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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 35 "The three elementary ro
w operations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg
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\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   mul
row(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,
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"A:=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#>%\"AG-%'MATRIXG6#
7&7&\"\"\"\"\"#\"\"$\"\"%7&\"\"&\"\"'\"\"(\"\")7&\"\"*\"#5\"#6\"#77&!#
B!#A!#@!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(E_3(4,1
,2,-5)&*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"
\"\"#\"\"$\"\"%7&\"\"!!\"%!\")!#77&\"\"*\"#5\"#6\"#77&!#B!#A!#@!#?" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm(E_3(4,1,3,-9)&* \" );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$\"
\"%7&\"\"!!\"%!\")!#77&F-F/!#;!#C7&!#B!#A!#@!#?" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "evalm(E_3(4,1,4,23)&* \" );" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$\"\"%7&\"\"!!\"%!\")!
#77&F-F/!#;!#C7&F-\"#C\"#[\"#s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "evalm(E_2(4,2,-1/4)&* \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$\"\"%7&\"\"!F(F)F*7&F-!\")!#;!#C7&F-\"
#C\"#[\"#s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalm(E_3(4,2
,1,-2)&* \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"
\"\"\"!!\"\"!\"#7&F)F(\"\"#\"\"$7&F)!\")!#;!#C7&F)\"#C\"#[\"#s" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(E_3(4,2,3,8)&* \");" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"!!\"\"!\"#7
&F)F(\"\"#\"\"$7&F)F)F)F)7&F)\"#C\"#[\"#s" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "evalm(E_3(4,2,4,-24)&* \");" }}{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 2 " r" }{TEXT 260 23 "educed row echelon form" }{TEXT 259 5 
" (or " }{TEXT 262 12 "Hermite form" }{TEXT 263 40 ") of A, that can b
e computed directly as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rr
ef(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"!!
\"\"!\"#7&F)F(\"\"#\"\"$7&F)F)F)F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 7 "X is a " }{TEXT 261 14 " random matrix" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "X:=randmatrix(4,4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"XG-%'MATRIXG6#7&7&!#7!#=\"#J!#E7&!#i\"\"\"!#Z!#\"*7
&F1!#h\"#T!#e7&!#!*\"#`!\"\"\"#%*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "evalm(X&*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MA
TRIXG6#7&7&\"$v(\"$](\"$D(\"$+(7&\"%8;\"%99\"%:7\"%;57&\"%^8\"%E7\"%,6
\"$w*7&!%'*>!%S>!%%)=!%G=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "XA h
as the" }{TEXT 264 5 " same" }{TEXT -1 13 " Hermite norm" }}}{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 0 "" 0 "" {TEXT 265 3 "Why" }{TEXT -1 2 " ?
" }}}}{MARK "20 0 1" 2 }{VIEWOPTS 1 1 0 1 1 1803 }