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{SECT 0 {PARA 256 "" 0 "" {TEXT 274 20 "REDUCED ECHELON FORM" }}{PARA 
257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 256 23 "Linear Alg
ebra Project " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
0 {PARA 263 "" 0 "" {TEXT 257 11 "Objective: " }}{PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 258 190 "To reduce a matrix to its echelon form b
y elementary row operations and use it to slove systems of linear equa
tions. While doing this you will begin to learn Maple's linear algebra
 features." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 306 "Computer will do the lengthy arithmetic operations for you. Th
is will help you to utilize your time and energy on understanding the \+
reduced echelon form of a matrix and the process for attaining it. You
 will find this very useful later on when you study the structure and \+
transformations of a vector space." }}}{PARA 264 "" 0 "" {TEXT -1 0 "
" }{TEXT 266 0 "" }{TEXT -1 0 "" }{TEXT 269 0 "" }}{SECT 0 {PARA 259 "
" 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT 264 15 "Solved Example:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT -1 0 "" }{TEXT 
257 119 "Solve the following system of equations using elementary row \+
operations to reduce its augmented matrix to echelon form." }}{PARA 
260 "" 0 "" {TEXT 273 1 " " }{XPPEDIT -1 0 "matrix([[2*x[1]+6*x[2]-x[3
] = 8], [3*x[1]+9*x[2]+0*x[3] = 15], [-2*x[1]-5*x[2]+6*x[3] = 1]]);" "
6#-%'matrixG6#7%7#/,(*&\"\"#\"\"\"&%\"xG6#F,F,F,*&\"\"'F,&F.6#F+F,F,&F
.6#\"\"$!\"\"\"\")7#/,(*&F6F,&F.6#F,F,F,*&\"\"*F,&F.6#F+F,F,*&\"\"!F,&
F.6#F6F,F,\"#:7#/,(*&F+F,&F.6#F,F,F7*&\"\"&F,&F.6#F+F,F7*&F1F,&F.6#F6F
,F,F," }}}{PARA 265 "" 0 "" {TEXT -1 0 "" }{TEXT 265 0 "" }}{SECT 0 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT 261 9 "Solution:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 " We a
ctivate the linear algebra package." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(linalg);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^
r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*W
ronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsub
G%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)cholesky
G%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'concatG%%condG%)cop
yintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%
(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvec
tsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibona
cciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genma
trixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ih
ermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimil
arG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)lin
solveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiply
G%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potenti
alG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspac
eG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smithG%,stackmatrixG%*
submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toe
plitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vector
G%*wronskianG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 54 "First let us write the augmented matrix of the system:" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "M:=matrix(3,4,[[2,6,-1,8],[
3,9,0,15],[-2,-5,6,1]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%
'matrixG6#7%7&\"\"#\"\"'!\"\"\"\")7&\"\"$\"\"*\"\"!\"#:7&!\"#!\"&F+\"
\"\"" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 
"Now we proceed to construct the row-reduced echelon matrix. " }}
{PARA 0 "" 0 "" {TEXT -1 40 "Multiply the first row of the matrix by \+
" }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 22 " to obt
ain matrix M1. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "\nM1:= mu
lrow(M,1,1/2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'matrixG6#
7%7&\"\"\"\"\"$#!\"\"\"\"#\"\"%7&F+\"\"*\"\"!\"#:7&!\"#!\"&\"\"'F*" }}
}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "\nM2:=pivot(M1,1,1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G
-%'matrixG6#7%7&\"\"\"\"\"$#!\"\"\"\"#\"\"%7&\"\"!F1#F+F.F+7&F1F*\"\"&
\"\"*" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{SECT 0 {PARA 262 "" 0 "" 
{TEXT 259 41 "Do you see what pivot operation has done?" }}{PARA 0 "" 
0 "" {TEXT -1 306 "Pivot: 1 in the first row and first column of the m
atrix M1 is used as a lever to turn the rest of the entries in column \+
one into 0. To do this the first row of M1 is multiplied by -3 and add
ed to the second row, and the first row is multiplied by 2 and added t
o the third row. The resulting matrix is M2. " }}{PARA 0 "" 0 "" 
{TEXT -1 135 "Matrices M1 and M2 are \"equivalent\",  meaning that the
 systems of equations that they represent have exactly the same set of
 solutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "\nM3:= swaprow(M2,2,3);\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#M3G-%'matrixG6#7%7&\"\"\"\"\"$#!\"\"\"\"#
\"\"%7&\"\"!F*\"\"&\"\"*7&F1F1#F+F.F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "\nM4:=pivot(M3,2,2);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#M4G-%'matrixG6#7%7&\"\"\"\"\"!#!#J\"\"#!#B7&F+F*\"\"&\"\"*7&F
+F+#\"\"$F.F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "\nM5:=mulr
ow(M4,3,2/3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M5G-%'matrixG6#7
%7&\"\"\"\"\"!#!#J\"\"#!#B7&F+F*\"\"&\"\"*7&F+F+F*F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "\nM6:=pivot(M5,3,3);\n" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#M6G-%'matrixG6#7%7&\"\"\"\"\"!F+\"\")7&F+F*F+!\"\"
7&F+F+F*\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 110 "M6 is the row reduced \+
echelon form of the augmented matrix M. This yields the solution to th
e given system as " }{XPPEDIT 18 0 "x[1] = 8;" "6#/&%\"xG6#\"\"\"\"\")
" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "x[2] = -1;" "6#/&%\"xG6#\"\"#,$\"
\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[3] = 2;" "6#/&%\"xG6#
\"\"$\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 82 "_______________________________________________
___________________________________" }}{PARA 261 "" 0 "" {TEXT 267 10 
"ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 0 {PARA 266 "" 0 "" {TEXT 268 10 "Problem 1:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "Use Maple as w
e did above to reduce the augmented matrix of the following system of \+
equations to its echelon form. Discuss the nature of solutions and fin
d a solution set for the system." }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}
{PARA 268 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "matrix([[2*x[1]+4*x[2]
-4*x[3]+4*x[4]+x[5] = 7], [x[1]+x[2]-3*x[3]+2*x[4]+x[5] = 2], [x[1]+2*
x[2]-2*x[3]+2*x[4]+3*x[5] = 6], [x[1]+2*x[2]-2*x[3]+2*x[4]-2*x[5] = 1]
])" "6#-%'matrixG6#7&7#/,,*&\"\"#\"\"\"&%\"xG6#F,F,F,*&\"\"%F,&F.6#F+F
,F,*&F1F,&F.6#\"\"$F,!\"\"*&F1F,&F.6#F1F,F,&F.6#\"\"&F,\"\"(7#/,,&F.6#
F,F,&F.6#F+F,*&F7F,&F.6#F7F,F8*&F+F,&F.6#F1F,F,&F.6#F>F,F+7#/,,&F.6#F,
F,*&F+F,&F.6#F+F,F,*&F+F,&F.6#F7F,F8*&F+F,&F.6#F1F,F,*&F7F,&F.6#F>F,F,
\"\"'7#/,,&F.6#F,F,*&F+F,&F.6#F+F,F,*&F+F,&F.6#F7F,F8*&F+F,&F.6#F1F,F,
*&F+F,&F.6#F>F,F8F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 270 10 
"Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 183 "Use Maple to reduce the a
ugmented matrix of the following system of equations to its echelon fo
rm. Discuss the nature of solutions and find a solution set for the sy
stem, if possible." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "
" {XPPEDIT 18 0 "matrix([[x[1]+4*x[2]+3*x[3] = 10], [2*x[1]+x[2]-x[3] \+
= -1], [3*x[1]-x[2]-4*x[3] = 11]]);" "6#-%'matrixG6#7%7#/,(&%\"xG6#\"
\"\"F-*&\"\"%F-&F+6#\"\"#F-F-*&\"\"$F-&F+6#F4F-F-\"#57#/,(*&F2F-&F+6#F
-F-F-&F+6#F2F-&F+6#F4!\"\",$F-FB7#/,(*&F4F-&F+6#F-F-F-&F+6#F2FB*&F/F-&
F+6#F4F-FB\"#6" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" 
}{TEXT 271 10 "Problem 3:" }}{PARA 0 "" 0 "" {TEXT -1 42 "Does the fol
lowing system have a solution?" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "(1, 0
, -3) = a*(1, 1, 2)+b*(-2, -1, 1);" "6#/6%\"\"\"\"\"!,$\"\"$!\"\",&*&%
\"aGF%6%F%F%\"\"#F%F%*&%\"bGF%6%,$F.F),$F%F)F%F%F%" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 272 10 
"Problem 4:" }}{PARA 0 "" 0 "" {TEXT -1 261 "Maple has a procedure tha
t automatically computes the row-reduced echelon form of a matrix, it \+
is denoted by \"gaussjord\". As an example let us use it on the matrix
 M which is the augmented matrix of the system of equations in the \"s
olved example\" we had above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "M:=matrix(3,4,[[2,6,-1,8],[3,9,0,15],[-2,-5,6,1]]);\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gaussjord(M);\n" }{TEXT -1 0 "" }}}
}{PARA 0 "" 0 "" {TEXT -1 56 "________________________________________
________________" }}{PARA 0 "" 0 "" {TEXT -1 93 "MSIP Grant #120A80089
-98: \"Three Urban Calculus Reform Programs: Adopting the Best\" 1998-
2001" }}}{MARK "0 0" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }