{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "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 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 
0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 
0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "Gau\337-Algorithmus" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "A:=matrix([[1,1,1,1,2],[2,2,4,4,4],[6,6,6,6,12],[3,3,
3,4,4],[4,4,4,4,8],[-1,-1,-1,-1,-2]]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7(7'\"\"\"F*F*F*\"\"#7'F+F+\"\"%F-F-7'\"\"'F/F
/F/\"#77'\"\"$F2F2F-F-7'F-F-F-F-\"\")7'!\"\"F6F6F6!\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "A1:=addrow(A,1,2,-2);\nA2:=addrow(
A1,1,3,-6);\nA3:=addrow(A2,1,4,-3);\nA4:=addrow(A3,1,5,-4);\nA5:=addro
w(A4,1,6,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'matrixG6#7(7'
\"\"\"F*F*F*\"\"#7'\"\"!F-F+F+F-7'\"\"'F/F/F/\"#77'\"\"$F2F2\"\"%F37'F
3F3F3F3\"\")7'!\"\"F7F7F7!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2
G-%'matrixG6#7(7'\"\"\"F*F*F*\"\"#7'\"\"!F-F+F+F-7'F-F-F-F-F-7'\"\"$F0
F0\"\"%F17'F1F1F1F1\"\")7'!\"\"F5F5F5!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A3G-%'matrixG6#7(7'\"\"\"F*F*F*\"\"#7'\"\"!F-F+F+F-7
'F-F-F-F-F-7'F-F-F-F*!\"#7'\"\"%F2F2F2\"\")7'!\"\"F5F5F5F0" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#A4G-%'matrixG6#7(7'\"\"\"F*F*F*\"\"#7'\"
\"!F-F+F+F-7'F-F-F-F-F-7'F-F-F-F*!\"#F.7'!\"\"F2F2F2F0" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#A5G-%'matrixG6#7(7'\"\"\"F*F*F*\"\"#7'\"\"!F-F+
F+F-7'F-F-F-F-F-7'F-F-F-F*!\"#F.F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "A6:=swaprow(A5,3,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#A6G-%'matrixG6#7(7'\"\"\"F*F*F*\"\"#7'\"\"!F-F+F+F-7'F-F-F-F*!\"
#7'F-F-F-F-F-F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gausse
lim(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7'\"\"\"F(F(F
(\"\"#7'\"\"!F+F)F)F+7'F+F+F+F(!\"#7'F+F+F+F+F+F.F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "B:=mulrow(A6,2,1/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'matrixG6#7(7'\"\"\"F*F*F*\"\"#7'\"\"!F-F*F*F-7
'F-F-F-F*!\"#7'F-F-F-F-F-F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "rank(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "nops(nullspace(A));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Elementare Zeilentransformat
ionen als Linksmultiplikation mit einer invertierbaren Matrix" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E4:=diag(1,1,1,1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#E4G-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F
+F+7&F+F+F*F+7&F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "
addrow(E4,1,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&
\"\"\"\"\"!F)F)7&F)F(F)F)7&\"\"#F)F(F)7&F)F)F)F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 17 "mulrow(E4,3,1/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"\"\"!F)F)7&F)F(F)F)7&F)F)#F(\"\"#
F)7&F)F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "mulrow(E4,3
,1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"\"\"!F
)F)7&F)F(F)F)7&F)F)#F(\"\"#F)7&F)F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "swaprow(E4,2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'matrixG6#7&7&\"\"\"\"\"!F)F)7&F)F)F(F)7&F)F(F)F)7&F)F)F)F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Eine Basis \+
des Kerns einer Matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "p
rint(A,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7(7'\"\"\"F(
F(F(\"\"#7'F)F)\"\"%F+F+7'\"\"'F-F-F-\"#77'\"\"$F0F0F+F+7'F+F+F+F+\"\"
)7'!\"\"F4F4F4!\"#-F$6#7(F'7'\"\"!F:F(F(F:7'F:F:F:F(F57'F:F:F:F:F:F<F<
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "nullspace(A);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#7'\"\"\"!\"\"\"\"!F*F*-F%6#7'F
*!\"#F.\"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Basis des Bildes" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'matrixG6#7(7'\"\"\"F(F(F(\"\"#7'F)F)\"\"%F+F+7'\"\"'F-F-F-\"#77'
\"\"$F0F0F+F+7'F+F+F+F+\"\")7'!\"\"F4F4F4!\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 90 "B1:=addcol(A,1,2,-1);\nB2:=addcol(B1,1,3,-1);\nB
3:=addcol(B2,1,4,-1);\nB4:=addcol(B3,1,5,-2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B1G-%'matrixG6#7(7'\"\"\"\"\"!F*F*\"\"#7'F,F+\"\"%F.
F.7'\"\"'F+F0F0\"#77'\"\"$F+F3F.F.7'F.F+F.F.\"\")7'!\"\"F+F7F7!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B2G-%'matrixG6#7(7'\"\"\"\"\"!F+F*
\"\"#7'F,F+F,\"\"%F.7'\"\"'F+F+F0\"#77'\"\"$F+F+F.F.7'F.F+F+F.\"\")7'!
\"\"F+F+F7!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B3G-%'matrixG6#7(
7'\"\"\"\"\"!F+F+\"\"#7'F,F+F,F,\"\"%7'\"\"'F+F+F+\"#77'\"\"$F+F+F*F.7
'F.F+F+F+\"\")7'!\"\"F+F+F+!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
B4G-%'matrixG6#7(7'\"\"\"\"\"!F+F+F+7'\"\"#F+F-F-F+7'\"\"'F+F+F+F+7'\"
\"$F+F+F*!\"#7'\"\"%F+F+F+F+7'!\"\"F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "B5:=addcol(B4,3,4,-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B5G-%'matrixG6#7(7'\"\"\"\"\"!F+F+F+7'\"\"#F+F-F+F+7
'\"\"'F+F+F+F+7'\"\"$F+F+F*!\"#7'\"\"%F+F+F+F+7'!\"\"F+F+F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "B6:=addcol(B5,4,5,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B6G-%'matrixG6#7(7'\"\"\"\"\"!F+F+F
+7'\"\"#F+F-F+F+7'\"\"'F+F+F+F+7'\"\"$F+F+F*F+7'\"\"%F+F+F+F+7'!\"\"F+
F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "B7:=swapcol(B6,2,
3);\nB8:=swapcol(B7,3,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B7G-%'
matrixG6#7(7'\"\"\"\"\"!F+F+F+7'\"\"#F-F+F+F+7'\"\"'F+F+F+F+7'\"\"$F+F
+F*F+7'\"\"%F+F+F+F+7'!\"\"F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#B8G-%'matrixG6#7(7'\"\"\"\"\"!F+F+F+7'\"\"#F-F+F+F+7'\"\"'F+F+F+F+
7'\"\"$F+F*F+F+7'\"\"%F+F+F+F+7'!\"\"F+F+F+F+" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "B9:=mulcol(B8,2,1/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B9G-%'matrixG6#7(7'\"\"\"\"\"!F+F+F+7'\"\"#F*F+F+F+7
'\"\"'F+F+F+F+7'\"\"$F+F*F+F+7'\"\"%F+F+F+F+7'!\"\"F+F+F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "colspan(A);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<%-%'vectorG6#7(\"\"!\"\"#F(F(F(F(-F%6#7(F(F(F(F)F
(F(-F%6#7(\"\"\"F)\"\"'\"\"$\"\"%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Test: Ist ein
 gegebener Vektor b in Bild(A)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "Basis:=submatrix(B9,1..6,1..3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&BasisG-%'matrixG6#7(7%\"\"\"\"\"!F+7%\"\"#F*F+7%\"\"'F+F+7%\"
\"$F+F*7%\"\"%F+F+7%!\"\"F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "b1:=transpose(matrix([[1,0,6,-1,4,1]]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#b1G-%'matrixG6#7(7#\"\"\"7#\"\"!7#\"\"'7#!\"\"7#\"\"
%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "concat(Basis,b1);nul
lspace(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7&\"\"\"\"
\"!F)F(7&\"\"#F(F)F)7&\"\"'F)F)F-7&\"\"$F)F(!\"\"7&\"\"%F)F)F27&F0F)F)
F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "also ist b1 nicht in Bild(A)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "b2:=transpose(matrix([[1,0,6,-1,4,-1]]));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#b2G-%'matrixG6#7(7#\"\"\"7#\"\"!7#\"\"'7#
!\"\"7#\"\"%F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "concat(Ba
sis,b2);nullspace(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7
(7&\"\"\"\"\"!F)F(7&\"\"#F(F)F)7&\"\"'F)F)F-7&\"\"$F)F(!\"\"7&\"\"%F)F
)F27&F0F)F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7&!\"\"
\"\"#\"\"%\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "also ist b2 i
n Bild(A)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Insbesondere ist als
o das Gleichungssystem Ax=b1 nicht l\366sbar, jedoch Ax=b2 l\366sbar.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 32 "Wir suchen eine L\366sung von Ax=b2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Ab:=concat(A,b2);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#AbG-%'matrixG6#7(7(\"\"\"F*F*F*\"\"#F*7(F+F+\"\"%
F-F-\"\"!7(\"\"'F0F0F0\"#7F07(\"\"$F3F3F-F-!\"\"7(F-F-F-F-\"\")F-7(F4F
4F4F4!\"#F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "A1:=addrow(
Ab,1,2,-2);\nA2:=addrow(A1,1,3,-6);\nA3:=addrow(A2,1,4,-3);\nA4:=addro
w(A3,1,5,-4);\nA5:=addrow(A4,1,6,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#A1G-%'matrixG6#7(7(\"\"\"F*F*F*\"\"#F*7(\"\"!F-F+F+F-!\"#7(\"\"'F
0F0F0\"#7F07(\"\"$F3F3\"\"%F4!\"\"7(F4F4F4F4\"\")F47(F5F5F5F5F.F5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'matrixG6#7(7(\"\"\"F*F*F*\"\"
#F*7(\"\"!F-F+F+F-!\"#7(F-F-F-F-F-F-7(\"\"$F1F1\"\"%F2!\"\"7(F2F2F2F2
\"\")F27(F3F3F3F3F.F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G-%'matr
ixG6#7(7(\"\"\"F*F*F*\"\"#F*7(\"\"!F-F+F+F-!\"#7(F-F-F-F-F-F-7(F-F-F-F
*F.!\"%7(\"\"%F3F3F3\"\")F37(!\"\"F6F6F6F.F6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A4G-%'matrixG6#7(7(\"\"\"F*F*F*\"\"#F*7(\"\"!F-F+F+F
-!\"#7(F-F-F-F-F-F-7(F-F-F-F*F.!\"%F/7(!\"\"F3F3F3F.F3" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#A5G-%'matrixG6#7(7(\"\"\"F*F*F*\"\"#F*7(\"\"!F-
F+F+F-!\"#7(F-F-F-F-F-F-7(F-F-F-F*F.!\"%F/F/" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "A6:=swaprow(A5,3,4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A6G-%'matrixG6#7(7(\"\"\"F*F*F*\"\"#F*7(\"\"!F-F+F+F
-!\"#7(F-F-F-F*F.!\"%7(F-F-F-F-F-F-F1F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "nullspace(A6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%-
%'vectorG6#7(\"\"!!\"#!\"$\"\"%F(\"\"\"-F%6#7(F,!\"\"F(F(F(F(-F%6#7(F(
F)F)\"\"#F,F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Eine L\366sung v
on Ax=b2 ist gegeben durch" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "x0:=matrix([[2], [0], [3], [-4], [0]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x0G-%'matrixG6#7'7#\"\"#7#\"\"!7#\"\"$7#!\"%F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "multiply(A,x0)=evalm(b2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7(7#\"\"\"7#\"\"!7#\"\"'
7#!\"\"7#\"\"%F.F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "also ist di
e L\366sungsmenge von Ax=b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
88 "evalm(x0)+span(matrix([[-2], [0], [-2], [2], [1]]), matrix([[-1], \+
[1], [0], [0], [0]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'matrixG
6#7'7#\"\"#7#\"\"!7#\"\"$7#!\"%F*\"\"\"-%%spanG6$-F%6#7'7#!\"#F*F7F(7#
F0-F%6#7'7#!\"\"F9F*F*F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Berechnung der Inversen ei
ner invertierbaren Matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "A:=matrix([[1,2,3,1],[0,1,2,3],[-1,0,1,2],[3,1,0,1]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&\"\"\"\"\"#\"\"$F*7&\"
\"!F*F+F,7&!\"\"F.F*F+7&F,F*F.F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "AE:=concat(A,diag(1,1,1,1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#AEG-%'matrixG6#7&7*\"\"\"\"\"#\"\"$F*F*\"\"!F-F-7*F-
F*F+F,F-F*F-F-7*!\"\"F-F*F+F-F-F*F-7*F,F*F-F*F-F-F-F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "AE1:=addrow(AE,1,3,1);\nAE2:=addrow
(AE1,1,4,-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AE1G-%'matrixG6#7&
7*\"\"\"\"\"#\"\"$F*F*\"\"!F-F-7*F-F*F+F,F-F*F-F-7*F-F+\"\"%F,F*F-F*F-
7*F,F*F-F*F-F-F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AE2G-%'matrix
G6#7&7*\"\"\"\"\"#\"\"$F*F*\"\"!F-F-7*F-F*F+F,F-F*F-F-7*F-F+\"\"%F,F*F
-F*F-7*F-!\"&!\"*!\"#!\"$F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 48 "AE3:=addrow(AE2,2,3,-2);\nAE4:=addrow(AE3,2,4,5);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$AE3G-%'matrixG6#7&7*\"\"\"\"\"#\"\"$F*F*\"\"
!F-F-7*F-F*F+F,F-F*F-F-7*F-F-F-!\"$F*!\"#F*F-7*F-!\"&!\"*F1F0F-F-F*" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AE4G-%'matrixG6#7&7*\"\"\"\"\"#\"
\"$F*F*\"\"!F-F-7*F-F*F+F,F-F*F-F-7*F-F-F-!\"$F*!\"#F*F-7*F-F-F*\"#8F0
\"\"&F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "AE5:=swaprow(A
E4,3,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AE5G-%'matrixG6#7&7*\"
\"\"\"\"#\"\"$F*F*\"\"!F-F-7*F-F*F+F,F-F*F-F-7*F-F-F*\"#8!\"$\"\"&F-F*
7*F-F-F-F1F*!\"#F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "AE6
:=mulrow(AE5,4,-1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AE6G-%'mat
rixG6#7&7*\"\"\"\"\"#\"\"$F*F*\"\"!F-F-7*F-F*F+F,F-F*F-F-7*F-F-F*\"#8!
\"$\"\"&F-F*7*F-F-F-F*#!\"\"F,#F+F,F4F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "AE7:=addrow(AE6,4,3,-13);\nAE8:=addrow(AE7,4,2,-3);\n
AE9:=addrow(AE8,4,1,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AE7G-%'
matrixG6#7&7*\"\"\"\"\"#\"\"$F*F*\"\"!F-F-7*F-F*F+F,F-F*F-F-7*F-F-F*F-
#\"\"%F,#!#6F,#\"#8F,F*7*F-F-F-F*#!\"\"F,#F+F,F7F-" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$AE8G-%'matrixG6#7&7*\"\"\"\"\"#\"\"$F*F*\"\"!F-F-7
*F-F*F+F-F*!\"\"F*F-7*F-F-F*F-#\"\"%F,#!#6F,#\"#8F,F*7*F-F-F-F*#F/F,#F
+F,F8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AE9G-%'matrixG6#7&7*\"\"
\"\"\"#\"\"$\"\"!#\"\"%F,#!\"#F,#F*F,F-7*F-F*F+F-F*!\"\"F*F-7*F-F-F*F-
F.#!#6F,#\"#8F,F*7*F-F-F-F*#F4F,#F+F,F;F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "AE10:=addrow(AE9,3,2,-2);\nAE11:=addrow(AE10,3,1,-3
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AE10G-%'matrixG6#7&7*\"\"\"\"
\"#\"\"$\"\"!#\"\"%F,#!\"#F,#F*F,F-7*F-F*F-F-#!\"&F,#\"#>F,#!#BF,F17*F
-F-F*F-F.#!#6F,#\"#8F,F*7*F-F-F-F*#!\"\"F,#F+F,F@F-" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%AE11G-%'matrixG6#7&7*\"\"\"\"\"#\"\"!F,#!\")\"\"$#
\"#JF/#!#QF/!\"$7*F,F*F,F,#!\"&F/#\"#>F/#!#BF/!\"#7*F,F,F*F,#\"\"%F/#!
#6F/#\"#8F/F*7*F,F,F,F*#!\"\"F/#F+F/FEF," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "AE12:=addrow(AE11,2,1,-2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%AE12G-%'matrixG6#7&7*\"\"\"\"\"!F+F+#\"\"#\"\"$#!\"(
F.#\"\")F.F*7*F+F*F+F+#!\"&F.#\"#>F.#!#BF.!\"#7*F+F+F*F+#\"\"%F.#!#6F.
#\"#8F.F*7*F+F+F+F*#!\"\"F.F,FCF+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "iA:=submatrix(AE12,1..4,5..8);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#iAG-%'matrixG6#7&7&#\"\"#\"\"$#!\"(F,#\"\")F,\"\"\"7
&#!\"&F,#\"#>F,#!#BF,!\"#7&#\"\"%F,#!#6F,#\"#8F,F17&#!\"\"F,F*FB\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "multiply(A,iA);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"\"\"!F)F)7&F)F(F
)F)7&F)F)F(F)7&F)F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "
inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&#\"\"#
\"\"$#!\"(F*#\"\")F*\"\"\"7&#!\"&F*#\"#>F*#!#BF*!\"#7&#\"\"%F*#!#6F*#
\"#8F*F/7&#!\"\"F*F(F@\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Summe und Durchschnitt von
 Untervektorr\344umen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "U
:=matrix([[4,2,6],[0,1,1],[2,1,3],[4,1,5],[4,2,6]]);\nV:=matrix([[2,4,
2],[0,3,2],[3,2,3],[1,3,1],[3,1,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"UG-%'matrixG6#7'7%\"\"%\"\"#\"\"'7%\"\"!\"\"\"F/7%F+F/\"\"$7%F*
F/\"\"&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'matrixG6#7'7%\"
\"#\"\"%F*7%\"\"!\"\"$F*7%F.F*F.7%\"\"\"F.F17%F.F1F-" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "A:=concat(U,V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matr
ixG6#7'7(\"\"%\"\"#\"\"'F+F*F+7(\"\"!\"\"\"F/F.\"\"$F+7(F+F/F0F0F+F07(
F*F/\"\"&F/F0F/7(F*F+F,F0F/F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "
Basis von U+V" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "colspan(A)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&-%'vectorG6#7'\"\"%\"\"!\"\"#F(
F(-F%6#7'F)F)\"\")!\"%F(-F%6#7'F)F(F)F/F)-F%6#7'F)F)F)\"#;!#C" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "Basis von ker(A)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "kerABasis:=nullspace(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*k
erABasisG<$-%'vectorG6#7(\"\"!#!\"\"\"\"#F+\"\"\"F.F,-F'6#7(F.F.F,F*F*
F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Berechnung eines Erzeugende
nsystems des Durchschnitts von U und V:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 124 "multiply(U,submatrix(convert(kerABasis[1],matrix),1.
.3,1..1)),multiply(U,submatrix(convert(kerABasis[2],matrix),1..3,1..1)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7'7#\"\"!F'F'F'F'-F$
6#7'7#!\"%7#!\"\"7#!\"#7#!\"$F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
40 "Der Durchschnitt ist also 1-dimensional." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "intbasis(colspan(U),colspan(V));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<#-%'vectorG6#7'!#C!\"'!#7!#=F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "W
elche Teilmengen eines Erzeugendensystems bilden eine Basis" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(A);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7'7(\"\"%\"\"#\"\"'F)F(F)7(\"\"!\"\"\"F-F
,\"\"$F)7(F)F-F.F.F)F.7(F(F-\"\"&F-F.F-7(F(F)F*F.F-F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "kerA:=concat(seq(convert(kerABasis[
j],matrix),j=1..nops(kerABasis)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%kerAG-%'matrixG6#7(7$\"\"\"\"\"!7$F*#!\"\"\"\"#7$F.F-7$F+F*F17$F+F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(combinat):" }}
{PARA 7 "" 1 "" {TEXT -1 47 "Warning, the name fibonacci has been rede
fined\n" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, the protected name Ch
i has been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 53 "Alle dim(Bild(A))=4-elementigen Teilmengen von \{1..6\}" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L:=choose(6,4);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%\"LG717&\"\"\"\"\"#\"\"$\"\"%7&F'F(F)\"\"&
7&F'F(F)\"\"'7&F'F(F*F,7&F'F(F*F.7&F'F(F,F.7&F'F)F*F,7&F'F)F*F.7&F'F)F
,F.7&F'F*F,F.7&F(F)F*F,7&F(F)F*F.7&F(F)F,F.7&F(F*F,F.7&F)F*F,F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 206 "for j from 1 to nops(L) do\n   T:=submatrix(kerA,c
onvert(\{1,2,3,4,5,6\} minus convert(L[j],set),list),1..2);\n   B:=sub
matrix(A,1..5,L[j]);\n   print(L[j],nullspace(T),convert(B,listlist),n
ops(colspan(B)));\nod:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"\"\"\"
#\"\"$\"\"%<#-%'vectorG6#7$F$\"\"!7'7&F'F%\"\"'F%7&F-F$F$F-7&F%F$F&F&7
&F'F$\"\"&F$7&F'F%F0F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"\"\"
\"#\"\"$\"\"&<#-%'vectorG6#7$F$\"\"!7'7&\"\"%F%\"\"'F07&F-F$F$F&7&F%F$
F&F%7&F0F$F'F&7&F0F%F1F$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"\"
\"\"#\"\"$\"\"'<#-%'vectorG6#7$F$\"\"!7'7&\"\"%F%F'F%7&F-F$F$F%7&F%F$F
&F&7&F0F$\"\"&F$7&F0F%F'F-F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"
\"\"\"#\"\"%\"\"&<\"7'7&F&F%F%F&7&\"\"!F$F,\"\"$7&F%F$F-F%7&F&F$F$F-7&
F&F%F-F$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"\"\"\"#\"\"%\"\"'<
\"7'7&F&F%F%F%7&\"\"!F$F,F%7&F%F$\"\"$F.7&F&F$F$F$7&F&F%F.F,F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"\"\"\"#\"\"&\"\"'<\"7'7&\"\"%F%F
+F%7&\"\"!F$\"\"$F%7&F%F$F%F.7&F+F$F.F$7&F+F%F$F-F+" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6&7&\"\"\"\"\"$\"\"%\"\"&<\"7'7&F&\"\"'\"\"#F&7&\"\"!F$
F.F%7&F,F%F%F,7&F&F'F$F%7&F&F+F%F$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6&7&\"\"\"\"\"$\"\"%\"\"'<\"7'7&F&F'\"\"#F+7&\"\"!F$F-F+7&F+F%F%F%7&F&
\"\"&F$F$7&F&F'F%F-F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"\"\"\"$
\"\"&\"\"'<\"7'7&\"\"%F'F+\"\"#7&\"\"!F$F%F,7&F,F%F,F%7&F+F&F%F$7&F+F'
F$F.F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"\"\"\"%\"\"&\"\"'<\"7'
7&F%\"\"#F%F+7&\"\"!F-\"\"$F+7&F+F.F+F.7&F%F$F.F$7&F%F.F$F-F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"#\"\"$\"\"%\"\"&<\"7'7&F$\"\"'F$
F&7&\"\"\"F-\"\"!F%7&F-F%F%F$7&F-F'F-F%7&F$F+F%F-F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6&7&\"\"#\"\"$\"\"%\"\"'<\"7'7&F$F'F$F$7&\"\"\"F,\"\"!F
$7&F,F%F%F%7&F,\"\"&F,F,7&F$F'F%F-F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6&7&\"\"#\"\"$\"\"&\"\"'<\"7'7&F$F'\"\"%F$7&\"\"\"F-F%F$7&F-F%F$F%7&F-
F&F%F-7&F$F'F-\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"#\"\"%
\"\"&\"\"'<\"7'7&F$F$F%F$7&\"\"\"\"\"!\"\"$F$7&F,F.F$F.7&F,F,F.F,7&F$F
.F,F-F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7&\"\"$\"\"%\"\"&\"\"'<\"7'
7&F'\"\"#F%F+7&\"\"\"\"\"!F$F+7&F$F$F+F$7&F&F-F$F-7&F'F$F-F.F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "88 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }