{VERSION 3 0 "IBM INTEL LINUX" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 256 "" 0 1 255 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 
255 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 259 "terminal" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 
}{CSTYLE "" -1 260 "terminal" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 
0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 
0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 
0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 
0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Norm
al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 
-1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 
0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 
0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 
7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "
" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 
-1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 
}}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 33 "Maple Tutorial for Linear
 Algebra" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "O
n most platforms, Maple V is started by double-clicking on the Maple i
con. However, in our Computer Lab in HH-3030/3056 you can simply type \+
" }{TEXT 256 6 "xmaple" }{TEXT -1 4 " or " }{TEXT 257 8 "xmaple &" }
{TEXT -1 155 " to start a Maple session. If you are familiar with the \+
Unix environment you can create a Maple subdirectory and start Maple f
rom within that subdirectory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 32 "At the top of the window is the " }{TEXT 
258 8 "menu bar" }{TEXT -1 26 " containing such menus as " }{TEXT 259 
4 "File" }{TEXT -1 5 " and " }{TEXT 260 4 "Edit" }{TEXT -1 40 ". Immed
iately below the menu bar is the " }{TEXT 261 8 "tool bar" }{TEXT -1 
138 ", which contains button-based shortcuts to common operations such
 as opening, saving, and printing. Immediately below the tool bar is t
he " }{TEXT 262 12 "context bar " }{TEXT -1 392 "which contains contro
ls specific to the task you are currently performing. The next area is
 large and displays your worksheet, the region in which you work. At t
he bottom of the window is the status bar, which displays system infor
mation. The worksheet should have opened with the prompt (>) in the up
per left corner. This worksheet can be viewed and downloaded from my h
ome page. The URL is " }{TEXT 263 33 " http://www.math.mun.ca/~drideou
t" }{TEXT -1 192 ". If you download the worksheet, you should execute \+
each of the commands, after the prompt (>), either one at a time or go
 to the menu bar, and then edit, and then execute the whole worksheet.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 343 "We w
ill introduce Maple commands which will be useful in Mathematics 2050 \+
or 2051. Maple V provides a wide range of special packages specially d
esigned for students. For example, there are packages for calculus, st
atistics, number theory, group theory, and graph theory. One of the mo
st frequently used packages is the linear algebra package, " }{TEXT 
278 6 "linalg" }{TEXT -1 89 ". This package provides a complete set of
 commands for working with vectors and matrices." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "First load the linear alg
ebra package using the " }{TEXT 264 4 "with" }{TEXT -1 121 " command. \+
If you do not wish to load the entire package, you can load selected r
outines from the package using the syntax" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 265 43 "with(packagename, routine1, ..., \+
routineN);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 50 "You can also use the long form of the routine name" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 30 "packagename[routi
nename](...);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 7 "" 1 "" {TEXT -1 32 "Wa
rning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning
, new definition for trace" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.Blo
ckDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*Wronski
anG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsubG%%ban
dG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$col
G%'coldimG%)colspaceG%(colspanG%*companionG%'concatG%%condG%)copyintoG
%*crossprodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(diver
geG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,e
ntermatrixG%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibonacciG%+
forwardsubG%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%
%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ihermite
G%*indexfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimilarG%'i
szeroG%)jacobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG
%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%nor
mG%*normalizeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potentialG%+r
andmatrixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(ro
wspanG%%rrefG%*scalarmulG%-singularvalsG%&smithG%,stackmatrixG%*submat
rixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG
%&traceG%*transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wro
nskianG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 126 "Note the many functions that are listed above. To find m
ore information on any topic you can look at the help pages by typing \+
" }{TEXT 268 6 "?topic" }{TEXT -1 15 ". For example, " }{TEXT 269 7 "?
addrow" }{TEXT -1 102 " will show you clearly, with examples, how to u
se this function. Experiment, by executing the command " }{TEXT 279 7 
"?addrow" }{TEXT -1 19 " on a command line." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Solving Systems of Eq
uations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 181 "An array is a generalization of the matrix data structur
e. The conversion between arrays and matrices is automatic, so followi
ng the creation of an array it can be used as a matrix." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Consider the 4x6 num
eric matrix created using the array command." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A:=array([[1,3,-2
,0,2,0],[2,6,-5,-2,4,-3],[0,0,5,10,0,15],[2,6,0,8,0,18]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7(\"\"\"\"\"$!\"#\"\"!\"
\"#F-7(F.\"\"'!\"&F,\"\"%!\"$7(F-F-\"\"&\"#5F-\"#:7(F.F0F-\"\")F-\"#=
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
260 "The above label A always refers to the above matrix unless we una
ssign the symbol A. Note the use of the semicolin (;). The semicolin i
s always needed in order to see the output. If you do not want to see \+
the output, end the command with the colin (:) instead." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"AG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The matrix is not display
ed. It is displayed using the " }{TEXT 274 5 "evalm" }{TEXT -1 9 " com
mand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalm(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrix
G6#7&7(\"\"\"\"\"$!\"#\"\"!\"\"#F+7(F,\"\"'!\"&F*\"\"%!\"$7(F+F+\"\"&
\"#5F+\"#:7(F,F.F+\"\")F+\"#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 12 "The command " }{TEXT 275 8 "print(A)
" }{TEXT -1 62 " will also work. The following command unassigns the s
ymbol A." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "A:='A';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGF$" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(A);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%\"AG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The ab
ove matrix can also be defined with the matrix command." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "A:=mat
rix(4,6,[1,3,-2,0,2,0,2,6,-5,-2,4,-3,0,0,5,10,0,15,2,6,0,8,0,18]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7(\"\"\"\"\"$!\"#
\"\"!\"\"#F-7(F.\"\"'!\"&F,\"\"%!\"$7(F-F-\"\"&\"#5F-\"#:7(F.F0F-\"\")
F-\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "B:=vector([0,-1,5
,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'vectorG6#7&\"\"!!\"
\"\"\"&\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 164 "If A is the coefficient matrix and B the constant matr
ix of a system of 4 equations in 6 unknowns, then one way to solve the
 system is to form the augmented matrix." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A_B:=augment(A,B);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$A_BG-%'matrixG6#7&7)\"\"\"\"\"$!
\"#\"\"!\"\"#F-F-7)F.\"\"'!\"&F,\"\"%!\"$!\"\"7)F-F-\"\"&\"#5F-\"#:F67
)F.F0F-\"\")F-\"#=F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 56 "Next we can row-reduce the matrix by using the \+
commands " }{TEXT 267 27 "addrow, mulrow, and swaprow" }{TEXT -1 83 ",
 as many times as is necessary. The symbol (%) always refers to the la
test output." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "addrow(%,1,2,-2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7&7)\"\"\"\"\"$!\"#\"\"!\"\"#F+F+7)F+F+!\"\"F*F+!\"$
F.7)F+F+\"\"&\"#5F+\"#:F17)F,\"\"'F+\"\")F+\"#=F5" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 17 "addrow(%,1,4,-2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7&7)\"\"\"\"\"$!\"#\"\"!\"\"#F+F+7)F+F+!\"
\"F*F+!\"$F.7)F+F+\"\"&\"#5F+\"#:F17)F+F+\"\"%\"\")!\"%\"#=\"\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "mulrow(%,2,-1);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)\"\"\"\"\"$!\"#\"\"!\"\"#F+F+7)
F+F+F(F,F+F)F(7)F+F+\"\"&\"#5F+\"#:F/7)F+F+\"\"%\"\")!\"%\"#=\"\"'" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "addrow(%,2,3,-5);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)\"\"\"\"\"$!\"#\"\"!\"\"#F+F
+7)F+F+F(F,F+F)F(7)F+F+F+F+F+F+F+7)F+F+\"\"%\"\")!\"%\"#=\"\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "addrow(%,2,4,-4);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)\"\"\"\"\"$!\"#\"\"!\"\"#F+F
+7)F+F+F(F,F+F)F(7)F+F+F+F+F+F+F+7)F+F+F+F+!\"%\"\"'F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "swaprow(%,3,4);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7&7)\"\"\"\"\"$!\"#\"\"!\"\"#F+F+7)F+F+F(
F,F+F)F(7)F+F+F+F+!\"%\"\"'F,7)F+F+F+F+F+F+F+" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "row_esc_form:=mulrow(%,3,-1/4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%-row_esc_formG-%'matrixG6#7&7)\"\"\"\"\"$!\"#\"
\"!\"\"#F-F-7)F-F-F*F.F-F+F*7)F-F-F-F-F*#!\"$F.#!\"\"F.7)F-F-F-F-F-F-F
-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
71 "We could have gotten to the same place in the computation by using
 the " }{TEXT 270 9 "gausselim" }{TEXT -1 66 " command, except that th
e leading numbers are not necessarily 1's." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gausselim(A_B);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)\"\"\"\"\"$!\"#\"\"!
\"\"#F+F+7)F+F+!\"\"F*F+!\"$F.7)F+F+F+F+!\"%\"\"'F,7)F+F+F+F+F+F+F+" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "
To put the matrix " }{TEXT 271 12 "row_esc_form" }{TEXT -1 75 " in red
uced row eschelon form, we continue with the following computations." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "addrow(row_esc_form,2,1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'matrixG6#7&7)\"\"\"\"\"$\"\"!\"\"%\"\"#\"\"'F,7)F*F*F(F,F*F)F(7)F*
F*F*F*F(#!\"$F,#!\"\"F,7)F*F*F*F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "addrow(%,3,1,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'matrixG6#7&7)\"\"\"\"\"$\"\"!\"\"%F*\"\"*F)7)F*F*F(\"\"#F*F)F(7)F*F
*F*F*F(#!\"$F.#!\"\"F.7)F*F*F*F*F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "As above, we can use one \+
command in Maple to do the same thing. Either use the command " }
{TEXT 272 4 "rref" }{TEXT -1 18 " or the following." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gaussjord
(A_B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7)\"\"\"\"\"$
\"\"!\"\"%F*\"\"*F)7)F*F*F(\"\"#F*F)F(7)F*F*F*F*F(#!\"$F.#!\"\"F.7)F*F
*F*F*F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 86 "As you may have already guessed, Maple has one command \+
to do all of the above at once." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "linsolve(A,B);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'vectorG6#7(,*\"\"$\"\"\"&%#_tG6#F)!\"$&F+6#\"
\"#!\"%&F+6#F(!\"*F*,(F)F)F.!\"#F2F-F.,&#!\"\"F0F)F2#F(F0F2" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "No
te the three free variables, " }{XPPEDIT 18 0 "_t[1],_t[2],_t[3];" "6%
&%#_tG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 77 ". Another way to sol
ve the system is to write the equations out, and use the " }{TEXT 273 
5 "solve" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eq1:=x1+3*x2-2*x3+2*x5=0;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,*%#x1G\"\"\"%#x2G\"\"$%#x3G!
\"#%#x5G\"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eq2:=
2*x1+6*x2-5*x3-2*x4+4*x5-3*x6=-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$eq2G/,.%#x1G\"\"#%#x2G\"\"'%#x3G!\"&%#x4G!\"#%#x5G\"\"%%#x6G!\"$!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eq3:=5*x3+10*x4+15*x6
=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,(%#x3G\"\"&%#x4G\"#5%#
x6G\"#:F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eq4:=2*x1+6*x2
+8*x4+18*x6=6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq4G/,*%#x1G\"\"#
%#x2G\"\"'%#x4G\"\")%#x6G\"#=F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "solve(\{eq1,eq2,eq3,eq4\},\{x1,x2,x3,x4,x5,x6\});" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#<(/%#x1G,*%#x2G!\"$%#x4G!\"%%#x6G!\"*\"\"$\"\"
\"/%#x3G,(F)!\"#F+F(F.F./F+F+/F)F)/F'F'/%#x5G,&F+#F-\"\"##!\"\"F:F." }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 
"The free variables here are x2, x4, and x6. Show that the two solutio
ns are equivalent. Another facility built into " }{TEXT 280 6 "linalg
" }{TEXT -1 94 " can extract the coefficient matrix and the constant m
atrix from a system of linear equations." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "A:=genmatrix([eq1,eq2,eq3,eq4],[x1,x2,x3,x4,x5,x6],'B
');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7(\"\"\"\"
\"$!\"#\"\"!\"\"#F-7(F.\"\"'!\"&F,\"\"%!\"$7(F-F-\"\"&\"#5F-\"#:7(F.F0
F-\"\")F-\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(B);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&\"\"!!\"\"\"\"&\"\"'" 
}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Matrix Algebra" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=matrix(3,3,a);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%-%\"aG6$\"\"\"F--F+6$F-\"\"#-F
+6$F-\"\"$7%-F+6$F0F--F+6$F0F0-F+6$F0F37%-F+6$F3F--F+6$F3F0-F+6$F3F3" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "B:=matrix(3,3,b);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%-%\"bG6$\"\"\"F-
-F+6$F-\"\"#-F+6$F-\"\"$7%-F+6$F0F--F+6$F0F0-F+6$F0F37%-F+6$F3F--F+6$F
3F0-F+6$F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "A+B;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"AG\"\"\"%\"BGF%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(A+B);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7%7%,&-%\"aG6$\"\"\"F,F,-%\"bGF+F,,&-F*6$F,\"\"#F,-F
.F1F,,&-F*6$F,\"\"$F,-F.F6F,7%,&-F*6$F2F,F,-F.F<F,,&-F*6$F2F2F,-F.F@F,
,&-F*6$F2F7F,-F.FDF,7%,&-F*6$F7F,F,-F.FIF,,&-F*6$F7F2F,-F.FMF,,&-F*6$F
7F7F,-F.FQF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(k*A);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%*&%\"kG\"\"\"-%\"a
G6$F*F*F**&F)\"\"\"-F,6$F*\"\"#F**&F)F/-F,6$F*\"\"$F*7%*&F)F/-F,6$F2F*
F**&F)F/-F,6$F2F2F**&F)F/-F,6$F2F6F*7%*&F)F/-F,6$F6F*F**&F)F/-F,6$F6F2
F**&F)F/-F,6$F6F6F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "eval
m(-A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%,$-%\"aG6$\"
\"\"F,!\"\",$-F*6$F,\"\"#F-,$-F*6$F,\"\"$F-7%,$-F*6$F1F,F-,$-F*6$F1F1F
-,$-F*6$F1F5F-7%,$-F*6$F5F,F-,$-F*6$F5F1F-,$-F*6$F5F5F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "transpose(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%-%\"aG6$\"\"\"F+-F)6$\"\"#F+-F)6$\"\"$F
+7%-F)6$F+F.-F)6$F.F.-F)6$F1F.7%-F)6$F+F1-F)6$F.F1-F)6$F1F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A&*B;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%#&*G6$%\"AG%\"BG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalm(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrix
G6#7%7%,(*&-%\"aG6$\"\"\"F-F--%\"bGF,F-F-*&-F+6$F-\"\"#F--F/6$F3F-F-F-
*&-F+6$F-\"\"$F--F/6$F9F-F-F-,(*&F*\"\"\"-F/F2F-F-*&F1F>-F/6$F3F3F-F-*
&F7F>-F/6$F9F3F-F-,(*&F*F>-F/F8F-F-*&F1F>-F/6$F3F9F-F-*&F7F>-F/6$F9F9F
-F-7%,(*&-F+F5F-F.F>F-*&-F+FBF-F4F>F-*&-F+FKF-F:F>F-,(*&FRF>F?F>F-*&FT
F>FAF>F-*&FVF>FDF>F-,(*&FRF>FHF>F-*&FTF>FJF>F-*&FVF>FMF>F-7%,(*&-F+F;F
-F.F>F-*&-F+FEF-F4F>F-*&-F+FNF-F:F>F-,(*&F\\oF>F?F>F-*&F^oF>FAF>F-*&F`
oF>FDF>F-,(*&F\\oF>FHF>F-*&F^oF>FJF>F-*&F`oF>FMF>F-" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "How can we deno
te the identity matrix? We cannot use the symbol " }{TEXT 276 1 "I" }
{TEXT -1 27 ", since it is reserved for " }{XPPEDIT 18 0 "sqrt(-1);" "
6#-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 34 ". Check the help page on the \+
word " }{TEXT 277 5 "alias" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "alias(Id=&*());" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"IG%#IdG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "zero:=matrix(3,3,0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%zeroG-%'matrixG6#7%7%\"\"!F*F*F)F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(zero+Id);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(A^2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%,(*$)-%\"aG6$\"\"\"F.\"\"#\"\"\"F.
*&-F,6$F.F/F.-F,6$F/F.F.F.*&-F,6$F.\"\"$F.-F,6$F9F.F.F.,(*&F+F.F2F0F.*
&F2F0-F,6$F/F/F.F.*&F7F0-F,6$F9F/F.F.,(*&F+F0F7F0F.*&F2F0-F,6$F/F9F.F.
*&F7F0-F,6$F9F9F.F.7%,(*&F4F0F+F0F.*&F?F0F4F0F.*&FGF0F:F0F.,(F1F.*$)F?
F/F0F.*&FGF0FBF0F.,(*&F4F0F7F0F.*&F?F0FGF0F.*&FGF0FJF0F.7%,(*&F:F0F+F0
F.*&FBF0F4F0F.*&FJF0F:F0F.,(*&F:F0F2F0F.*&FBF0F?F0F.*&FJF0FBF0F.,(F6F.
FTF.*$)FJF/F0F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=matri
x(2,2,[a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6
#7$7$%\"aG%\"bG7$%\"cG%\"dG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$*&%
\"dG\"\"\",&*&%\"aG\"\"\"F)F.F.*&%\"bGF.%\"cGF.!\"\"!\"\",$*&F0F*F+F3F
27$,$*&F1F*F+F3F2*&F-F*F+F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "A:=matrix(3,3,[1,-2,2,2,1,1,1,0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"!\"#\"\"#7%F,F*F*7%F*\"\"!F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(A^(-1));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"#!\"%7%!\"\"F,\"\"$7%F
,!\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "evalm(%%&*%)
; # Note (%%) refers to second last output. The stuff after the symbol
 # is a comment and is not executed on the command line." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }
}}}}{MARK "3 19 1" 0 }{VIEWOPTS 0 0 0 3 4 1802 }