0emF'/FQQ2verythinmath spaceF'FRFU-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/%+den omalignGQ'centerF'/%)numalignGFbq/%)bevelledGF=F]oFco-F[o6%-F^o6#-Fhp6 (F`o-Fao6$Q\"6F'F9F]qF`qFcqFeq-F^o6#-F#6$F^pF`oF]o/%%openGQ\"[F'/%&clo seGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "backsub(aa,bb) ; inverse(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$ 6%-I$mtdGF$6#-I#mnGF$6$Q\"9F'/%,mathvariantGQ'normalF'-F56#-F,6$-I#moG F$60Q*&uminus0;F'F;/%&fenceGQ&falseF'/%*separatorGFH/%)stretchyGFH/%*s ymmetricGFH/%(largeopGFH/%.movablelimitsGFH/%'accentGFH/%%formGQ'prefi xF'/%'lspaceGQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGQ\"1F'/% (maxsizeGQ)infinityF'-F86$Q#36F'F;-F56#-F86$Q#30F'F;-F26%F>-F56#-F86$Q $192F'F;-F56#-F,6$FB-F86$Q$180F'F;-F26%FaoF]p-F56#Fap/%%openGQ\"[F'/%& closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulena meG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6 %-I$mtdGF$6#-I#mnGF$6$Q\"9F'/%,mathvariantGQ'normalF'-F56#-F,6$-I#moGF $60Q*&uminus0;F'F;/%&fenceGQ&falseF'/%*separatorGFH/%)stretchyGFH/%*sy mmetricGFH/%(largeopGFH/%.movablelimitsGFH/%'accentGFH/%%formGQ'prefix F'/%'lspaceGQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGQ\"1F'/%( maxsizeGQ)infinityF'-F86$Q#36F'F;-F56#-F86$Q#30F'F;-F26%F>-F56#-F86$Q$ 192F'F;-F56#-F,6$FB-F86$Q$180F'F;-F26%FaoF]p-F56#Fap/%%openGQ\"[F'/%&c loseGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "linsolve(a,b );" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Ty pesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$ 6#-I#mnGF$6$Q\"9F'/%,mathvariantGQ'normalF'-F56#-F,6$-I#moGF$60Q*&umin us0;F'F;/%&fenceGQ&falseF'/%*separatorGFH/%)stretchyGFH/%*symmetricGFH /%(largeopGFH/%.movablelimitsGFH/%'accentGFH/%%formGQ'prefixF'/%'lspac eGQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGQ\"1F'/%(maxsizeGQ) infinityF'-F86$Q#36F'F;-F56#-F86$Q#30F'F;-F26%F>-F56#-F86$Q$192F'F;-F5 6#-F,6$FB-F86$Q$180F'F;-F26%FaoF]p-F56#Fap/%%openGQ\"[F'/%&closeGQ\"]F '" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 62 "1.1.14. Gauss-Jordan kik \303\274sz\303\266b\303\266l\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sysl ibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6(-I$mtdGF$6#-I#mnGF$6$Q\"1F '/%,mathvariantGQ'normalF'-F56#-I&mfracGF$6(F7-F86$Q\"2F'F;/%.linethic knessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFK/%)bevelledGQ&falseF '-F56#-FA6(F7-F86$Q\"3F'F;FFFIFLFNF4-F56#-F86$Q\"0F'F;FX-F26(F>FQ-F56# -FA6(F7-F86$Q\"4F'F;FFFIFLFNFXF4FX-F26(FQFin-F56#-FA6(F7-F86$Q\"5F'F;F FFIFLFNFXFXF4/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "c1:=pivot(c,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " -I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#c1F '/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'norma lF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(la rgeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/th ickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(m fencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6(-I$mtdGF$6#-I#mnGF$6$FTF9-F^o 6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\"1F'/%+denomalignGQ 'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F`o-Fao6$Q\"3F'F9F[p F^pFapFcpF]o-F^o6#-Fao6$Q\"0F'F9F\\q-F[o6(F\\q-F^o6#-Ffo6(F`o-Fao6$Q#1 2F'F9F[pF^pFapFcpFcq-F^o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ' prefixF'/FNQ$0emF'/FQQ2verythinmathspaceF'FRFUFeoF]oF\\q-F[o6(F\\qFcq- F^o6#-Ffo6(-Fao6$Q\"4F'F9-Fao6$Q#45F'F9F[pF^pFapFcp-F^o6#-F#6$F^rFgpF \\qF]o/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "mulrow(c1,2,12); 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A m\305\261veletig\303\251nyek \303\266sszehasonl\303 \255t\303\241sa." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 66 "*1.1.16. A h\303\241romsz\303\266gm\303\241trixo k m\303\263dszere." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "u:=LUdecomp(a,P='p',L='l');" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6%Q\"uF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$ 60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/% *symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&inf ixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsiz eGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6# -I#mnGF$6$FTF9-F^o6#-I&mfracGF$6(F`o-Fao6$Q\"2F'F9/%.linethicknessGQ\" 1F'/%+denomalignGQ'centerF'/%)numalignGF`p/%)bevelledGF=-F^o6#-Ffo6(F` o-Fao6$Q\"3F'F9F[pF^pFapFcp-F[o6%-F^o6#-Fao6$Q\"0F'F9-F^o6#-Ffo6(F`o-F ao6$Q#12F'F9F[pF^pFapFcpFcq-F[o6%F^qF^q-F^o6#-Ffo6(F`o-Fao6$Q$180F'F9F [pF^pFapFcp/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "print(p); print(l); evalm(p&*l&*u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"1F'/% ,mathvariantGQ'normalF'-F56#-F86$Q\"0F'F;F>-F26%F>F4F>-F26%F>F>F4/%%op enGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$ 6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F56# -F86$Q\"0F'F;F>-F26%-F56#-I&mfracGF$6(F7-F86$Q\"2F'F;/%.linethicknessG Q\"1F'/%+denomalignGQ'centerF'/%)numalignGFR/%)bevelledGQ&falseF'F4F>- F26%-F56#-FH6(F7-F86$Q\"3F'F;FMFPFSFUF4F4/%%openGQ\"[F'/%&closeGQ\"]F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Type settingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6# -I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF'-F56#-I&mfracGF$6(F7-F86$Q\"2 F'F;/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFK/%)be velledGQ&falseF'-F56#-FA6(F7-F86$Q\"3F'F;FFFIFLFN-F26%F>FQ-F56#-FA6(F7 -F86$Q\"4F'F;FFFIFLFN-F26%FQFZ-F56#-FA6(F7-F86$Q\"5F'F;FFFIFLFN/%%open GQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f orwardsub(l,b); backsub(u,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfen cedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtab leGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariantGQ'normalF' -F56#-F86$Q\"0F'F;F>-F26%-F56#-F,6$-I#moGF$60Q*&uminus0;F'F;/%&fenceGQ &falseF'/%*separatorGFO/%)stretchyGFO/%*symmetricGFO/%(largeopGFO/%.mo vablelimitsGFO/%'accentGFO/%%formGQ'prefixF'/%'lspaceGQ$0emF'/%'rspace GQ2verythinmathspaceF'/%(minsizeGF:/%(maxsizeGQ)infinityF'-I&mfracGF$6 (F7-F86$Q\"2F'F;/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numa lignGF_p/%)bevelledGFOF4F>-F26%-F56#-Feo6(F7-F86$Q\"6F'F;FjoF]pF`pFbp- F56#-F,6$FIF7F4/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I%mrowGF$6#-I'mtableGF$6%-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"9F'/%,ma thvariantGQ'normalF'-F56#-F,6$-I#moGF$60Q*&uminus0;F'F;/%&fenceGQ&fals eF'/%*separatorGFH/%)stretchyGFH/%*symmetricGFH/%(largeopGFH/%.movable limitsGFH/%'accentGFH/%%formGQ'prefixF'/%'lspaceGQ$0emF'/%'rspaceGQ2ve rythinmathspaceF'/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F86$Q#36F'F; -F56#-F86$Q#30F'F;-F26%F>-F56#-F86$Q$192F'F;-F56#-F,6$FB-F86$Q$180F'F; -F26%FaoF]p-F56#Fap/%%openGQ\"[F'/%&closeGQ\"]F'" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 39 "*1.1.17. Numerikus megjegyz\303\251sek." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 25 "1.1.18. Vektort\303\251r." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "1.1.19. P\303\251ld\303\241k." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "v:=vector(3,[5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/ I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"vF'/%'italicG Q%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fence GQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%. movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspac eF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6% -I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"5F'F9-F\\o6#-F_o6$Q\" 6F'F9-F\\o6#-I%msubGF$6%F+-F#6#-F_o6$Q\"3F'F9/%/subscriptshiftGQ\"0F'/ %%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v[3]:=7; print(v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"5F' /%,mathvariantGQ'normalF'-F26#-F56$Q\"6F'F8-F26#-F56$Q\"7F'F8/%%openGQ \"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u:= vector(3,i->i+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"uF'/%'italicGQ%trueF' /%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&false F'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablel imitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rs paceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtabl eGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"2F'F9-F\\o6#-F_o6$Q\"3F'F9-F \\o6#-F_o6$Q\"4F'F9/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalm(2*u+v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I'mtableG F$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"9F'/%,mathvariantGQ'normalF'-F2 6#-F56$Q#12F'F8-F26#-F56$Q#15F'F8/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "w:=vector(4,0); vectdim(w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesett ingGI(_syslibGF'6%-I#miGF$6%Q\"wF'/%'italicGQ%trueF'/%,mathvariantGQ'i talicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/ %)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentG F=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeG Q\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6&- I$mtdGF$6#-I#mnGF$6$Q\"0F'F9F[oF[oF[o/%%openGQ\"[F'/%&closeGQ\"]F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "w:=randvector(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"wF'/ %'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF '/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(larg eopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thic kmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfe ncedGF$6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q#23F'F9-F\\o6# -F_o6$Q#75F'F9-F\\o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ'prefi xF'/FNQ$0emF'/FQQ2verythinmathspaceF'FRFU-F_o6$Q#92F'F9/%%openGQ\"[F'/ %&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "type(w,'v ector'); type(w,'matrix');" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*p rotectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "1.1.20 Alt\303\251r." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 29 "1.1.21 . P\303\251ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 49 "1.1.22. Line\303\241ris kombin\303\241ci \303\263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm(-87*z-7/3*u+16/3*v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I%mrowGF$6%-I#mnGF$6$Q#22F'/%,ma thvariantGQ'normalF'-I#moGF$60Q(−F'F;/%&fenceGQ&falseF'/%*separa torGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimitsGFD/%' accentGFD/%%formGQ&infixF'/%'lspaceGQ0mediummathspaceF'/%'rspaceGFV/%( minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F56%-F86$Q#87F'F;-F?60Q1&Invisib leTimes;F'F;FBFEFGFIFKFMFOFQ/FUQ$0emF'/FXFboFYFfn-I#miGF$6%Q\"zF'/%'it alicGQ%trueF'/FFin-F26#-F56%-F86$Q #28F'F;F>Fin/%%openGQ\"[F'/%&closeGQ\"]F'" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "1.1.23. Megjegyz\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "z:=vector([1,0,0] ); a:=augment(z,u,v,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+ modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"zF'/%'italicGQ% trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ &falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.mo vablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF '/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I 'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$FTF9-F\\o6#-F_o6$Q\"0F'F9F ao/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mr owG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"aF'/%'i talicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/% &fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeop GF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickma thspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfence dGF$6%-F#6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9-F^o6#-Fa o6$Q\"2F'F9-F^o6#-Fao6$Q\"5F'F9-F^o6#-Fao6$Q#23F'F9-F[o6&-F^o6#-Fao6$Q \"0F'F9-F^o6#-Fao6$Q\"3F'F9-F^o6#-Fao6$Q\"6F'F9-F^o6#-Fao6$Q#75F'F9-F[ o6&Fdp-F^o6#-Fao6$Q\"4F'F9-F^o6#-Fao6$Q\"7F'F9-F^o6#-F#6$-F660Q*&uminu s0;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/FNQ$0emF'/FQQ2verythinmathspaceF'FR FU-Fao6$Q#92F'F9/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b:=stackmatrix(u,v,w,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6%Q\"bF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#: =F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symm etricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/ %'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)i nfinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6&-I$mtrGF$6%-I$mtdGF$6#-I#mn GF$6$Q\"2F'F9-F^o6#-Fao6$Q\"3F'F9-F^o6#-Fao6$Q\"4F'F9-F[o6%-F^o6#-Fao6 $Q\"5F'F9-F^o6#-Fao6$Q\"6F'F9-F^o6#-Fao6$Q\"7F'F9-F[o6%-F^o6#-Fao6$Q#2 3F'F9-F^o6#-Fao6$Q#75F'F9-F^o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH /FKQ'prefixF'/FNQ$0emF'/FQQ2verythinmathspaceF'FRFU-Fao6$Q#92F'F9-F[o6 %-F^o6#-Fao6$FTF9-F^o6#-Fao6$Q\"0F'F9Fas/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "col(a,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sysli bGF'6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"2F'/%,mathvaria ntGQ'normalF'-F26#-F56$Q\"3F'F8-F26#-F56$Q\"4F'F8/%%openGQ\"[F'/%&clos eGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "col(a,2..3);" } }{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesettin gGI(_syslibGF'6%-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I# mnGF$6$Q\"2F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"3F'F;-F56#-F86$Q\"4 F'F;/%%openGQ\"[F'/%&closeGQ\"]F'-I#moGF$60Q\",F'F;/%&fenceGQ&falseF'/ %*separatorGQ%trueF'/%)stretchyGFT/%*symmetricGFT/%(largeopGFT/%.movab lelimitsGFT/%'accentGFT/%%formGQ&infixF'/%'lspaceGQ$0emF'/%'rspaceGQ3v erythickmathspaceF'/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-F,6%-F/6#- F26%-F56#-F86$Q\"5F'F;-F56#-F86$Q\"6F'F;-F56#-F86$Q\"7F'F;FHFK" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "row(a,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF' 6%-I'mtableGF$6#-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$Q\"0F'/%,mathvariantGQ 'normalF'-F26#-F56$Q\"3F'F8-F26#-F56$Q\"6F'F8-F26#-F56$Q#75F'F8/%%open GQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "r ow(a,2..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6& -I$mtdGF$6#-I#mnGF$6$Q\"0F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"3F'F; 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T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*1.1.25. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 50 "1.1.26. Gener\303\241torrendszer, dime nzi\303\263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 91 "1.1.27. Line\303\241ris f\303\274ggetlens\303\251g \+ \303\251s f\303\274gg\305\221s\303\251g." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.1.28. T\303\251tel." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.1.29. B\303\241zis." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "basis([z,u,v,w]); basis([u,v,w,z]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "7%I\"zG6\"I\"uGF$I\"vGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%I\"uG6\"I\"vGF$I\"wGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "intbasis([z,u],[u,v,w]); intbasis([z,u],[v,w]); intbasis([z,u],[v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I +modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6%-F#6%-I'mtableG F$6#-I$mtrGF$6%-I$mtdGF$6#-F,6$-I#moGF$60Q*&uminus0;F'/%,mathvariantGQ 'normalF'/%&fenceGQ&falseF'/%*separatorGFD/%)stretchyGFD/%*symmetricGF D/%(largeopGFD/%.movablelimitsGFD/%'accentGFD/%%formGQ'prefixF'/%'lspa ceGQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGQ\"1F'/%(maxsizeGQ )infinityF'-I#mnGF$6$Q$937F'F?-F76#-F,6$F;-F[o6$Q%1077F'F?-F76#-F,6$F; -F[o6$Q%1436F'F?/%%openGQ\"[F'/%&closeGQ\"]F'-F<60Q\",F'F?FB/FFQ%trueF 'FGFIFKFMFO/FRQ&infixF'FT/FXQ3verythickmathspaceF'FZFgn-F#6%-F16#-F46% -F76#-F[o6$Q\"2F'F?-F76#-F[o6$Q\"3F'F?-F76#-F[o6$Q\"4F'F?F\\pF_p/F]pQ \"|frF'/F`pQ\"|hrF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+m odulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-F#6%-I'mtableGF$ 6#-I$mtrGF$6%-I$mtdGF$6#-F,6$-I#moGF$60Q*&uminus0;F'/%,mathvariantGQ'n ormalF'/%&fenceGQ&falseF'/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/ %(largeopGFD/%.movablelimitsGFD/%'accentGFD/%%formGQ'prefixF'/%'lspace GQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGQ\"1F'/%(maxsizeGQ)i nfinityF'-I#mnGF$6$Q$937F'F?-F76#-F,6$F;-F[o6$Q%1077F'F?-F76#-F,6$F;-F [o6$Q%1436F'F?/%%openGQ\"[F'/%&closeGQ\"]F'/F]pQ\"|frF'/F`pQ\"|hrF'" } }{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sumbasis([z,u],[v,w]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%I\"zG6\"I\"uGF$I\"vGF$" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.1.30. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.1.31. K\303\266vetkezm\303\251ny." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.1.32. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.1.33. K\303\266vetkezm \303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.1.34. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 " " {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.1.35. K\30 3\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.1.36. K\303\266vetkezm\303\251ny." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.1.37. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 13 "1.1.38. Rang." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ba sis(a,colspace); basis(a,rowspace);" }}{PARA 11 "" 1 "" {XPPMATH 20 "- I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowGF$6'- F#6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariant GQ'normalF'-F76#-F:6$Q\"0F'F=F@/%%openGQ\"[F'/%&closeGQ\"]F'-I#moGF$60 Q\",F'F=/%&fenceGQ&falseF'/%*separatorGQ%trueF'/%)stretchyGFQ/%*symmet ricGFQ/%(largeopGFQ/%.movablelimitsGFQ/%'accentGFQ/%%formGQ&infixF'/%' lspaceGQ$0emF'/%'rspaceGQ3verythickmathspaceF'/%(minsizeGF " 0 "" {MPLTEXT 1 0 32 "print(a); backsub(gausselim(a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I%mrowG F$6#-I'mtableGF$6%-I$mtrGF$6&-I$mtdGF$6#-I#mnGF$6$Q\"1F'/%,mathvariant GQ'normalF'-F56#-F86$Q\"2F'F;-F56#-F86$Q\"5F'F;-F56#-F86$Q#23F'F;-F26& -F56#-F86$Q\"0F'F;-F56#-F86$Q\"3F'F;-F56#-F86$Q\"6F'F;-F56#-F86$Q#75F' F;-F26&FO-F56#-F86$Q\"4F'F;-F56#-F86$Q\"7F'F;-F56#-F,6$-I#moGF$60Q*&um inus0;F'F;/%&fenceGQ&falseF'/%*separatorGFcp/%)stretchyGFcp/%*symmetri cGFcp/%(largeopGFcp/%.movablelimitsGFcp/%'accentGFcp/%%formGQ'prefixF' /%'lspaceGQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(minsizeGF:/%(maxsiz eGQ)infinityF'-F86$Q#92F'F;/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sys libGF'6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I%mrowGF$6$-I#moGF$60Q*&u minus0;F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGF@/%) stretchyGF@/%*symmetricGF@/%(largeopGF@/%.movablelimitsGF@/%'accentGF@ /%%formGQ'prefixF'/%'lspaceGQ$0emF'/%'rspaceGQ2verythinmathspaceF'/%(m insizeGQ\"1F'/%(maxsizeGQ)infinityF'-I#mnGF$6$Q$219F'F;-F26#-F56$F7-Fg n6$Q$359F'F;-F26#-Fgn6$Q$192F'F;/%%openGQ\"[F'/%&closeGQ\"]F'" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "1.1.39. 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K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.1.42. T \303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "1.1.43. Koordin\303\241t\303\241k." }}{PARA 0 "" 0 " " {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "1.1.44. Izom orfia." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.1.45. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 43 "*1.1.46. Vektorterek direkt \+ \303\266sszege." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 " " 0 "" {TEXT 207 36 "*1.1.48. K\303\266vetkezm\303\251ny." }}{PARA 0 " " 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*1.1.49 . P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 " " 0 "" {TEXT 207 31 "1.1.50. Affin sokas\303\241gok." }}{PARA 0 "" 0 " " {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.1.51. 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K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 " " {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 35 "1.1.53. K\30 3\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "x:=vector([1+t1,2-t2,t1+t2-1]); x0= map2(subs,[t1=0,t2=0],x);\n" }{MPLTEXT 1 0 15 "y:=evalm(x-x0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)s tretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\" 1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6%-I$m tdGF$6#-F#6%-I#mnGF$6$FTF9-F660Q\"+F'F9F;F>F@FBFDFFFHFJ/FNQ0mediummath spaceF'/FQFgoFRFU-F,6%Q#t1F'F/F2-F\\o6#-F#6%-Fao6$Q\"2F'F9-F660Q(&minu s;F'F9F;F>F@FBFDFFFHFJFfoFhoFRFU-F,6%Q#t2F'F/F2-F\\o6#-F#6'FioFcoFfpFc pF`o/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I% mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#x0F'/% 'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q\"=F'/F3Q'normalF' /%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(large opGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thick mathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfen cedGF$6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$FTF9-F\\o6#-F_o6 $Q\"2F'F9-F\\o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/F NQ$0emF'/FQQ2verythinmathspaceF'FRFUF^o/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'it alicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/% )stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF =/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ \"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6%-I $mtdGF$6#-F,6%Q#t1F'F/F2-F\\o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH /FKQ'prefixF'/FNQ$0emF'/FQQ2verythinmathspaceF'FRFU-F,6%Q#t2F'F/F2-F \\o6#-F#6%F^o-F660Q\"+F'F9F;F>F@FBFDFFFHFJ/FNQ0mediummathspaceF'/FQFip FRFUF^p/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "y1:=map2(subs,[t1=1,t2=0],y); y2:=map2(subs,[t1=0,t2= 1],y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,T ypesettingGI(_syslibGF'6%-I#miGF$6%Q#y1F'/%'italicGQ%trueF'/%,mathvari antGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separa torGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%' accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(m insizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mt rGF$6%-I$mtdGF$6#-I#mnGF$6$FTF9-F\\o6#-F_o6$Q\"0F'F9F[o/%%openGQ\"[F'/ %&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenam eG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#y2F'/%'italicGQ%trueF'/%, mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/ %*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimi tsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspac eGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF $6#-I$mtrGF$6%-I$mtdGF$6#-I#mnGF$6$Q\"0F'F9-F\\o6#-F#6$-F660Q*&uminus0 ;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/FNQ$0emF'/FQQ2verythinmathspaceF'FRFU -F_o6$FTF9-F\\o6#F_p/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 25 "z:=evalm(x0+t1*y1+t2*y2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6% -I#miGF$6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q #:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*sy mmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF '/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ )infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6%-I$mtdGF$6#-F#6%-I# mnGF$6$FTF9-F660Q\"+F'F9F;F>F@FBFDFFFHFJ/FNQ0mediummathspaceF'/FQFgoFR FU-F,6%Q#t1F'F/F2-F\\o6#-F#6%-Fao6$Q\"2F'F9-F660Q(−F'F9F;F>F@FBF DFFFHFJFfoFhoFRFU-F,6%Q#t2F'F/F2-F\\o6#-F#6'FioFcoFfpFcpF`o/%%openGQ\" [F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "x:=ve ctor([1+t1+t3,2+t2+t3,-1,2]); x0:=map2(subs,[t1=0,t2=0,t3=0],x);\n" } {MPLTEXT 1 0 15 "y:=evalm(x-x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%m rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"xF'/%' italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/ %&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeo pGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickm athspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfenc edGF$6%-I'mtableGF$6#-I$mtrGF$6&-I$mtdGF$6#-F#6'-I#mnGF$6$FTF9-F660Q\" +F'F9F;F>F@FBFDFFFHFJ/FNQ0mediummathspaceF'/FQFgoFRFU-F,6%Q#t1F'F/F2Fc o-F,6%Q#t3F'F/F2-F\\o6#-F#6'-Fao6$Q\"2F'F9Fco-F,6%Q#t2F'F/F2FcoF\\p-F \\o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/FNQ$0emF'/FQ Q2verythinmathspaceF'FRFUF`o-F\\o6#Fcp/%%openGQ\"[F'/%&closeGQ\"]F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesettin gGI(_syslibGF'6%-I#miGF$6%Q#x0F'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%) stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF= /%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ \"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6&-I $mtdGF$6#-I#mnGF$6$FTF9-F\\o6#-F_o6$Q\"2F'F9-F\\o6#-F#6$-F660Q*&uminus 0;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/FNQ$0emF'/FQQ2verythinmathspaceF'FRF UF^oFao/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " -I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"yF '/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'norma lF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(la rgeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/th ickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(m fencedGF$6%-I'mtableGF$6#-I$mtrGF$6&-I$mtdGF$6#-F#6%-F,6%Q#t1F'F/F2-F6 60Q\"+F'F9F;F>F@FBFDFFFHFJ/FNQ0mediummathspaceF'/FQFgoFRFU-F,6%Q#t3F'F /F2-F\\o6#-F#6%-F,6%Q#t2F'F/F2FcoFio-F\\o6#-I#mnGF$6$Q\"0F'F9Fcp/%%ope nGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 " y1:=map2(subs,[t1=1,t2=0,t3=0],y); y2:=map2(subs,[t1=0,t2=1,t3=0],y); \n" }{MPLTEXT 1 0 34 "y3:=map2(subs,[t1=0,t2=0,t3=1],y);" }}{PARA 11 " " 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslib GF'6%-I#miGF$6%Q#y1F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moG F$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF= /%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&i nfixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxs izeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6&-I$mtdGF$6#-I# mnGF$6$FTF9-F\\o6#-F_o6$Q\"0F'F9FaoFao/%%openGQ\"[F'/%&closeGQ\"]F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesettin gGI(_syslibGF'6%-I#miGF$6%Q#y2F'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%) stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF= /%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ \"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6&-I $mtdGF$6#-I#mnGF$6$Q\"0F'F9-F\\o6#-F_o6$FTF9F[oF[o/%%openGQ\"[F'/%&clo seGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#y3F'/%'italicGQ%trueF'/%,mathv ariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*sep aratorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF= /%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/ %(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I $mtrGF$6&-I$mtdGF$6#-I#mnGF$6$FTF9F[o-F\\o6#-F_o6$Q\"0F'F9Fao/%%openGQ \"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "bas is([y1,y2,y3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I#y1G6\"I#y2GF$" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "z:=evalm(x0+u1*y1+u2*y2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'it alicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/% )stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF =/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ \"1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-I'mtableGF$6#-I$mtrGF$6&-I $mtdGF$6#-F#6%-I#mnGF$6$FTF9-F660Q\"+F'F9F;F>F@FBFDFFFHFJ/FNQ0mediumma thspaceF'/FQFgoFRFU-F,6%Q#u1F'F/F2-F\\o6#-F#6%-Fao6$Q\"2F'F9Fco-F,6%Q# u2F'F/F2-F\\o6#-F#6$-F660Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ'prefixF'/FN Q$0emF'/FQQ2verythinmathspaceF'FRFUF`o-F\\o6#F`p/%%openGQ\"[F'/%&close GQ\"]F'" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 47 "1.1.54. Affin sokas \303\241g dimenzi\303\263ja." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT 206 46 "1.2. Line\303\241ris lek\303\251 pez\303\251sek" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;with(linalg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7^rI.BlockDiagonalG6\"I,GramSchmidtGF$I,JordanBlockGF$I)L UdecompGF$I)QRdecompGF$I*WronskianGF$I'addcolGF$I'addrowGF$I$adjGF$I(a djointGF$I&angleGF$I(augmentGF$I(backsubGF$I%bandGF$I&basisGF$I'bezout GF$I,blockmatrixGF$I(charmatGF$I)charpolyGF$I)choleskyGF$I$colGF$I'col dimGF$I)colspaceGF$I(colspanGF$I*companionGF$I'concatGF$I%condGF$I)cop yintoGF$I*crossprodGF$I%curlGF$I)definiteGF$I(delcolsGF$I(delrowsGF$I$ detGF$I%diagGF$I(divergeGF$I(dotprodGF$I*eigenvalsGF$I,eigenvaluesGF$I -eigenvectorsGF$I+eigenvectsGF$I,entermatrixGF$I&equalGF$I,exponential GF$I'extendGF$I,ffgausselimGF$I*fibonacciGF$I+forwardsubGF$I*frobenius GF$I*gausselimGF$I*gaussjordGF$I(geneqnsGF$I*genmatrixGF$I%gradGF$I)ha damardGF$I(hermiteGF$I(hessianGF$I(hilbertGF$I+htransposeGF$I)ihermite GF$I*indexfuncGF$I*innerprodGF$I)intbasisGF$I(inverseGF$I'ismithGF$I*i ssimilarGF$I'iszeroGF$I)jacobianGF$I'jordanGF$I'kernelGF$I*laplacianGF $I*leastsqrsGF$I)linsolveGF$I'mataddGF$I'matrixGF$I&minorGF$I(minpolyG F$I'mulcolGF$I'mulrowGF$I)multiplyGF$I%normGF$I*normalizeGF$I*nullspac eGF$I'orthogGF$I*permanentGF$I&pivotGF$I*potentialGF$I+randmatrixGF$I+ randvectorGF$I%rankGF$I(ratformGF$I$rowGF$I'rowdimGF$I)rowspaceGF$I(ro wspanGF$I%rrefGF$I*scalarmulGF$I-singularvalsGF$I&smithGF$I,stackmatri xGF$I*submatrixGF$I*subvectorGF$I)sumbasisGF$I(swapcolGF$I(swaprowGF$I *sylvesterGF$I)toeplitzGF$I&traceGF$I*transposeGF$I,vandermondeGF$I*ve cpotentGF$I(vectdimGF$I'vectorGF$I*wronskianGF$" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 49 "1.2.1. Line\303 \241ris lek\303\251pez\303\251sek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" } }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 72 "1.2.2. P\303\251ld\303\241k li ne\303\241ris lek\303\251pez\303\251sekre." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "1.2.3. T\303\251tel. " }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 95 "1.2.4. Line\303\241ris lek\303\251pez\303\251sek \303\266sszeg e \303\251s skal\303\241rszorosa." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "1.2.5. T\303\2 51tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 58 "1.2.6. Line\303\241ris lek\303\251pez\303\251sek szorzat a." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "1.2.7. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "1.2.8. K\303\266vetkezm\303\251n y." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "1.2.9. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.10. K\303\266vetkezm\303\251 ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.11. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 72 "1.2.12. Vektorrendsz er \303\251s lek\303\251pez\303\251s m\303\241trixa." }}{PARA 0 "" 0 " " {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a:=matrix ([[1,1,2,5,0],[0,1,1,3,-1],[1,0,1,2,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$;F&\"\"&E\\[l06$F'F'F&6$\"\"#F&\"\"!6$F 'F-F.6$F-F-F&6$F-F'F&6$F'F&F&6$F-\"\"%F'6$F-F)!\"\"6$F&F'F-6$F&F&F&6$F &F-F&6$F&F)F.6$F'F4F-6$F&F4F)6$F'F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "colspace(a); rank(a); kernel(a); coldim(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$=6\"6#;\"\"\"\"\"$E\\[l$F'F'\"\"#\"\"!F(F'=F $F%E\\[l$F'F+F*F'F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "<%=6\"6#;\"\"\"\"\"&E\\[l&F'!\"\"\"\"#F* \"\"$F'F(\"\"!\"\"%F-=F$F%E\\[l&F'!\"#F+!\"$F,F-F(F-F.F'=F$F%E\\[l&F'F *F+F'F,F-F(F'F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.13. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "a:=matrix([[1,2,3],[2,3,4],[3,4,5]]); x:=vector([1,0,1]); mult iply(a,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l* 6$F'F'\"\"&6$\"\"#F&F,6$F'F,\"\"%6$F,F,F'6$F,F'F.6$F'F&F'6$F&F'F'6$F&F &F&6$F&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F &F&\"\"#\"\"!F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E \\[l$F&\"\"%\"\"#\"\"'F'\"\")" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.14. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.15. T\303\251tel." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 50 "1.2.16. \303\201tt\303\251r\303\251sm\303\241trix." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 88 "1.2.17. P \303\251ld\303\241k: egyszer\305\261 b\303\241zistranszform\303\241ci \303\263k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT 207 21 "1.2.18. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.19. K\303\266vetkezm\30 3\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.20. K\303\266vetkezm\303\251ny." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 77 "1.2.21. Ekvivalens transzform\303\241ci\303\263k, ekvivalens m\303 \241trixok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "t:=matrix([[1,1,0],[0,1,1],[1,0,1]]); rank(t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'F&6$\"\" #F&\"\"!6$F'F+F,6$F+F+F&6$F+F'F&6$F'F&F&6$F&F'F,6$F&F&F&6$F&F+F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "b:=evalm(inverse(t)&*a&*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'#\"#6\"\"#6$F,F&F&6$F'F,# \"\"*F,6$F,F,#F&F,6$F,F'#F'F,6$F'F&\"\"&6$F&F'#\"\"(F,6$F&F&F'6$F&F,#F 6F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "s:=matrix([[1,-2,0], [0,1,-2],[-2,0,1]]); rank(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$; \"\"\"\"\"$F%E\\[l*6$F'F'F&6$\"\"#F&\"\"!6$F'F+F,6$F+F+F&6$F+F'!\"#6$F 'F&F06$F&F'F,6$F&F&F&6$F&F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(inverse(s)&*evalm(a &*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F' F'#!#Z\"\"(6$\"\"#F&#!#QF,6$F'F.#!#LF,6$F.F.#!#JF,6$F.F'#!#XF,6$F'F&#! #SF,6$F&F'#!#bF,6$F&F&#!#[F,6$F&F.#!#TF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "augment(s,evalm(a&*t)); gaussjord(%,3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$;F&\"\"'E\\[l36$F'F'F&6$\"\"#F& \"\"!6$F'F-F.6$F-F-F&6$F-F'!\"#6$F'F&F26$F-\"\"%F)6$F&F)\"\"&6$F-F7F76 $F&F'F.6$F&F&F&6$F&F-F26$F&F7F'6$F'F5\"\")6$F'F)\"\"*6$F-F)\"\"(6$F&F5 F56$F'F7FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$;F&\"\"' E\\[l36$F'F'F&6$\"\"#F&\"\"!6$F'F-F.6$F-F-F&6$F-F'F.6$F'F&F.6$F-\"\"%# !#Q\"\"(6$F&F)#!#bF76$F-\"\"&#!#JF76$F&F'F.6$F&F&F&6$F&F-F.6$F&F<#!#TF 76$F'F4#!#SF76$F'F)#!#ZF76$F-F)#!#XF76$F&F4#!#[F76$F'F<#!#LF7" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 38 "1.2.22. Line\303\241ris form\303 \241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f:=matrix(1,4); x:=vector(4); multiply(f,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"F&;F&\"\"%E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"%E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"F&E\\[l\"F&,**&&I\"fGF#6$F&F&F&&I\"xGF#6#F&F &F&*&&F+6$F&\"\"#F&&F.6#F3F&F&*&&F+6$F&\"\"$F&&F.6#F9F&F&*&&F+6$F&\"\" %F&&F.6#F?F&F&" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*1.2.23. T\3 03\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 " " {TEXT 207 52 "1.2.24. Biline\303\241ris lek\303\251pez\303\251sek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "a:=matrix(2,3); x:=vector(2); y:=vector(3); innerprod(x,a,y); \n" }{MPLTEXT 1 0 15 "evalm(x&*a&*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"#;F&\"\"$E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6 \"6#;\"\"\"\"\"#E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\" \"\"$E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*(&I\"yG6\"6#\"\"\"F(&I \"xGF&F'F(&I\"aGF&6$F(F(F(F(*(F$F(&F*6#\"\"#F(&F,6$F1F(F(F(*(&F%F0F(F) F(&F,6$F(F1F(F(*(F5F(F/F(&F,6$F1F1F(F(*(&F%6#\"\"$F(F)F(&F,6$F(F>F(F(* (FF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&,&*&&I\"xG 6\"6#\"\"\"F*&I\"aGF(6$F*F*F*F**&&F'6#\"\"#F*&F,6$F1F*F*F*F*&I\"yGF(F) F*F**&,&*&F&F*&F,6$F*F1F*F**&F/F*&F,6$F1F1F*F*F*&F5F0F*F**&,&*&F&F*&F, 6$F*\"\"$F*F**&F/F*&F,6$F1FDF*F*F*&F56#FDF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x:=vector(3); innerprod(x,y);" }}{PARA 11 "" 1 " " {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&&I\"xG6\"6#\"\"\"F(&I\"yGF&F'F(F(*&&F%6#\"\"#F(&F*F-F(F(*&&F%6 #\"\"$F(&F*F2F(F(" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.25. T \303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.26. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 46 "1.2.27. Kva dratikus lek\303\251pez\303\251sek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "b:=matrix([[1,2,3],[2,3,4] ,[5,6,7]]); q:=innerprod(x,b,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\" 6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"(6$\"\"#F&F,6$F'F,\"\"'6$F,F,F'6$F,F' \"\"%6$F'F&\"\"&6$F&F'F'6$F&F&F&6$F&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*$)&I\"xG6\"6#\"\"\"\"\"#F)F)*(\"\"%F)F%F)&F&6#F*F)F)*(\"\")F)F% F)&F&6#\"\"$F)F)*&F3F))F-F*F)F)*(\"#5F)F-F)F1F)F)*&\"\"(F))F1F*F)F)" } }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.28. T\303\251tel." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "s:=evalm(1/2*b+1/2*transpose(b)); a:=evalm(1/2*b-1/2*transpose(b ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F' \"\"(6$\"\"#F&F,6$F'F,\"\"&6$F,F,F'6$F,F'F.6$F'F&\"\"%6$F&F'F26$F&F&F& 6$F&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$ F'F'\"\"!6$\"\"#F&F*6$F'F,F&6$F,F,F*6$F,F'!\"\"6$F'F&F&6$F&F'F06$F&F&F *6$F&F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "innerprod(x,s, x); innerprod(x,a,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*$)&I\"xG6\"6 #\"\"\"\"\"#F)F)*(\"\"%F)F%F)&F&6#F*F)F)*(\"\")F)F%F)&F&6#\"\"$F)F)*&F 3F))F-F*F)F)*(\"#5F)F-F)F1F)F)*&\"\"(F))F1F*F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 38 "1.2.29. \+ P\303\251lda: vektori szorzat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a:=matrix([[0,k,-j],[-k,0,i] ,[j,-i,0]]); b:=innerprod(x,a,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6 \"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"!6$\"\"#F&,$I\"kGF#!\"\"6$F'F,,$I\"i GF#F/6$F,F,F*6$F,F'F26$F'F&I\"jGF#6$F&F',$F6F/6$F&F&F*6$F&F,F." }} {PARA 11 "" 1 "" {XPPMATH 20 ",.*(&I\"yG6\"6#\"\"\"F(&I\"xGF&6#\"\"#F( I\"kGF&F(!\"\"*(F$F(&F*6#\"\"$F(I\"jGF&F(F(*(&F%F+F(&F*F'F(F-F(F(*(F5F (F0F(I\"iGF&F(F.*(&F%F1F(F6F(F3F(F.*(F:F(F)F(F8F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "i:=vector([1,0,0]); j:=vector([0,1,0]); k: =vector([0,0,1]); evalm(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\" \"\"\"\"$E\\[l$F&F&\"\"#\"\"!F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6 \"6#;\"\"\"\"\"$E\\[l$F&\"\"!\"\"#F&F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&\"\"!\"\"#F)F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&,&*&&I\"yGF#6#\"\"#F&&I\"xGF#6# F'F&!\"\"*&&F,F1F&&F0F-F&F&F.,&*&&F,6#F&F&F/F&F&*&F4F&&F0F9F&F2F',&*&F 8F&F5F&F2*&F+F&F;F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "cr ossprod(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l $F&,&*&&I\"yGF#6#\"\"#F&&I\"xGF#6#F'F&!\"\"*&&F,F1F&&F0F-F&F&F.,&*&&F, 6#F&F&F/F&F&*&F4F&&F0F9F&F2F',&*&F8F&F5F&F2*&F+F&F;F&F&" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 128 "1.2.30. T\303\251tel: szimmetrikus \303 \251s altern\303\241l\303\263 biline\303\241ris lek\303\251pez\303\251 sek m\303\241trixa." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 94 "1.2.31. Szimmetrikus biline\303\241ris f orm\303\241k \303\241tl\303\263s alakra hoz\303\241sa." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "print(s ); addcol(s,1,2,-2); addrow(%,1,2,-2); addcol(%,1,3,-4);\n" }{MPLTEXT 1 0 21 "s1:=addrow(%,1,3,-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"sG6 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\" \"(6$\"\"#F&F,6$F'F,!\"$6$F,F,!\"\"6$F,F'\"\"&6$F'F&\"\"%6$F&F'F46$F&F &F&6$F&F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E \\[l*6$F'F'\"\"(6$\"\"#F&\"\"!6$F'F,!\"$6$F,F,!\"\"6$F,F'F/6$F'F&\"\"% 6$F&F'F46$F&F&F&6$F&F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\" \"\"\"$F%E\\[l*6$F'F'!\"*6$\"\"#F&\"\"!6$F'F,!\"$6$F,F,!\"\"6$F,F'F/6$ F'F&\"\"%6$F&F'F-6$F&F&F&6$F&F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6 \"6$;\"\"\"\"\"$F%E\\[l*6$F'F'!\"*6$\"\"#F&\"\"!6$F'F,!\"$6$F,F,!\"\"6 $F,F'F/6$F'F&F-6$F&F'F-6$F&F&F&6$F&F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "addcol(s1,2,3,-3); s2:=addrow(%,2,3,-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"!6$\"\"#F&F*6 $F'F,!\"$6$F,F,!\"\"6$F,F'F*6$F'F&F*6$F&F'F*6$F&F&F&6$F&F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"!6$\"\"#F& F*6$F'F,F*6$F,F,!\"\"6$F,F'F*6$F'F&F*6$F&F'F*6$F&F&F&6$F&F,F*" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.2.32. Megjegyz" }{TEXT 207 10 "\303\251s." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 26 "*1.2.33. Antiszi mmetrikus " }{TEXT 207 73 "biline\303\241ris form\303\241k \303\241tl \303\263s alakra hoz\303\241sa." }{TEXT 207 1 " " }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 62 "1.2.34. Definit \303\251s indefinit kvadratikus f orm\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "definite(s,'positive_def'); definite(s,'positive _semidef');\n" }{MPLTEXT 1 0 31 "definite(s,'negative_semidef');" }} {PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I& falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "princ axis:=proc(b) local a,i,j,k,n,x;\n" }{MPLTEXT 1 0 42 "if not(type(b,'m atrix'(rational,square))\n" }{MPLTEXT 1 0 73 " or type(b,'matrix'(flo at,square))) then ERROR(`Invalid argument`) fi;\n" }{MPLTEXT 1 0 15 "n :=coldim(b);\n" }{MPLTEXT 1 0 38 "for i to n do for j from i+1 to n do \n" }{MPLTEXT 1 0 56 " if b[i,j]<>b[j,i] then ERROR(`Invalid argument `) fi;\n" }{MPLTEXT 1 0 9 "od; od;\n" }{MPLTEXT 1 0 13 "a:=copy(b);\n" }{MPLTEXT 1 0 15 "for k to n do\n" }{MPLTEXT 1 0 20 " if a[k,k]=0 th en\n" }{MPLTEXT 1 0 60 " for j from k+1 to n do if a[j,j]<>0 then b reak fi; od;\n" }{MPLTEXT 1 0 60 " if j<=n then a:=swapcol(a,k,j); \+ a:=swaprow(a,k,j); fi;\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 20 " if a[k,k]=0 then\n" }{MPLTEXT 1 0 60 " for j from k+1 to n do if \+ a[k,j]<>0 then break fi; od;\n" }{MPLTEXT 1 0 58 " if j<=n then a:= addcol(a,j,k); a:=addrow(a,j,k); fi;\n" }{MPLTEXT 1 0 7 " fi;\n" } {MPLTEXT 1 0 21 " if a[k,k]<>0 then\n" }{MPLTEXT 1 0 28 " for j fr om k+1 to n do\n" }{MPLTEXT 1 0 65 " x:=-a[k,j]/a[k,k]; a:=addcol (a,k,j,x);a:=addrow(a,k,j,x);\n" }{MPLTEXT 1 0 9 " od;\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 18 "od; evalm(a); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"bG6\"6(I\"aGF%I\"iGF%I\"jGF%I\"kGF%I\"nGF%I\"x GF%F%F%C(@$45-I%typeG%*protectedG6$F$-._I'linalgG6+F3I(_syslibGF%F:F:F :F:F:F:F:I'matrixGF%6$I)rationalGF3I'squareGF%-F26$F$-F66$I&floatGF3F> -I&ERRORGF36#I1Invalid~argumentGF%>F+-_F8I'coldimGF%F#?(F(\"\"\"FMF+I% trueGF3?(F),&F(FMFMFMFMF+FN@$0&F$6$F(F)&F$6$F)F(FD>F'-I%copyGF%F#?(F*F MFMF+FNC%@$/&F'6$F*F*\"\"!C$?(F),&F*FMFMFMFMF+FN@$0&F'6$F)F)Fjn[@$1F)F +C$>F'-_F8I(swapcolGF%6%F'F*F)>F'-_F8I(swaprowGF%Fjo@$FgnC$?(F)F]oFMF+ FN@$0&F'6$F*F)FjnFbo@$FdoC$>F'-_F8I'addcolGF%6%F'F)F*>F'-_F8I'addrowGF %F\\q@$0FhnFjn?(F)F]oFMF+FNC%>F,,$*&FdpFMFhn!\"\"Fhq>F'-Fjp6&F'F*F)F,> F'-F_qF[r-I&evalmG6$F3F:6#F'F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "princaxis(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$ ;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"!6$\"\"#F&F*6$F'F,F*6$F,F,!\"\"6$F,F'F*6 $F'F&F*6$F&F'F*6$F&F&F&6$F&F,F*" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "1.2.35. Sylvester-f" }{TEXT 207 45 "\303\251le tehetetlens\303 \251gi t\303\251tel" }{TEXT 207 2 ". " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "*1.2.36. Konjug" }{TEXT 207 55 "\303\241lt biline\303\24 1ris lek\303\251pez\303\251sek." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 10 "*1.2.37. T" }{TEXT 207 11 "\303\251tel" }{TEXT 207 1 "." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 36 "*1.2.38. K\303\266vetkezm\303\251ny." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "*1.2.39. Hermite-form" }{TEXT 207 10 "\303\241k." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 56 "*1.2.40. Multiline\303\241ris lek\303\251pez\303\251sek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*1.2.41. T\303 \251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*1.2.42. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 33 "*1.2.43. Tenzorok, tenzorszo rzat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "*1.2.44. Seg\303\251dt\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 78 "*1.2.45. Tenzo rok koordin\303\241t\303\241inak transzform\303\241ci\303\263ja." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "*1.2.46. Einstein-konvenci\303\263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "1.2.47. Permut\303\241ci \303\263k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT 207 21 "1.2.48. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.49. K\303\266vetkezm\30 3\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 69 "1.2.50. Ter\303\274let, t\303\251rfogat \303\251s determ in\303\241ns." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.51. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.52. K\303\266vetkezm \303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 36 "*1.2.53. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.54. K\3 03\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 42 "1.2.55. M\303\241trix determin\303\241ns a." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a:=matrix(4,4); det(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%F%E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",R**&I \"aG6\"6$\"\"\"F(F(&F%6$\"\"#F+F(&F%6$\"\"$F.F(&F%6$\"\"%F1F(F(**F$F(F )F(&F%6$F.F1F(&F%6$F1F.F(!\"\"**F$F(&F%6$F.F+F(&F%6$F+F.F(F/F(F7**F$F( F9F(&F%6$F+F1F(F5F(F(**F$F(&F%6$F1F+F(F;F(F3F(F(**F$F(FAF(F>F(F,F(F7** &F%6$F+F(F(&F%6$F(F+F(F,F(F/F(F7**FEF(FGF(F3F(F5F(F(**FEF(F9F(&F%6$F(F .F(F/F(F(**FEF(F9F(&F%6$F(F1F(F5F(F7**FEF(FAF(FKF(F3F(F7**FEF(FAF(FNF( F,F(F(**&F%6$F.F(F(FGF(F;F(F/F(F(**FSF(FGF(F>F(F5F(F7**FSF(F)F(FKF(F/F (F7**FSF(F)F(FNF(F5F(F(**FSF(FAF(FKF(F>F(F(**FSF(FAF(FNF(F;F(F7**&F%6$ F1F(F(FGF(F;F(F3F(F7**FenF(FGF(F>F(F,F(F(**FenF(F)F(FKF(F3F(F(**FenF(F )F(FNF(F,F(F7**FenF(F9F(FKF(F>F(F7**FenF(F9F(FNF(F;F(F(" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.56. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a:=matrix( 1,1); det(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"F&F%E\\[l! " }}{PARA 11 "" 1 "" {XPPMATH 20 "&I\"aG6\"6$\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a:=matrix(2,2); det(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"#F%E\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&&I\"aG6\"6$\"\"\"F(F(&F%6$\"\"#F+F(F(*&&F%6$F(F+F(&F% 6$F+F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a:=matrix( 3,3); det(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E \\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*(&I\"aG6\"6$\"\"\"F(F(&F%6$ \"\"#F+F(&F%6$\"\"$F.F(F(*(F$F(&F%6$F+F.F(&F%6$F.F+F(!\"\"*(&F%6$F+F(F (&F%6$F(F+F(F,F(F4*(F6F(&F%6$F(F.F(F2F(F(*(&F%6$F.F(F(F8F(F0F(F(*(F>F( F;F(F)F(F4" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.57. T\303\25 1tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.58. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 55 "1.2.59. R\303 \251szm\303\241trix, aldetermin\303\241ns." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.60. T\303\251tel ." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.61. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.62. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a: =hilbert(3); det(a); gausselim(a,'ra','da'); da;" }}{PARA 11 "" 1 "" {XPPMATH 20 "=I*symmetricG6\"6$;\"\"\"\"\"$F&E\\[l'6$F(F(#F'\"\"&6$\" \"#F.#F'F(6$F.F(#F'\"\"%6$F'F(F/6$F'F'F'6$F'F.#F'F." }}{PARA 11 "" 1 " " {XPPMATH 20 "#\"\"\"\"%g@" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\" \"\"\"\"$F%E\\[l*6$F'F'#F&\"$!=6$\"\"#F&\"\"!6$F'F-F.6$F-F-#F&\"#76$F- F'F16$F'F&F.6$F&F'#F&F'6$F&F&F&6$F&F-#F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"\"\"%g@" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 38 "1 .2.63. Kifejt\303\251si t\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "delcols(a,2..3);delrows( a,1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$;F&F&E\\[l $6$\"\"#F&#F&F+6$F'F&#F&F'6$F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6 \"6$;\"\"\"\"\"#;F&\"\"$E\\[l'6$F'F&#F&F)6$F'F'#F&\"\"%6$F'F)#F&\"\"&6 $F&F)F.6$F&F&#F&F'6$F&F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "delcols(delrows(a,1..1),1..1);A11:=minor(a,1,1);A12:=minor(a,1,2); A13:=minor(a,1,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"# F%E\\[l%6$F'F&#F&\"\"%6$F'F'#F&\"\"&6$F&F&#F&\"\"$6$F&F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"#F%E\\[l%6$F'F&#F&\"\"%6$F'F'#F &\"\"&6$F&F&#F&\"\"$6$F&F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$; \"\"\"\"\"#F%E\\[l%6$F'F&#F&\"\"$6$F'F'#F&\"\"&6$F&F&#F&F'6$F&F'#F&\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"#F%E\\[l%6$F'F&# F&\"\"$6$F'F'#F&\"\"%6$F&F&#F&F'6$F&F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a[1,1]*det(A11)-a[1,2]*det(A12)+a[1,3]*det(A13);" }} {PARA 11 "" 1 "" {XPPMATH 20 "#\"\"\"\"%g@" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT 207 74 "*1.2.64. Line\303\241ris transzform\303\241ci\303\263 \+ determin\303\241nsa." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 58 "*1.2.65. Determin\303\241nsok szorz\303 \241st\303\251tele." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 57 "1.2.66. Determin\303\241nsok szorz\303\2 41st\303\251tele." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 92 "1.2.67. Me gjegyz\303\251s: line\303\241ris transzform\303\241ci\303\263 determin \303\241nsa." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.68. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.69. T\303\251tel." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "b:=adjoint(a); evalm(a&*b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=I* symmetricG6\"6$;\"\"\"\"\"$F&E\\[l'6$F(F(#F'\"#76$\"\"#F.#\"\"%\"#X6$F .F(#!\"\"F,6$F'F(#F'\"#s6$F'F'#F'\"$S#6$F'F.#F4\"#g" }}{PARA 11 "" 1 " " {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'#F&\"%g@6$\"\"#F&\"\"!6 $F'F-F.6$F-F-F*6$F-F'F.6$F'F&F.6$F&F'F.6$F&F&F*6$F&F-F." }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.70. Line\303\241ris egyenletek." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a:=matrix([[1,1,2,5,0],[0,1,1,3,-1],[1,0,1,2,1]]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$;F&\"\"&E\\[l06$F'F'F&6$\"\"#F& \"\"!6$F'F-F.6$F-F-F&6$F-F'F&6$F'F&F&6$F-\"\"%F'6$F-F)!\"\"6$F&F'F-6$F &F&F&6$F&F-F&6$F&F)F.6$F'F4F-6$F&F4F)6$F'F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "colspace(a); rank(a); kernel(a); coldim(a);" }} {PARA 11 "" 1 "" {XPPMATH 20 "<$=6\"6#;\"\"\"\"\"$E\\[l$F'F'\"\"#\"\"! F(F'=F$F%E\\[l$F'F+F*F'F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%=6\"6#;\"\"\"\"\"&E\\[l&F'!\"\"\"\" #F*\"\"$F'F(\"\"!\"\"%F-=F$F%E\\[l&F'!\"#F+!\"$F,F-F(F-F.F'=F$F%E\\[l& F'F*F+F'F,F-F(F'F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b:=vector([0,0,0]); linsolve (a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&\"\" !\"\"#F)F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"&E\\[l& F&,(&I#_tGF#6#F&!\"\"*&\"\"#F&&F+6#F/F&F-&F+6#\"\"$F-F/,(F*F-*&F4F&F0F &F-F2F&F4F*F'F2\"\"%F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b :=vector([1,1,1]); linsolve(a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6 \"6#;\"\"\"\"\"$E\\[l$F&F&\"\"#F&F'F&" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 37 "1.2.71. \303\201ll\303\255t\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 43 "1.2.72. A szu perpoz\303\255ci\303\263 elve." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.73. K\303\266vetkezm\303\251 ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.74. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.75. K\303\266vet kezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 " " 0 "" {TEXT 207 35 "1.2.76. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 35 "1.2.77. \+ K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b:=vector([0,0,0]); linsolve (a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&\"\" !\"\"#F)F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"&E\\[l& F&,(&I#_tGF#6#F&!\"\"*&\"\"#F&&F+6#F/F&F-&F+6#\"\"$F-F/,(F*F-*&F4F&F0F &F-F2F&F4F*F'F2\"\"%F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b :=vector([1,1,0]); linsolve(a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6 \"6#;\"\"\"\"\"$E\\[l$F&F&\"\"#F&F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"&E\\[l&F&,(&I#_tGF#6#\"\"#!\"\"*&F-F&&F+6#F&F&F.& F+6#\"\"$F.F-,*F&F&F*F.*&F4F&F0F&F.F2F&F4F*F'F2\"\"%F0" }}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 33 "1.2.78. Invari\303\241ns alterek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 86 "1.2.79. Saj\30 3\241t\303\251rt\303\251k, saj\303\241tvektor, saj\303\241talt\303\251 r." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 33 "1.2.80. Karakterisztikus polinom." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=matrix([[1,2,I ],[2,-I,3],[-1,2*I,4]]);\n" }{MPLTEXT 1 0 61 "evalm(-charmat(a,lambda) ); det(%); (-1)^3*charpoly(a,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"%6$\"\"#F&F,6$F'F,*&F,F&^#F&F&6$F ,F,,$F/!\"\"6$F,F'F'6$F'F&F26$F&F'F/6$F&F&F&6$F&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F',&I'lambdaGF#!\"\"\"\" %F&6$\"\"#F&F/6$F'F/*&F/F&^#F&F&6$F/F/,&F+F,F2F,6$F/F'F'6$F'F&F,6$F&F' F26$F&F&,&F+F,F&F&6$F&F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*$)I'lam bdaG6\"\"\"$\"\"\"!\"\"*&\"\"&F()F%\"\"#F(F(*&,$^#F(F(F(F,F(F)*&*&\"#5 F(F0F(F(F%F(F(,&\"#DF)F2F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*$)I'l ambdaG6\"\"\"$\"\"\"!\"\"*&\"\"&F()F%\"\"#F(F(*&,$^#F(F(F(F,F(F)*&*&\" #5F(F0F(F(F%F(F(,&\"#DF)F2F)F(" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.2.81. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "evalf(solve(%,lambda)); %[1]+%[2]+% [3]; trace(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&$\"+q.\\&R%!\"*\" \"\"*&$\"+e&y:N\"F&F'^#F'F'F',&$\"+bBoMCF&F'*&$\"+ " 0 "" {MPLTEXT 1 0 20 "evalf(eigenvals(a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&$\"+q.\\&R%!\"*\"\"\"*&$\"+e&y:N\"F&F'^#F'F'F',&$\"+bBoMCF&F'*&$ \"+ " 0 "" {MPLTEXT 1 0 21 "evalf(eigenvects(a));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%7%,&$\"+q.\\&R%!\"*\"\"\"*&$\"+e&y:N\" F'F(^#F(F(F($F(\"\"!<#=6\"6#;F(\"\"$E\\[l$F(F-\"\"#,&$\"+O^]a>F'F(*&$ \"+\\\"*p^p!#5F(F,F(!\"\"F4,&$\"+(Q=>u#F'F(*&$\"*,*>N^F'F(F,F(F(7%,&$ \"+bBoMCF'F(*&$\"+F-<#=F1F2E\\[l$F(F-F6,&$\"+!yaT2&F=F( *&$\"+_ie%p%F=F(F,F(F>F4,&$\"+hm=3HF=F>*&$\"+#)R^)>%F=F(F,F(F>7%,&$\"+ BF*&$\"+VE%=7\"F'F(F,F(F>F-<#=F1F2E\\[l$F(F-F6,&$\"+81#>@\"F'F> *&$\"+7Yr`eF=F(F,F(F>F4,&$\"*4G+*[F=F(*&$\"+k\\JjSF=F(F,F(F(" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 38 "1.2.82. Algebrai multiplicit\303 \241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "matrix([[1,1,0],[0,1,0],[0,0,5]]); eigenvects(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"&6$ \"\"#F&\"\"!6$F'F,F-6$F,F,F&6$F,F'F-6$F'F&F-6$F&F'F-6$F&F&F&6$F&F,F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"&\"\"\"<#=6\"6#;F%\"\"$E\\[l$F %\"\"!\"\"#F-F+F%7%F%F.<#=F(F)E\\[l$F%F%F.F-F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=matrix([[1,1,0],[0,1,0],[0,0,1]]); eigenve cts(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F 'F'F&6$\"\"#F&\"\"!6$F'F+F,6$F+F+F&6$F+F'F,6$F'F&F,6$F&F'F,6$F&F&F&6$F &F+F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"\"\"\"$<$=6\"6#;F#F$E\\[l $F#F#\"\"#\"\"!F$F,=F'F(E\\[l$F#F,F+F,F$F#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "matrix([[1,1,0],[0,1,1],[0,0,1]]); eigenvects(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'F&6$\" \"#F&\"\"!6$F'F+F,6$F+F+F&6$F+F'F&6$F'F&F,6$F&F'F,6$F&F&F&6$F&F+F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"\"\"\"$<#=6\"6#;F#F$E\\[l$F#F#\"\" #\"\"!F$F," }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "*1.2.83. Cayley- Hamilton-t" }{TEXT 207 12 "\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a:=matrix([[1,0,0],[1 ,1,0],[0,0,2]]);" }{MPLTEXT 1 0 34 " p:=charpoly(a,lambda); expand(p); " }{MPLTEXT 1 0 123 "\nL:=PolynomialTools[CoefficientList](p,lambda); \nL:=zip((x,y)->x*y,L,[a&^i$i=0..nops(L)-1]);\np:=convert(L,`+`); eval m(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F '\"\"#6$F*F&F&6$F'F*\"\"!6$F*F*F&6$F*F'F-6$F'F&F-6$F&F'F-6$F&F&F&6$F&F *F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&),&I'lambdaG6\"\"\"\"F'!\"\"\" \"#F',&F%F'F)F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**$)I'lambdaG6\"\" \"$\"\"\"F(*&\"\"%F()F%\"\"#F(!\"\"*&\"\"&F(F%F(F(F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&!\"#\"\"&!\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&,$*&\"\"#\"\"\"-I#&^G6\"6$I\"aGF)\"\"!F&!\"\",$*&\"\"&F&-F(6$F+F &F&F&,$*&\"\"%F&-F(6$F+F%F&F--F(6$F+\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"\"#\"\"\"-I#&^G6\"6$I\"aGF(\"\"!F%!\"\"*&\"\"&F%-F' 6$F*F%F%F%*&\"\"%F%-F'6$F*F$F%F,-F'6$F*\"\"$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"!6$\"\"#F&F*6$F'F,F*6 $F,F,F*6$F,F'F*6$F'F&F*6$F&F'F*6$F&F&F*6$F&F,F*" }}}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 50 "1.2.84. Fels\305\221 h\303\241romsz\303\266g alak ." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "t:=matrix([[1,0,0],[0,1,2],[0,1,1]]); tm:=inverse(t); b:=evalm(tm&*a&*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\" \"$F%E\\[l*6$F'F'F&6$\"\"#F&\"\"!6$F'F+F&6$F+F+F&6$F+F'F+6$F'F&F,6$F&F 'F,6$F&F&F&6$F&F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\" $F%E\\[l*6$F'F'!\"\"6$\"\"#F&\"\"!6$F'F,F&6$F,F,F*6$F,F'F,6$F'F&F-6$F& F'F-6$F&F&F&6$F&F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\" \"$F%E\\[l*6$F'F'\"\"!6$\"\"#F&!\"\"6$F'F,F-6$F,F,F'6$F,F'F,6$F'F&F&6$ F&F'F*6$F&F&F&6$F&F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "e igenvects(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"\"\"\"#<#=6\"6# ;F$\"\"$E\\[l$F$\"\"!F%!\"\"F+F$7%F%F$<#=F(F)E\\[l$F$F-F%!\"#F+F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "f11:=vector([0,-2,1]); e11: =vector([1,0,0]); e12:=vector([0,1,0]); e13:=vector([0,0,1]); basis([f 11,e11,e12,e13]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E \\[l$F&\"\"!\"\"#!\"#F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\" \"\"\"$E\\[l$F&F&\"\"#\"\"!F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6 #;\"\"\"\"\"$E\\[l$F&\"\"!\"\"#F&F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&\"\"!\"\"#F)F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%I$f11G6\"I$e11GF$I$e12GF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "t1:=matrix([[0,1,0],[-2,0,1],[1,0,0]]); t1m:=inverse( t1);\nbb:=evalm(t1m&*b&*t1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$; \"\"\"\"\"$F%E\\[l*6$F'F'\"\"!6$\"\"#F&!\"#6$F'F,F*6$F,F,F*6$F,F'F&6$F 'F&F&6$F&F'F*6$F&F&F*6$F&F,F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$; \"\"\"\"\"$F%E\\[l*6$F'F'\"\"#6$F*F&F&6$F'F*F&6$F*F*\"\"!6$F*F'F.6$F'F &F.6$F&F'F&6$F&F&F.6$F&F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\" \"\"\"\"$F%E\\[l*6$F'F'F&6$\"\"#F&\"\"!6$F'F+F&6$F+F+F&6$F+F'F,6$F'F&F ,6$F&F'!\"\"6$F&F&F+6$F&F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b1:=matrix([[1,0],[1,1]]); eigenvects(b1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"#F%E\\[l%6$F'F&F&6$F'F'F&6$F&F&F&6$F&F' \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"\"\"\"#<#=6\"6#;F#F$E\\[l #F#\"\"!F$F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f22:=vector ([0,1]); e22:=vector([1,0]); e23:=vector([0,1]);\nbasis([f22,e22,e23]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"#E\\[l#F&\"\"!F'F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"#E\\[l#F&F&F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"#E\\[l#F&\"\"!F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I$f22G6\"I$e22GF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "t2:=matrix([[0,1],[1,0]]); t2m:=inverse(t2 ); bb1:=evalm(t2m&*b1&*t2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\" \"\"\"\"#F%E\\[l%6$F'F&F&6$F'F'\"\"!6$F&F&F+6$F&F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"#F%E\\[l%6$F'F&F&6$F'F'\"\"!6$F&F&F+6 $F&F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"#F%E\\[l%6$F 'F&\"\"!6$F'F'F&6$F&F&F&6$F&F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b2:=matrix([[1]]); eigenvects(b2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"F&F%E\\[l\"6$F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"\"F#<#=6\"6#;F#F#E\\[l\"F#F#" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "f23:=1*e22; print(f23);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$e22G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$e22G6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "print(f11); f12:=0*e11+1*e12 ; print(f12); f13:=1*e11+0*e12; print(f13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$f11G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$e12G6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "I$e12G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$e11G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$e11G6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "t:=matrix([[0,0,1],[-2,1,0],[1,0,0] ]); tm:=inverse(t);\nprint(evalm(tm&*b&*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"!6$\"\"#F&!\"#6$F'F,F *6$F,F,F&6$F,F'F*6$F'F&F&6$F&F'F&6$F&F&F*6$F&F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'\"\"!6$\"\"#F&F*6$F'F,F*6 $F,F,F&6$F,F'F,6$F'F&F&6$F&F'F&6$F&F&F*6$F&F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F'F&6$\"\"#F&\"\"!6$F'F+F,6 $F+F+F&6$F+F'F&6$F'F&F,6$F&F'F&6$F&F&F+6$F&F+!\"\"" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 8 "1.2.85. " }{TEXT 207 27 "K\303\266vetkezm\303\2 51ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 33 "1.2.86. Seg\303\251dt\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 8 "1.2.87. " } {TEXT 207 25 "\303\201tl\303\263s alak." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "a:=matrix([[2,0,0],[ 0,1,0],[0,0,3]]);\nt:=matrix([[1,-4,3],[1,-2,2],[3,-3,-2]]); tm:=inver se(t);\nb:=evalm(tm&*a&*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\" \"\"\"\"$F%E\\[l*6$F'F'F'6$\"\"#F&\"\"!6$F'F+F,6$F+F+F&6$F+F'F,6$F'F&F ,6$F&F'F,6$F&F&F+6$F&F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\" \"\"\"$F%E\\[l*6$F'F'!\"#6$\"\"#F&F&6$F'F,!\"$6$F,F,F*6$F,F'F,6$F'F&F' 6$F&F'F'6$F&F&F&6$F&F,!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\" \"\"\"\"$F%E\\[l*6$F'F'#!\"#\"#86$\"\"#F&#!\")F,6$F'F.#\"\"*F,6$F.F.# \"#6F,6$F.F'#!\"\"F,6$F'F&#!\"$F,6$F&F'#F.F,6$F&F&#!#5F,6$F&F.#\"# " 0 "" {MPLTEXT 1 0 17 "L:=eigenvects(b);" }{MPLTEXT 1 0 19 " b1:=op(1,L[1][3 ]);" }{MPLTEXT 1 0 19 " b2:=op(1,L[2][3]);" }{MPLTEXT 1 0 19 " b3:=op( 1,L[3][3]);" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 23 "tb:=augment(b1,b2, b3); " }{MPLTEXT 1 0 26 "evalm(inverse(tb)&*b&*tb);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%7%\"\"$\"\"\"<#=6\"6#;F%F$E\\[l$F%!\"#\"\"#F%F$F-7%F- F%<#=F(F)E\\[l$F%#\"#5F$F-#\"\")F$F$F%7%F%F%<#=F(F)E\\[l$F%#\"#<\"\"*F -#\"#6F " 0 "" {MPLTEXT 1 0 21 "restart;with(linalg);" }} {PARA 11 "" 1 "" {XPPMATH 20 "7^rI.BlockDiagonalG6\"I,GramSchmidtGF$I, JordanBlockGF$I)LUdecompGF$I)QRdecompGF$I*WronskianGF$I'addcolGF$I'add rowGF$I$adjGF$I(adjointGF$I&angleGF$I(augmentGF$I(backsubGF$I%bandGF$I &basisGF$I'bezoutGF$I,blockmatrixGF$I(charmatGF$I)charpolyGF$I)cholesk yGF$I$colGF$I'coldimGF$I)colspaceGF$I(colspanGF$I*companionGF$I'concat GF$I%condGF$I)copyintoGF$I*crossprodGF$I%curlGF$I)definiteGF$I(delcols GF$I(delrowsGF$I$detGF$I%diagGF$I(divergeGF$I(dotprodGF$I*eigenvalsGF$ I,eigenvaluesGF$I-eigenvectorsGF$I+eigenvectsGF$I,entermatrixGF$I&equa lGF$I,exponentialGF$I'extendGF$I,ffgausselimGF$I*fibonacciGF$I+forward subGF$I*frobeniusGF$I*gausselimGF$I*gaussjordGF$I(geneqnsGF$I*genmatri xGF$I%gradGF$I)hadamardGF$I(hermiteGF$I(hessianGF$I(hilbertGF$I+htrans poseGF$I)ihermiteGF$I*indexfuncGF$I*innerprodGF$I)intbasisGF$I(inverse GF$I'ismithGF$I*issimilarGF$I'iszeroGF$I)jacobianGF$I'jordanGF$I'kerne lGF$I*laplacianGF$I*leastsqrsGF$I)linsolveGF$I'mataddGF$I'matrixGF$I&m inorGF$I(minpolyGF$I'mulcolGF$I'mulrowGF$I)multiplyGF$I%normGF$I*norma lizeGF$I*nullspaceGF$I'orthogGF$I*permanentGF$I&pivotGF$I*potentialGF$ I+randmatrixGF$I+randvectorGF$I%rankGF$I(ratformGF$I$rowGF$I'rowdimGF$ I)rowspaceGF$I(rowspanGF$I%rrefGF$I*scalarmulGF$I-singularvalsGF$I&smi thGF$I,stackmatrixGF$I*submatrixGF$I*subvectorGF$I)sumbasisGF$I(swapco lGF$I(swaprowGF$I*sylvesterGF$I)toeplitzGF$I&traceGF$I*transposeGF$I,v andermondeGF$I*vecpotentGF$I(vectdimGF$I'vectorGF$I*wronskianGF$" }}} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 13 "1.3.1. Norma." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "1.3.2. P\303\251ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x:=vector([ 1,1,1]);\n" }{MPLTEXT 1 0 74 "norm(x,1), norm(x,1.5), norm(x,2), norm( x,3), norm(x,infinity), norm(x);\n" }{MPLTEXT 1 0 9 "evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&F&\"\"#F&F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"$$\"+BQ3!3#!\"**$)F##\"\"\"\"\" #F**$)F##F*F#F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"\"$\"\"!$\"+ BQ3!3#!\"*$\"+330K " 0 "" {MPLTEXT 1 0 25 "y:=vector([1,2,3+4*I]);\n" }{MPLTEXT 1 0 81 "no rm(y,1), norm(y,1.5), norm(y,2), norm(y,3), norm(y,infinity), norm(y); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&F &\"\"#F)F',&F'F&*&\"\"%F&^#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6( \"\")$\"+n;d%3'!\"**$)\"#I#\"\"\"\"\"#F+*$)\"$M\"#F+\"\"$F+\"\"&F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6($\"\")\"\"!$\"+n;d%3'!\"*$\"+vbAxaF($ \"+Z*Hs6&F($\"\"&F%F-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%? G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 28 "1.3.3. Bels\305\221 szorz at." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "1.3.4. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "1.3.5. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "dot prod(x,y); dotprod(y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"'\"\" \"*&\"\"%F$^#F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"'\"\"\"* &\"\"%F$^#F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }} }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 25 "1.3.6. Megjegyz\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 77 "1.3.7. Sz\303\266ge k, euklid\303\251szi t\303\251r, unit\303\251r t\303\251r." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "y:= vector([1,-1,0]); angle(x,y); angle(y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&F&\"\"#!\"\"F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"\"#F%I#PiG%*protectedGF%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"\"#F%I#PiG%*protectedGF%F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 31 "1.3.8. Defin\303\255ci\303\263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "z:=vect or([2,3,4]); dotprod(x,y); dotprod(y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&\"\"#F)F'F'\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 22 "1.3.9. Vegyes szorzat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "dotprod(x,crossprod(y,z)); det(augment(x,y,z) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"$" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 69 "1.3.10. \303\201lt al\303\241nos\303\255tott Pythagorasz-t\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.3.11. K\303 \266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 51 "1.3.12. Gram-Schmidt-ortogonaliz\303\241 l\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "basis([z,y,x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7 %I\"zG6\"I\"yGF$I\"xGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " GramSchmidt([z,y,x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%7%\"\"#\"\"$ \"\"%7%#\"#J\"#H#!#EF*#F&F*7%#F&\"#>F/#!\"&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "L:=map(normalize,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%=6\"6#;\"\"\"\"\"$E\\[l$F',$*&#\"\"#\"#HF')F.#F'F-F'F'F -,$*&#F(F.F'F/F'F'F(,$*&#\"\"%F.F'F/F'F'=F$F%E\\[l$F',$*(#\"#J\"%`;F') \"#dF0F'F/F'F'F-,$*(#\"#EF>F'F?F'F/F'!\"\"F(,$*(#F7F>F'F?F'F/F'F'=F$F% E\\[l$F',$*(#F7F@F')F(F0F')\"#>F0F'F'F-FKF(,$*(#\"\"&F@F'FNF'FOF'FE" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map(norm,L,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"\"F#F#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.3. 13. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.3.14. K\303\266vetkezm\303\251 ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.15. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 80 "1.3.16. Az approxim\303\241ci\30 3\263 alapfeladata norm\303\241lt t\303\251rben." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 94 "1.3.17. Legjob b line\303\241ris approxim\303\241ci\303\263 bels\305\221 szorzat t\30 3\251rben." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT 207 72 "1.3.18. K\303\266vetkezm\303\251ny: Bessel-egyenl\305 \221tlens\303\251g." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 58 "1.3.19. Ortogon\303\241lis felbont\303\2 41si t\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.3.20. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.3.2 1. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 61 "1.3.22. El\305\221\303\241ll\303 \255t\303\241si t\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 49 "1.3.23. Adjung\303\241lt lek\303 \251pez\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.24. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 56 "1.3.25. Adjung\303\2 41lt oper\303\241tor m\303\241trixa." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a:=matrix([[1,2,I],[2,-I, 3]]); htranspose(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\" \"#;F&\"\"$E\\[l'6$F&F&F&6$F&F'F'6$F'F)F)6$F&F)^#F&6$F'F',$F/!\"\"6$F' F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$;F&\"\"#E\\[l' 6$F'F&,$^#F&!\"\"6$F&F&F&6$F&F)F)6$F'F)F'6$F)F)F-6$F)F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "d:=matrix(L); ds:=htranspose(d);" } }{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&,$*(#\" \"%\"#dF&)F'#F&\"\"#F&)\"#>F0F&F&6$F&F&,$*&#F1\"#HF&)F8F0F&F&6$F&F1,$* &#F'F8F&F9F&F&6$F1F',$*(#F-\"%`;F&)F.F0F&F9F&F&6$F'F1F*6$F&F',$*&#F-F8 F&F9F&F&6$F'F',$*(#\"\"&F.F&F/F&F2F&!\"\"6$F1F1,$*(#\"#EFBF&FCF&F9F&FN 6$F1F&,$*(#\"#JFBF&FCF&F9F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$; \"\"\"\"\"$F%E\\[l*6$F'F&,$*&#\"\"%\"#HF&)F.#F&\"\"#F&F&6$F&F&,$*&#F1F .F&F/F&F&6$F&F1,$*(#\"#J\"%`;F&)\"#dF0F&F/F&F&6$F1F',$*(#F-F=F&)F'F0F& )\"#>F0F&F&6$F'F1,$*(#F-F;F&F " 0 "" {MPLTEXT 1 0 27 "evalm(d&*ds); evalm(ds&*d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&\"\"! 6$F&F&F&6$F&\"\"#F*6$F-F'F*6$F'F-F*6$F&F'F*6$F'F'F&6$F-F-F&6$F-F&F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&\"\"!6$ F&F&F&6$F&\"\"#F*6$F-F'F*6$F'F-F*6$F&F'F*6$F'F'F&6$F-F-F&6$F-F&F*" }}} }{SECT 0 {PARA 5 "" 0 "" {TEXT 207 90 "1.3.26. \303\226nadjung\303\241 lt \303\251s norm\303\241lis transzform\303\241ci\303\263k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "b: =matrix([[1,2,I],[2,3,-3*I],[-I,3*I,-1]]); bs:=htranspose(b);" }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&,$^#F&! \"\"6$F&F&F&6$F&\"\"#F/6$F/F',$*&F'F&F+F&F,6$F'F/F26$F&F'F+6$F'F'F,6$F /F/F'6$F/F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E \\[l*6$F'F&,$^#F&!\"\"6$F&F&F&6$F&\"\"#F/6$F/F',$*&F'F&F+F&F,6$F'F/F26 $F&F'F+6$F'F'F,6$F/F/F'6$F/F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalm(b&*bs); evalm(bs&*b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "= 6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&*&\"\"'F&^#F&F&6$F&F&F+6$F&\"\"#\"\"&6$ F/F',$*&\"\"%F&F,F&!\"\"6$F'F/F36$F&F',$F*F56$F'F'\"#66$F/F/\"#A6$F/F& F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&*& \"\"'F&^#F&F&6$F&F&F+6$F&\"\"#\"\"&6$F/F',$*&\"\"%F&F,F&!\"\"6$F'F/F36 $F&F',$F*F56$F'F'\"#66$F/F/\"#A6$F/F&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "c:=matrix([[I,2,1],[-2,3*I,4],[-1,-4,0]]); cs:=htrans pose(c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$ F'F&!\"\"6$F&F&^#F&6$F&\"\"#F.6$F.F'\"\"%6$F'F.!\"%6$F&F'F&6$F'F'\"\"! 6$F.F.*&F'F&F,F&6$F.F&!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\" \"\"\"\"$F%E\\[l*6$F'F&F&6$F&F&,$^#F&!\"\"6$F&\"\"#!\"#6$F/F'!\"%6$F'F /\"\"%6$F&F'F-6$F'F'\"\"!6$F/F/,$*&F'F&F,F&F-6$F/F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalm(c&*cs); evalm(cs&*c);" }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&,&\"\")! \"\"^#F&F&6$F&F&\"\"'6$F&\"\"#,&\"\"%F&*&F+F&F-F&F,6$F1F',&F1F&*&\"#7F &F-F&F,6$F'F1,&F1F&F7F&6$F&F',&F+F,F-F,6$F'F'\"#<6$F1F1\"#H6$F1F&,&F3F &F4F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F &,&\"\")!\"\"^#F&F&6$F&F&\"\"'6$F&\"\"#,&\"\"%F&*&F+F&F-F&F,6$F1F',&F1 F&*&\"#7F&F-F&F,6$F'F1,&F1F&F7F&6$F&F',&F+F,F-F,6$F'F'\"#<6$F1F1\"#H6$ F1F&,&F3F&F4F&" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.27. P\30 3\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "a:=matrix([[1,2,I],[2,-I,3],[-1,2*I,4]]); as:=htrans pose(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$ F'F&!\"\"6$F&F&F&6$F&\"\"#F-6$F-F'F'6$F'F-*&F-F&^#F&F&6$F&F'F16$F'F'\" \"%6$F-F-,$F1F*6$F-F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\" \"\"$F%E\\[l*6$F'F&,$^#F&!\"\"6$F&F&F&6$F&\"\"#F/6$F/F',$*&F/F&F+F&F,6 $F'F/F'6$F&F'F,6$F'F'\"\"%6$F/F/F+6$F/F&F/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "evalm(a&*as); evalm(as&*a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&!\"\"6$F&F&\"\"'6$F&\"\"# ,&F.F&*&\"\"&F&^#F&F&F&6$F.F'\"\")6$F'F.F46$F&F'F*6$F'F'\"#@6$F.F.\"#9 6$F.F&,&F.F&F0F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F% E\\[l*6$F'F&,&\"\"#F&^#F&!\"\"6$F&F&\"\"'6$F&F+,&F+F&*&\"\"%F&F,F&F-6$ F+F',$*&F'F&F,F&F-6$F'F+F66$F&F',&F+F&F,F&6$F'F'\"#E6$F+F+\"\"*6$F+F&, &F+F&F2F&" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 42 "1.3.28.Polariz\30 3\241ci\303\263s formula." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.3.29. K\303\266vetkezm\303\251ny." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.3.30. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.31. T\303\251tel." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "1.3.32. Megjegyz\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "orthog(d); orthog(ds);" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.33. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 50 "1.3.34. Fels\305\221 h\303\241ro msz\303\266g alak." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a:=map(evalf,a); as:=map(evalf,as);\n" } {MPLTEXT 1 0 45 "eigenvects(as); e3:=normalize(op(1,%[1][3]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&$!\"\"\" \"!6$F&F&$F&F,6$F&\"\"#$F0F,6$F0F'$F'F,6$F'F0*&F1F&^#F&F&6$F&F'*&F.F&F 6F&6$F'F'$\"\"%F,6$F0F0,$F8F+6$F0F&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&,$*&$F&\"\"!F&^#F&F&!\"\"6$F&F&F,6$F& \"\"#$F2F-6$F2F',$*&F3F&F.F&F/6$F'F2$F'F-6$F&F'$F/F-6$F'F'$\"\"%F-6$F2 F2F+6$F2F&F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%,&$\"+o.\\&R%!\"*\" \"\"*&$\"+c&y:N\"F'F(^#F(F(!\"\"F(<#=6\"6#;F(\"\"$E\\[l$F(,&$\"+\\HI9? !#5F(*&$\"+W\\c\\>F8F(F,F(F-\"\"#,&$\"+^?bn[F8F(*&$\"*p!4%*zF8F(F,F(F- F3,&$\"+,![0`&F8F(*&$\"+C&*RVxF8F(F,F(F(7%,&$\"+@F83TF8F-*&$\"+S5gvTF8F(F,F(F(F<,&$\" +m:Q*G#F8F(*&$\"+K5d$*pF8F(F,F(F-F3,&$\"+uDfjBF8F-*&$\"++x?RCF8F(F,F(F (7%,&$\"+ZBoMCF'F(*&$\"+;ftH7F'F(F,F(F(F(<#=F0F1E\\[l$F(,&$\"+b#QJH\"F 8F-*&$\"+9j[M\")F8F(F,F(F-F<,&$\"+()GT^CF8F(*&$\"+d\"ywC%F8F(F,F(F-F3, &$\"+$e,_C$F8F-*&$\"+)>m_w%F8F(F,F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&,&$\"+4n<==!#5F&*&$\"+P?ufi0#*\\F,F&*&$\"+j% \\%*)pF,F&F0F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f11:=ve ctor([1,0,0]); c1:=dotprod(f11,e3);\n" }{MPLTEXT 1 0 33 "g11:=normaliz e(evalm(f11-c1*e3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\" \"$E\\[l$F&F&\"\"#\"\"!F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&$\"+4n< ==!#5\"\"\"*&$\"+P?uf " 0 "" {MPLTEXT 1 0 66 "f12:=vector([0,1 ,0]); c1:=dotprod(f12,e3); c2:=dotprod(f12,g11);\n" }{MPLTEXT 1 0 40 " g12:=normalize(evalm(f12-c1*e3-c2*g11));" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&\"\"!\"\"#F&F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&$\"+mSh$R%!#5\"\"\"*&$\"+J;t:s!#6F&^#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&$\"+u)f&p&*!#6!\"\"*&$\"+NXhNmF%\"\"\"^#F*F*F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&,&$\"+7&35 ,*!#?F&*&$\"+%Qc#enF,F&^#F&F&!\"\"\"\"#,&$\"+t&H!y))!#5F&*&$\"\"!F9F&F 0F&F&F',&$\"+*ycD.#F6F1*&$\"+r-3HTF6F&F0F&F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "a1:=matrix(2,2):a1[1,1]:=dotprod(multiply(a,g11), g11):\n" }{MPLTEXT 1 0 40 "a1[2,1]:=dotprod(multiply(a,g11),g12):\n" } {MPLTEXT 1 0 40 "a1[1,2]:=dotprod(multiply(a,g12),g11):\n" }{MPLTEXT 1 0 40 "a1[2,2]:=dotprod(multiply(a,g12),g12):\n" }{MPLTEXT 1 0 30 "pr int(a1);a1s:=htranspose(a1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#a1G6\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"#F%E\\[l%6$F&F&,&$ \"+24!=.\"!\"*F&*&$\"+)y&RF5!#5F&^#F&F&F&6$F&F',&$\"+bn " 0 "" {MPLTEXT 1 0 37 "eigenvects(a1s); f2:=op(1,%[1][3]);\n" }{MPLTEXT 1 0 42 "e2:=normalize(evalm(f2[1]*g11+f2[2]*g12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,&$\"+cBoMC!\"*\"\"\"*&$\"+;ftH7F'F(^#F(F(F(F(<#=6\"6 #;F(\"\"#E\\[l#F(,&$\"+tQ*)4w!#5F(*&$\"+?6x%\\#F7F(F,F(!\"\"F2,&$\"+pj @ZfF7F(*&$\"+h%*\\Wq!#6F(F,F(F(7%,&$\"+AF " 0 "" {MPLTEXT 1 0 65 "f21:=vector([1,0,0 ]); c1:=dotprod(f21,e2); c2:=dotprod(f21,e3);\n" }{MPLTEXT 1 0 38 "e1: =normalize(evalm(f21-c1*e2-c2*e3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "= 6\"6#;\"\"\"\"\"$E\\[l$F&F&\"\"#\"\"!F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&$\"+6WDit!#5\"\"\"*&$\"+Poe8CF%F&^#F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 ",&$\"+4n<==!#5\"\"\"*&$\"+P?uf " 0 "" {MPLTEXT 1 0 49 "ta:=augment(e1,e2,e3); evalm(inverse(ta)&*a&*ta);" } }{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&,&$\"+ \"QOK$G!#6F&*&$\"+;mCaB!#5F&^#F&F&F&6$F&F&,&$\"+zi!Rz&F1F&*&$\"\"!F9F& F2F&F&6$F&\"\"#,&$\"+6WDitF1F&*&$\"+Poe8CF1F&F2F&!\"\"6$F;F',&$\"+mSh$ R%F1F&*&$\"+J;t:sF-F&F2F&FB6$F'F;,&$\"+:Cj=7F1FB*&$\"+'yXDP%F1F&F2F&FB 6$F&F',&$\"+4n<==F1F&*&$\"+P?ufi0#*\\F1F&*&$\" +j%\\%*)pF1F&F2F&F&6$F;F;,&$\"+M&zhQ%F1F&*&$\"+v8)=f$F-F&F2F&F&6$F;F&, &$\"+aWv@qF1FB*&$\"+!R(e\"R$F1F&F2F&FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&,&$\"#8!#5F&*&$\"\"%F-F&^#F&F&F&6$F& F&,&$\"+?F " 0 "" {MPLTEXT 1 0 35 "print(b); Lb:=evalf(eigenvects(b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"bG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%,& $\"+J\"p\\1&!\"*\"\"\"*&$F(F'F(^#F(F(!\"\"$F(\"\"!<#=6\"6#;F(\"\"$E\\[ l$F(F-\"\"#,&$\"+wu!3f#F'F(*&$\"+7syez!#>F(F+F(F,F4,&$\"+\\zRw*)F=F(*& $\"+?ek;6F'F(F+F(F(7%,&$\"+\\\"*R+MF'F,*&$\"+SSDgcF=F(F+F(F,F-<#=F1F2E \\[l$F(F-F6,&$\"+gN'eA\"F'F,*&$\"*NG>#zF=F(F+F(F,F4,&$\"+Fa^$>'F=F,*&$ \"+G?n[>F'F(F+F(F(7%,&$\"+>+VN8F'F(*&$\"+/a-m6!#=F(F+F(F(F-<#=F1F2E\\[ l$F(F-F6,&$\"+qz\\'H\"!#5F,*&$\"+Gf\\))[F=F(F+F(F(F4,&$\"*v[QM#F=F,*&$ \"+LhHZfFboF(F+F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "b1:= op(1,Lb[1][3]); b2:=op(1,Lb[2][3]); b3:=op(1,Lb[3][3]);\n" }{MPLTEXT 1 0 57 "tb:=augment(normalize(b1),normalize(b2),normalize(b3));\n" } {MPLTEXT 1 0 26 "evalm(inverse(tb)&*b&*tb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&$F&\"\"!\"\"#,&$\"+wu!3f#!\"*F& *&$\"+7syez!#>F&^#F&F&!\"\"F',&$\"+\\zRw*)F3F&*&$\"+?ek;6F/F&F4F&F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&$F&\"\"!\"\"# ,&$\"+gN'eA\"!\"*!\"\"*&$\"*NG>#z!#>F&^#F&F&F0F',&$\"+Fa^$>'F4F0*&$\"+ G?n[>F/F&F5F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E \\[l$F&$F&\"\"!\"\"#,&$\"+qz\\'H\"!#5!\"\"*&$\"+Gf\\))[!#>F&^#F&F&F&F' ,&$\"*v[QM#F4F0*&$\"+LhHZfF/F&F5F&F0" }}{PARA 11 "" 1 "" {XPPMATH 20 " =6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&,&$\"++6%*)*H!#>F&*&$\"+3BiIP!#5F&^#F& F&F&6$F&F&$\"+p\"=4M$F16$F&\"\"#$\"+$3vS)RF16$F7F',&$\"+\"*eY26F1!\"\" *&$\"+MJuvTF-F&F2F&F&6$F'F7,&$\"+\\IanCF-F>*&$\"+scljxF1F&F2F&F&6$F&F' $\"+l$z>a)F16$F'F',&$\"+m26-?!#?F>*&$\"+&3o,3&F1F&F2F&F>6$F7F7,&$\"+lC $R)[F1F>*&$\"+Nd:cJFPF&F2F&F>6$F7F&,&$\"+pdnb')F1F&*&$\"+!ol*eEF-F&F2F &F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&, &$\"#8!#5F&*&$\"+W>C!y'!#>F&^#F&F&!\"\"6$F&F&,&$\"+K\"p\\1&!\"*F&*&$\" \"!F;F&F2F&F&6$F&\"\"#,&$F=F8F3*&$\"+(3X,[&F1F&F2F&F&6$F=F',&$F&F8F&*& $\"+q%\\@#*)F1F&F2F&F&6$F'F=,&$\"#6F-F&*&$\"+093rdF1F&F2F&F36$F&F',&$F 'F-F3*&$\"+kk&eP\"!#=F&F2F&F&6$F'F',&$\"+>+VN8F8F&*&$\"\"&F1F&F2F&F&6$ F=F=,&$\"+]\"*R+MF8F3*&$\"\"'F1F&F2F&F36$F=F&,&$FgnF-F&*&$\"+2!)R;()F1 F&F2F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "print(c); evalf (eigenvects(c));" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"cG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%,&$\"\"#!\"*!\"\"*&$\"+T/$o7*!#5\"\"\"^#F-F- F-$F-\"\"!<#=6\"6#;F-\"\"$E\\[l$F-,&$\"+Gyv/PF'F(*&$\"+F[j!p\"F'F-F.F- F-F&F/F6,&$\"+:*yB&=F'F(*&$\"+V<)[B$F,F-F.F-F-7%,&$\"+guR8>!#=F-*&$\"+ <5OgjF'F-F.F-F-F/<#=F3F4E\\[l$F-,&$\"*\\%o37F'F-*&$\"+T[$Q%QF,F-F.F-F( F&F/F6,&$\")CUVgF'F-*&$\"+<^)*ykF,F-F.F-F-7%,&$\"+SDg'3#FHF-*&$\"+jS/t KF'F-F.F-F(F/<#=F3F4E\\[l$F-,&$\"*>t-3$F'F-*&$\"+O[$4/&F,F-F.F-F-F&F/F 6,&$\"+&fO,a\"F,F-*&$\"+tU@;8F'F-F.F-F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map(evalf,d); Ld:=eigenvects(%);\n" }{MPLTEXT 1 0 44 "abs(Ld[1][1]); abs(Ld[2][1]); abs(Ld[3][1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"$F%E\\[l*6$F'F&$\"+I%H\")H&!#56$F&F&$\"+ jn!Rr$F,6$F&\"\"#$\"+Z,'3d&F,6$F1F'$\"+e$z$Q)*!#66$F'F1F*6$F&F'$\"+GN \"yU(F,6$F'F'$!+)yhEi'F,6$F1F1$!+&eY\\R'F,6$F1F&$\"+,SuCwF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%,&$\"+D3&=l*!#5!\"\"*&$\"+!*)ych#F'\"\"\"^ #F,F,F,F,<#=6\"6#;F,\"\"$E\\[l$F,,&$\"+)Q@7l%F'F,*&$\"+R%GwS%F'F,F-F,F (\"\"#,&$\"+yfC&R%F'F3$!+s`!)QUF'7%,&F%F(F)F(F,<#=F0F1E\\[l$F,,&F6F,F8F,F;,&F=F(F@F ,F3,&FDF,FGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+-+++5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"*)********!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+-+++5!\"*" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 " 1.3.37. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 77 "1.3.38. Szimmetrikus transzfo rm\303\241ci\303\263 \303\241tl\303\263s alakja." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 61 "2. T\303\266 bbv\303\241ltoz\303\263s f\303\274ggv\303\251nyek" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 65 "3. F\303\274ggv\303\251nysorozatok \303\251s f\303\274ggvenysorok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }