{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "MS Sans Serif" 1 8 128 0 128 1 
0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 
0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "MS Sans Serif" 1 9 0 128 128 1 2 2 2 
0 0 2 0 0 0 1 }{CSTYLE "Roter Text" -1 256 "Tahoma" 0 0 255 0 0 1 2 1 
1 0 0 2 0 0 0 0 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 
{CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 
4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times
" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }
{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 
2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output
" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }
3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" 
-1 -1 "Helvetica" 1 9 128 0 128 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 
0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier
" 1 8 0 128 128 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Seitenumbruch" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 
1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 1 2 0 1 }{PSTYLE "Normal" 
-1 259 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 
1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 
-1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 
0 1 }{PSTYLE "Heading 3" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 
1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0
" -1 262 1 {CSTYLE "" -1 -1 "Helvetica" 1 9 128 0 128 1 2 2 2 2 2 2 1 
1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 263 1 
{CSTYLE "" -1 -1 "Courier" 1 8 0 128 128 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 
0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Seitenumbruch" -1 264 1 {CSTYLE "" -1 
-1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 1 2 
0 1 }{PSTYLE "Normal" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" 
-1 266 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 
1 0 0 8 4 1 0 1 0 2 2 0 1 }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 " Beispiele zu 5.15" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 4 " (a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 
1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee
n redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "
Gegeben sei die folgende Matrix A mit komplexen Eintr\344gen:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=3:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "A:=matrix(n,n, (k,l)->k+l*I);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AGK%'matrixG6#7%7%^$\"\"\"F+^$F+\"\"#^$F+\"\"$7%^$F
-F+^$F-F-^$F-F/7%^$F/F+^$F/F-^$F/F/Q(pprint06\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 184 "Ich berechne die Determinante mit elementaren Zeile
numformungen und benutze danach Regeln aus 5.13. Nach Regel 5.13. (e) \+
\344ndert sich die Determinante nicht bei Operationen vom Typ III." }}
{PARA 0 "" 0 "" {TEXT -1 35 "Folgende Abk\374rzungen f\374hre ich ein:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Z3:=addrow; Z1:=mulrow;Z4:=swap
row;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Z3G%'addrowG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#Z1G%'mulrowG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#Z4G%(swaprowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A1:=
Z1(A,1,(1+I)^(-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1GK%'matrix
G6#7%7%\"\"\"^$#\"\"$\"\"##F*F.^$F.F*7%F0^$F.F.^$F.F-7%^$F-F*^$F-F.^$F
-F-Q(pprint16\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Bei dieser Ope
ration \344ndert sich die Determinante nach Regel 5.13 (f). Es ist " }
{TEXT 258 27 "   det(A)=(1+I) * det (A1)." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "A2:=Z3(A1,1,2,-A1[2,1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A2GK%'matrixG6#7%7%\"\"\"^$#\"\"$\"\"##F*F.^$F.F*7%
\"\"!^$#!\"\"F.F4^$F5F57%^$F-F*^$F-F.^$F-F-Q(pprint26\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A3:=Z3(A2,1,3,-A2[3,1]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#A3GK%'matrixG6#7%7%\"\"\"^$#\"\"$\"\"##F*
F.^$F.F*7%\"\"!^$#!\"\"F.F4^$F5F57%F2F6^$!\"#F9Q(pprint36\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A4:=Z1(A3,2,A3[2,2]^(-1));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4GK%'matrixG6#7%7%\"\"\"^$#\"\"$
\"\"##F*F.^$F.F*7%\"\"!F*F.7%F2^$!\"\"F5^$!\"#F7Q(pprint46\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Hier hat sich wieder etwas ge\344n
dert (5.13 (f)): " }{TEXT 257 49 "    det(A1)=det(A2)=det(A3)= (-1/2-I
/2) * det(A4)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A5:=Z3(A4,
2,3,-A4[3,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5GK%'matrixG6#7%
7%\"\"\"^$#\"\"$\"\"##F*F.^$F.F*7%\"\"!F*F.7%F2F2F2Q(pprint56\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Endergebnis: " }{TEXT 259 20 "    \+
                " }{TEXT 260 8 "det (A) " }{TEXT 261 59 " =(1+I) * (-1
/2-I/2) * det(A5) = (1+I) * (-1/2-I/2) * 0 =  " }{TEXT 262 1 "0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 4 " (b)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 
80 "Warning, the protected names norm and trace have been redefined an
d unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 20 "Die folgende Matrix " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "n:=3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=matrix(
n,n, (k,l)->alpha^(k*l));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGK%'
matrixG6#7%7%%&alphaG*$)F*\"\"#\"\"\"*$)F*\"\"$F.7%F+*$)F*\"\"%F.*$)F*
\"\"'F.7%F/F6*$)F*\"\"*F.Q(pprint66\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 29 "habe Eintr\344ge aus dem K\366rper " }{XPPEDIT 18 0 "F[4]
 = \{0, 1, alpha, alpha+1\};" "6#/&%\"FG6#\"\"%<&\"\"!\"\"\"%&alphaG,&
F+F*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Es gelten in
 " }{XPPEDIT 18 0 "F[4]" "6#&%\"FG6#\"\"%" }{TEXT -1 29 " die folgende
n Rechenregeln: " }}{PARA 0 "" 0 "" {TEXT -1 75 "1+1=0 und deswegen a+
a=1*a+1*a=(1+1)*a = 0 f\374r alle Elemente a des K\366rpers." }}{PARA 
0 "" 0 "" {XPPEDIT 18 0 "alpha^2+alpha+1 = 0;" "6#/,(*$%&alphaG\"\"#\"
\"\"F&F(F(F(\"\"!" }{TEXT -1 20 "   , und deswegen   " }{XPPEDIT 18 0 
"alpha^2 = alpha+1;" "6#/*$%&alphaG\"\"#,&F%\"\"\"F(F(" }{TEXT -1 2 " \+
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
36 " So wird Maple gesagt:   rechne mit " }{XPPEDIT 18 0 "alpha;" "6#%
&alphaG" }{TEXT -1 11 "  so, dass " }{XPPEDIT 18 0 "alpha^2+alpha+1 = \+
0" "6#/,(*$%&alphaG\"\"#\"\"\"F&F(F(F(\"\"!" }{TEXT -1 2 " :" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "alias(alpha=RootOf(x^2+x+1));" }{TEXT -1 
1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&alphaG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 34 "Zuerst rechne ich die Eintr\344ge in " }{XPPEDIT 
18 0 "A;" "6#%\"AG" }{TEXT -1 47 " so um, dass man erkennt, welche Ele
mente aus  " }{XPPEDIT 18 0 "F[4];" "6#&%\"FG6#\"\"%" }{TEXT -1 15 "  \+
gemeint sind:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=map(modp ,simpl
ify(A,RootOf),2);" }{TEXT -1 212 "  map(modp,  ....  , 2) zwingt Maple
 in allen Eintr\344gen der Matrix A so zu rechnen, dass die Regel \"1+
1=0\" beachtet wird (modulo 2). simplify(  ...  , RootOf) zwingt Maple
 so zu rechnen dass die Vereinbarung f\374r " }{XPPEDIT 18 0 "alpha;" 
"6#%&alphaG" }{TEXT -1 19 "  eingehalten wird." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AGK%'matrixG6#7%7%%&alphaG,&\"\"\"F,F*F,F,7%F+F*F,7
%F,F,F,Q(pprint76\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Jetzt wir
d wie im Beispiel (a) mit elementaren Umformungen bearbeitet. Ich benu
tze die gleichen Abk\374rzungen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 
"Z3:=addrow; Z1:=mulrow;Z4:=swaprow;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#Z3G%'addrowG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Z1G%'mulrowG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Z4G%(swaprowG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 25 "A1:=Z1(A,1,(alpha)^(-1));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#A1GK%'matrixG6#7%7%\"\"\"*&%&alphaG!\"\",&F*F*F,F
*F**&F*F*F,F-7%F.F,F*7%F*F*F*Q(pprint86\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 124 "Auch hier, wie bei jedem weiteren Schritt,  muss dann da
rauf bestanden werden, dass obige Rechenregeln eingehalten werden.  " 
}{XPPEDIT 18 0 "A1;" "6#%#A1G" }{TEXT -1 31 "  hat danach die folgende
 Form:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A1:=map(modp ,simplify(A1
,RootOf),2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1GK%'matrixG6#7%7%
\"\"\"%&alphaG,&F*F*F+F*7%F,F+F*7%F*F*F*Q(pprint96\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 34 "Meine weiteren Rechenschritte sind" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 60 "A2:=map(modp ,simplify(   Z3(A1,1,2,-A1[2,1]
)   ,RootOf),2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2GK%'matrixG6#
7%7%\"\"\"%&alphaG,&F*F*F+F*7%\"\"!F,F,7%F*F*F*Q)pprint106\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A3:=map(modp ,simplify(   Z3
(A2,1,3,-A2[3,1])   ,RootOf),2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#A3GK%'matrixG6#7%7%\"\"\"%&alphaG,&F*F*F+F*7%\"\"!F,F,7%F.F,F+Q)pprin
t116\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A4:=map(modp ,sim
plify(   Z1(A3,2,A3[2,2]^(-1))   ,RootOf),2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A4GK%'matrixG6#7%7%\"\"\"%&alphaG,&F*F*F+F*7%\"\"!F*
F*7%F.F,F+Q)pprint126\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "
A5:=map(modp ,simplify(   Z3(A4,2,3,-A4[3,2])   ,RootOf),2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#A5GK%'matrixG6#7%7%\"\"\"%&alphaG,&F*F*F+
F*7%\"\"!F*F*7%F.F.F*Q)pprint136\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 82 "Die Auswertung verl\344uft genau wie im Beispiel (a) mit Hilfe \+
von 5.13 (e) und (f): " }}{PARA 0 "" 0 "" {TEXT -1 9 "det(A) = " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 7 "* (1+  " }{XPPEDIT 
18 0 "alpha;" "6#%&alphaG" }{TEXT -1 15 ") * det(A5) =  " }{XPPEDIT 
18 0 "alpha;" "6#%&alphaG" }{TEXT -1 7 "* (1+  " }{XPPEDIT 18 0 "alpha
;" "6#%&alphaG" }{TEXT -1 6 ")  =1." }}{PARA 0 "" 0 "" {TEXT -1 120 "H
ier ist \"zuf\344llig\" det(A)=det(A5), da sich die beiden Z1-Umformun
gen in ihrer Auswirkung auf die Determinante aufheben." }}{PARA 0 "" 
0 "" {TEXT -1 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Nat\374rli
ch kann Maple das auch:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify
(det(A),RootOf) mod 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "4 15 2 0" 22 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }