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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 266 16 "Linear equations" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=randmatrix(4,6);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7&7(!#&*!#Z\"#^\"#Z\"\"&\"#L7(\"
\"*\"#E!#b\"#P!#%*!#l7(\"#!*!#Q!#Y\"#G\"#))\"#k7(!#A!#P\"#o!#%)\"#!)\"
#B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "b:=convert(randmatrix
(1,4),vector);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'VECTORG6#7&
!#?!\"(\"\"%!#x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x:=vecto
r(5,[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%&arrayG6$;\"\"\"\"
\"&7\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 6 "Ax = b" }{TEXT -1 15 " \+
has solutions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(lin
solve(A,b),6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7(,($!'P
]L!\"&\"\"\"&%#_tG6#F+$\"')*3HF*&F-6#\"\"#$\"'Ie=F*,($\"'/cLF*F+F,$!'!
)H5F*F1$!'KQ5F*,($!'eEQF*F+F,$\"'rBPF*F1$\"'$R0#F*F,,(F,$!'Z\"z\"F*$\"
'\"4#HF*F+F1$!'`-?F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "e
valf(linsolve(A,b,'r',theta),6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%
'VECTORG6#7(,($!'P]L!\"&\"\"\"&%&thetaG6#F+$\"')*3HF*&F-6#\"\"#$\"'Ie=
F*,($\"'/cLF*F+F,$!'!)H5F*F1$!'KQ5F*,($!'eEQF*F+F,$\"'rBPF*F1$\"'$R0#F
*F,,(F,$!'Z\"z\"F*$\"'\"4#HF*F+F1$!'`-?F*F1" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 2 "r;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "augment(A,b);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7)!#&*!#Z\"#^\"#Z\"\"&\"#L!#?7)\"
\"*\"#E!#b\"#P!#%*!#l!\"(7)\"#!*!#Q!#Y\"#G\"#))\"#k\"\"%7)!#A!#P\"#o!#
%)\"#!)\"#B!#x" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 
270 24 "reduced row echelon form" }{TEXT -1 5 " (or " }{TEXT 271 12 "H
ermite form" }{TEXT -1 12 ")  of  (A,b)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "evalf(rref(\"),6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'MATRIXG6#7&7)$\"\"\"\"\"!F*F*F*$\"'zB;!\"&$\"'R$R\"F-$\"'e#R\"F-7)
F*F(F*F*$!'O[d!\"'$!'3G6F5$\"'+x;F-7)F*F*F(F*$\"'ey?F-$\"'[3@F-$\"'wWA
F-7)F*F*F*F($\"'+#e&F5$\"'\"y6\"F-$\"'XI;F-" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 65 "B:=A: for j from 1 to 6 do B[4,j]:=2*A[1,j]-4*A[2,
j]+3*A[3,j] od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(B);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7(!#&*!#Z\"#^\"#Z\"
\"&\"#L7(\"\"*\"#E!#b\"#P!#%*!#l7(\"#!*!#Q!#Y\"#G\"#))\"#k7(\"#W!$7$\"
$%=\"#I\"$]'\"$=&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 4 "Bx=b" }
{TEXT -1 24 " does not have solutions" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "linsolve(B,b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "evalf(rref(augment(B,b)),6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'MATRIXG6#7&7)$\"\"\"\"\"!F*F*$!'Grp!\"'$\"'lM7!\"&$\"'GThF-F*7)F*
F(F*$!'<I\"*F-$!'[%3\"F0$!'RL6F0F*7)F*F*F($!'T=7F0$\"'Y)R\"F0$\"'GluF-
F*7)F*F*F*F*F*F*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 6 "Three " }
{TEXT 267 25 "elementary row operations" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 97 "E_1 := proc(n, i, j)\015\n   # swap rows i and j\015
\n   swaprow(array(identity, 1..n,1..n), i, j);\015\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "E_2 := proc(n, i, alpha)\015\n   #
 multiply row i times alpha\015\n   mulrow(array(identity, 1..n,1..n),
 i, alpha);\015\nend:\015\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
119 "E_3 := proc(n, i, j, alpha)\015\n   # add alpha*row i to row j\n \+
  addrow(array(identity, 1..n,1..n), i, j, alpha);\015\n   end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "C:=matrix([[1,2,3,4],[5,6,7,
8],[9,10,11,12],3*[1,2,3,4]+2*[5,6,7,8]-4*[9,10,11,12]]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"CG-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$\"\"%7&\"
\"&\"\"'\"\"(\"\")7&\"\"*\"#5\"#6\"#77&!#B!#A!#@!#?" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "evalm(E_3(4,1,4,23) &* E_3(4,1,3,-9) &* E
_3(4,1,2,-5) &* C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7
&\"\"\"\"\"#\"\"$\"\"%7&\"\"!!\"%!\")!#77&F-F/!#;!#C7&F-\"#C\"#[\"#s" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "evalm(E_3(4,2,4,-24) &* E
_3(4,2,3,8) &* E_3(4,2,1,-2) &* E_2(4,2,-1/4) &* \" );" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"!!\"\"!\"#7&F)F(\"\"#\"
\"$7&F)F)F)F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 11 "This is the" 
}{TEXT 257 1 " " }{TEXT 273 1 "r" }{TEXT 260 64 "educed row echelon fo
rm  of  C, that can be computed directly as" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 8 "rref(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATR
IXG6#7&7&\"\"\"\"\"!!\"\"!\"#7&F)F(\"\"#\"\"$7&F)F)F)F)F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 7 "X is a " }{TEXT 261 14 " random matrix" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "X:=randmatrix(4,4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'MATRIXG6#7&7&!#]!#$*\"#9!#F7&
F*\"#e\"#Y!#%)7&!#5F1!#M!#m7&\"#$)!#')\"#R\"#q" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "evalm(X&*C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'MATRIXG6#7&7&\"$K#\"#w!#!)!$O#7&\"%'e#\"%cD\"%ED\"%'\\#7&\"$#y\"$
)e\"$%R\"$+#7&!%1;!%+:!%%R\"!%)G\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 10 "XC has the" }{TEXT 264 5 " same" }{TEXT -1 20 " Hermite norm (w
hy?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(\");" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"!!\"\"!\"#7&F)F(\"
\"#\"\"$7&F)F)F)F)F/" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 7 "Inverse
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G:=matrix(3,3,[1,2,3,4,
5,6,7,8,8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%'MATRIXG6#7%7%
\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")F3" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "Ginv:=inverse(G);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%GinvG-%'MATRIXG6#7%7%#!\")\"\"$#\"\")F,!\"\"7%#\"#5F
,#!#8F,\"\"#7%F/F5F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eva
lm(G &* Ginv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"
\"\"\"!F)7%F)F(F)7%F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "evalm(Ginv &* G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7
%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "b:=vector(3,[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"bG-%&arrayG6$;\"\"\"\"\"$7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "linsolve(G,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#
7%,(&%\"bG6#\"\"\"#!\")\"\"$&F)6#F.!\"\"&F)6#\"\"##\"\")F.,(F(#\"#5F.F
/F4F2#!#8F.,(F(F1F/F1F2F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "evalm(Ginv &* b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7
%,(&%\"bG6#\"\"\"#!\")\"\"$&F)6#F.!\"\"&F)6#\"\"##\"\")F.,(F(#\"#5F.F/
F4F2#!#8F.,(F(F1F/F1F2F4" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 11 "Det
erminant" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(G);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "det(E_1(10,1,3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "det(E_2(10,2,alpha
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&alphaG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "det(E_3(10,4,7,alpha));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "A nonsin
gular matrix, and its inverse, are products of elementary row matrices
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "evalm(E_3(3,1,3,-7) &* \+
E_3(3,1,2,-4) &* G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%
7%\"\"\"\"\"#\"\"$7%\"\"!!\"$!\"'7%F,F.!#8" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "evalm(E_3(3,2,1,-2) &* E_3(3,2,3,6) &* E_2(3,2,-1/3
) &* \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"\"
\"!!\"\"7%F)F(\"\"#7%F)F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "evalm(E_3(3,3,1,1) &* E_3(3,3,2,-2) &* E_2(3,3,-1) &* \");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"\"\"!F)7%F)F(F)7
%F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "evalm(E_3(3,3,1
,1) &* E_3(3,3,2,-2) &* E_2(3,3,-1) &* E_3(3,2,1,-2) &* E_3(3,2,3,6) &
* E_2(3,2,-1/3) &* E_3(3,1,3,-7) &* E_3(3,1,2,-4));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'MATRIXG6#7%7%#!\")\"\"$#\"\")F*!\"\"7%#\"#5F*#!#8F
*\"\"#7%F-F3F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "evalm(E_
3(3,1,2,4) &* E_3(3,1,3,7) &* E_2(3,2,-3) &* E_3(3,2,3,-6) &* E_3(3,2,
1,2) &* E_2(3,3,-1) &* E_3(3,3,2,2) &* E_3(3,3,1,-1));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7
%\"\"(\"\")F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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