MVR5
I/context/cfpushR6$%"aG'%"LG<$%%nameG%&stackG6#%"tG6#%inCopyright~(c)~1997~by~W
aterloo~Maple~Inc.~All~rights~reserved.G6"C%-%'ASSERTG6#-%)assignedG6#F*@$4-%%t
ypeG6$9%F*-%'assignG6$F<-&F*6#.%$newGF/-&F*6#.%%pushG6$9$F<F/F/F/6",
I*getMatrixRF/6#%"AGF/F/C(-%'printfG6#%VPlease~enter~a~matrix.~A:=matrix([[1,2,
3],[3,2,2]]);|+G>8$-%)readstatGF/>%'finishGFW@$5/Fen%%exitG/Fen%%EXITG-%'RETURN
G6#%4Execution~SuspendedG?(F/"""FaoF/4-F:6$FW%'matrixGC&-FS6#%UPlease~try~again
:~eg:~A:=matrix([[1,2,3],[3,2,2]]);|+G>FWFX>FenFW@$FgnF\o-F]o6#FWF/6#FenF/F/,
I5CM_RemoveAssumptionsR6#%"fGF+F-F/C&-%(readlibG6#%'assumeG@$-F:6$FK%.HasAssump
tionGC$-F>6$-%*substringG6$FK;Fao!"#.F`q-F]o6#F`q>FW-%'indetsGF[q@%0FW<"-%$mapG
6$9!FW%%NULLGF/F/F/6",
I+getIntegerRF/6#%"nGF/F/C(-FS6#%DPlease~enter~an~integer.~eg:~n:=3;|+G>FWFX>Fe
nFW@$FgnF\o?(F/FaoFaoF/4-F:6$FW%(integerGC&-FS6#%=Please~try~again:~eg:~n:=3;|+
G>FWFX>FenFW@$FgnF\oF]pF/F_pF/F/,
I5help/text/transtrans-%%TEXTG6,%"~G%MHELP~SCREEN~for~:~Transpose~of~the~transp
oseGF_t%3PACKAGE~:~MatricesG%hoCONTENTS~:~This~function~checks~if~the~transpose
~of~the~transpose~is~equal~toG%8~~~~~~~~~~~the~matrix~AGF_t%T~~~~~~~~~~~~To~Cal
l~:~>~A~:=~matrix([[1,2],[3,3]]);G%G~~~~~~~~~~~~~~~~~~~~~~>~transtrans(A);GF_tF
/,
I0context/cfparseR6#'%#CFG%)CF_CLASSG6=%(arglistG%(ActionsG%+ActionNameG%+Actio
nKeysG%)AncestorG%-ConditionalsG%0ConditionalKeysG%%DataG%,DescriptionG%%fileG%
%headG%+HelpStringG%(IfBlockG%%MenuG%$KeyG%&LocCFG%+MenuStringG%,MenuEntriesG%+
RecognizerG%-ReplacementsG%)SubMenusG%-SubMenusKeysG%#t0G%%tailG%%TestG%)warnin
gsG%"xGF-F/CK@$0%4context/InitializedG%%trueG--Ffp6#.%-context/initGF/-Ffp6#%1c
ontext/cmglobalG-Ffp6#%.context/cfgetG-Ffp6#%8context/cfparse/contentG-Ffp6#%6c
ontext/cfparse/menusG-Ffp6#%;context/cfparse/recognizerG>83FK>%0_EnvContextData
G-%&tableGF/>8%-Fjq6$Fbx.%*CF_ACTIONG>8:-%%timeGF/-%)userinfoG6%Fao<#Fft-%(spri
ntfG6%QF%7.3f~:~Replacing~%d~Actions~by~Keys.6",&F_yFaoF^y!""-%%nopsG6#Fhx>8'Fe
x?&8>FhxF\wC&>82-Fdw6#-Fgw6$Q+MenuString6"Fbz>&F`z6#Fbz-%*CM_ACTIONG6#Fez>&Fdx6
#F_[lFbz-Fjw6$F_[lFex>87<#-%$seqG6$/&FdxF^[lFbz/Fbz-F_r6$%#opG7#-%(entriesG6#F`
z>Fbx-%%subsG6$Fh[lFbx>8)-Fjq6$Fbx.%/CF_CONDITIONALG-Fby6%FaoFdy-Ffy6%QK%7.3f~:
~Replacing~%d~Conditionals~by~Keys.FiyFjy-F]z6#F\]l>Fh[l-F_r6$%Bcontext/cfparse
/splitconditionalsGF\]l>F\]l-Fi\l6$Fh[lF\]l>FbxFh\l>8*Fex?&FbzF\]lF\wC'>Fez-Fdw
6#Q*ConditionFiy>&Fa^lF^[l-%/CM_CONDITIONALGFa[l>8+-F_r6$RF+F/F/F/@$-F:6$FKF`[l
&Fdx6#-%/CM_ACTION_DATAG6#-Fb\l6#FKF/F/F/Fbz>&Fdx6#Fj^lFbz-Fjw6$Fj^lFex>Fh[l<#-
F[\l6$F]\l/Fbz-F_r6$Fb\l7#-Fe\l6#Fa^l>Fbx-%)CM_CLASSG6#-Fb\l6#Fh\l>88-Fjq6$Fbx.
%+CF_SUBMENUG-Fby6%FaoFdy-Ffy6$QD%7.3f~:~Replacing~Submenus~by~Keys.FiyFjy>89Fe
x?&FbzFaalF\wC%>FezFhz>&F\blF^[l-%+CM_SUBMENUGFa[l>&Fdx6#FablFbz>Fh[l<#-F[\l6$F
]\l/Fbz-F_r6$Fb\l7#-Fe\l6#F\bl>FbxFh\l>Fbx-Fi\l6$<%/F[uF\al/%(CF_CODEG%(CM_CODE
G/%-CF_SEPARATORG%-CM_SEPARATORGFbx-Fby6%FaoFdy-Ffy6$QD%7.3f~:~Building~Menu~Te
mplates~...FiyFjy-F]x6%Fbx7"Fddl>8=-%*interfaceG6#.%*warnlevelG-Fhdl6#/Fjdl""!-
Fby6%FaoFdy-Ffy6$Q@%7.3f~:~Building~Recognizer~...FiyFjy>86-F`x6$Fbx-%%evalG6#F
dx>%,ContextDataGFiel-Fhdl6#/F[elFfdl-Fby6%FaoFdy-Ffy6$Q>%7.3f~:~CONTEXTMENU~Co
mpletedFiyFjy-F]o6#-Fjel6#FfelF/6$FdxF]flF/Fiy,
I-context/initR6"6#%.ReadlibDefineG6$%FCopyRight~1997~by~Waterloo~Maple~Inc.G%f
nCopyright~(c)~1997~Waterloo~Maple~Inc.~All~rights~reserved.GF]glCco@$/F[wF\w-F
]o6#Fbr>8$Ffp-Figl6#.%3context/autoassignG-Figl6#.%9context/BuildInputStringG-F
igl6#.%8context/InitializeProcsG-Figl6#.Fgw-Figl6#.Fft-Figl6#.%?context/cfparse
/classtoifblockG-Figl6#.%<context/cfparse/act_TRIPLESG-Figl6#.%=context/cfparse
/act_completeG-Figl6#.%?context/cfparse/act_LINEARFUNCG-Figl6#.%<context/cfpars
e/conditionalG-Figl6#.Fjw-Figl6#.F]x-Figl6#.F`x-Figl6#.F">%*CM_GlobalG-Ffp6#.Fd
w-Figl6#.%<convert/CONTEXTMENU/TRIPLESG-Figl6#.%?convert/CONTEXTMENU/LINEARFUNC
G-Figl6#%6context/defaultconfigG-Figl6#.%5context/fixpiecewiseG-Figl6#.%0contex
t/fullkeyG-Figl6#.%3context/isparsableG>%.CM_IsParsableG.-Ffp6#Ff\m-Figl6#.%1co
ntext/jointextG-Figl6#.%3context/multiparseG-Figl6#.Ficl-Figl6#.F`p-Figl6#.%+CM
_GetVarsG-Figl6#.%0CM_GetArrayVarsG-Figl6#.%,CM_GetFuncsG-Figl6#.%.CM_InertFunc
sG-Figl6#.%4convert/CONTEXTMENUG-Figl6#.%<convert/context/ListToStackG-Figl6#.%
<convert/context/StackToListG-Figl6#.%/type/CF_ACTIONG-Figl6#.%.type/CF_CLASSG-
Figl6#.%-type/CF_CODEG-Figl6#.%1type/CF_CODETYPEG-Figl6#.%4type/CF_CONDITIONALG
-Figl6#.%2type/CF_SEPARATORG-Figl6#.%0type/CF_SUBMENUG-Figl6#.%/type/CM_ACTIONG
-Figl6#.%.type/CM_BUILDG-Figl6#.%.type/CM_CLASSG-Figl6#.%-type/CM_CODEG-Figl6#.
%4type/CM_CONDITIONALG-Figl6#.%0type/CM_IFBLOCKG-Figl6#.%3type/CM_InertFuncsG-F
igl6#.%2type/CM_SEPARATORG-Figl6#.%0type/CM_SUBMENUG-Figl6#.%1type/CM_MathFuncG
-Figl6#.%/type/CM_MatrixG-Figl6#.%0type/CM_NOPRINTG-Figl6#.%2type/CM_PIECEWISEG
-Figl6#.%3type/HasAssumptionG-Figl6#.%9type/context/Unbounded@@G-Ffp6#.F*-Figl6
#.%+APPENDPLOTG-Figl6#.%,APPENDPLOTSG-Figl6#.%+DELETEPLOTG-Figl6#.%.EDITSMARTPL
OTG-Figl6#.%4SMARTPLOTADD_OptionG-Figl6#.%/SMARTPLOTColorG-Figl6#.%4SMARTPLOTCo
lor_ListG-Figl6#.%3SMARTPLOTGetColorsG-Figl6#.%6SMARTPLOTFindCoords2DG-Figl6#.%
6SMARTPLOTFindCoords3DG-Figl6#.%-SMARTPLOTFixG>F[wF\wFbrF]gl6%F][mFi\mF[wF]glF]
gl,
I)mainmenuRF/6#%)responseG6#%SCopyright~1996~by~Ananda~Gunawardena~&~Elias~Deeb
aGF/C,-%&blankG6#Fao-FS6#%aoChoose,~from~the~following~options,~the~appropriate
~route~to~proceed.|+G-FS6#%]oAt~the~prompt~sign,~enter~your~selection~followed~
by~a~semi-colon|+G-FS6#%S~~~1.~Find~the~inverse~of~the~coefficient~matrix.|+G-F
S6#%W~~~2.~Find~the~determinant~of~the~coefficient~matrix.|+G-FS6#%jo~~~3.~Find
~the~row~echelon~form~of~the~matrix~to~find~a~solution~to~the~system|+G>FWFX>Fe
nFW@$FgnC$-%&printGF^o-F]o6#F_tF]pF/F_pF/F/,
I'linmat=F/%&falseGE\[l+%)GeometryGRF/63%"SG%"TG%#S1G%#S2G%#T1G%"iG%"jG%#g1G%#g
2G%#x1G%#x2G%#y1G%#y2G%#t1G%#t2G%%min1G%&indexGFdhmF/C*-F^jm6#%]pThe~purpose~of
~this~function~is~to~demonstrate~the~effect~of~matrix~multiplicationG-F^jm6#%`p
on~a~2D~image.~The~image~must~be~entered~as~a~set~of~n~elements~where~n~is~the~
numberG-F^jm6#%^pof~points~that~make~up~the~image.~The~matrix~is~a~2x2~matrix.~
The~function~displaysG-F^jm6#%jothe~original~image~and~the~transformed~image.~T
his~function~can~be~used~to~viewG-F^jm6#%Dreflections,~rotations,~and~scalingGF
ghm@)/9#""#C$@&-F:6$&9"Fihm%$setG>FWFa]n-F:6$Fa]nFeo>FhxFa]n@&-F:6$&Fb]n6#F\]nF
c]n>FWF[^n-F:6$F[^nFeo>FhxF[^n/F[]nFao@'Fe]nC'>FhxFa]n-FS6#%Aplease~enter~the~i
mage~as~a~set|+G-F^jm6#%FFor~example:~~S:=|fr[1,2],[3,2],[2,1]|hr;G>FWFX?(F/Fao
FaoF/4-F:6$FWFc]nC$-FS6#%>A~set~is~expected.~Try~again|+G>FWFXF_]nC'>FWFa]n-FS6
#%;please~enter~the~matrix~T|+G-F^jm6#%HFor~example:~A:=matrix([[1,1],[-1,1]]);
G>FhxFX?(F/FaoFaoF/4-F:6$FhxFeoC$-FS6#%<Matrix~expected.~Try~again|+G>FhxFX-%&E
RRORG6#%MIncorrect~arguments.~Type~?Geometry~for~helpG/F[]nF_elC*Fe^nFh^n>FWFX?
(F/FaoFaoF/F]_nC$Fa_n>FWFXFg_nFj_n>FhxFX?(F/FaoFaoF/F_`nC$Fc`n>FhxFX-Fh`n6#%Lto
o~many~arguments.~Type~?Geometry~for~helpG@%5550-%'rowdimGF^z-%'coldimGF^z4-F:6
$&FWFihm%%listG2""$-F]z6#Fdbn2FgbnF]bnFg`nC$-%%withG6#%&plotsG@&3/F]bnF\]n/Fhbn
F\]nC4>8&-Feo6$F\]n-F]zF^p?(F\]lFaoFaoF\]nF\w?(Fa^lFaoFaoFicnF\w>&Ffcn6$F\]lFa^
l-%&evalfG6$&&FWFi`lFg]lF\]n>8(-%)multiplyG6$FhxFfcn>F`zF]r?(Fa^lFaoFaoFicnF\w>
F`z-%&unionG6$F`z<#-%$colG6$FednFa^l>Fbx"$***?(F\]lFaoFaoFicnF\w@$2&&FWFg]lFihm
FbxC$>84F\]l>FbxFhen>8-&&FW6#F\fnFihm>8/&FafnF\^n>FbxFden?(F\]lFaoFao-F]zFf\lF\
w@$2&&F`zFg]lFihmFbxC$>F\fnF\]l>FbxF[gn>8.&&F`zFbfnFihm>80&FcgnF\^n>81-%)textpl
otG6%7%,&F_fnFaoF[zFao,&FdfnFaoF\]nFao%/original~imageG/%&alignG<$%&ABOVEG%%LEF
TG/%&colorG%$redG>Fez-Fjgn6%7%,&FagnFaoF[zFao,&FegnFaoFdqFao%2transformed~image
G/Fahn<$Fdhn%&BELOWG/Ffhn%%blueG>F]_l-%*pointplotG6%FW/Ffhn%$REDG/%*thicknessG"
#5>8,-Ffin6%F`z/Ffhn%%BLUEGFjin-%(displayG6#<&F]_lF^jnFezFhgn3/F]bnFgbn/FhbnFgb
nF/F/F_pF/%*trinverseGRF/6+FP%%flagG%#c1G%#r1G%'transAG%&invtAG%%invAG%(transiA
G%$ansGFdhmF/C-Fghm-F^jm6#%boThe~purpose~of~this~program~is~to~check~if~the~inv
erse~of~the~transposeG-F^jm6#%Wof~a~matrix~A~is~the~transpose~of~the~inverse;~t
hat~isG-%)continueGF/@$5FjnFhn-F]oF/@&F[anC&-FS6#%ZEnter~the~matrix.~|frfor~exa
mple,~A:=matrix([[1,2],[2,3]])|+G>FWFX?(F/FaoFaoF/FboC$-FS6#%?Matrix~is~Expecte
d.~Try~Again|+G>FWFX-F^jm6$%;The~matrix~you~entered~is~G/FW-%&evalmGF^pFa^nC$>F
W-%(convertGFf]n-F^jm6$%6The~given~matrix~is~:GF`]oF\\o@$F_\oF`\o>F`z-F^bnF^p>F
fcn-F`bnF^p@%3/F`zFfcn0-%$detGF^pF_elC6>Fedn-&%'linalgG6#%*transposeGF^p>F\]l-&
Fi^o6#%(inverseG6#Fedn>Fa^l-F^_oF^p>F]_l-Fh^oFi`l-F^jm6$%IThe~inverse~of~the~tr
anspose~inv(trA)~=~GF\]lF\\o@$F_\oF`\o-F^jm6$%KThe~transpose~of~the~inverse~tr(
inv(A))~=~GF]_lF\\o@$F_\oF`\o-FS6#%^oIs~the~transpose~of~the~inverse~equal~to~t
he~inverse~of~transpose?|+G-FS6#%CEnter~your~answer~as:~yes;~or~no;|+G>F^jnFX?(
F/FaoFaoF/53/F^jn%#noG0-%%normG6$-Fb]o6#,&F\]lFaoF]_lF[zF\]nF_el3/F^jn%$yesGFj`
oC1-FS6#%TTry~to~find~a~matrix~for~which~this~property~holds|+GFghm-FS6#%2Enter
~the~matrix|+GFX>FW-Ff]o6$%"%GFeo-F^jm6$%7You~entered~the~matrixGFW>F`zF\^o>Ffc
nF^^o@$F`^oC*>FednFg^o>F\]lF]_o>Fa^lFc_o>F]_lFe_oFf_oF\\o@$F_\oF`\oFj_oF\\o@$F_
\oF`\oFghm-FS6#%JDoes~this~property~hold~for~this~matrix?|+G-FS6#%CType~your~an
swer~as~:~yes;~or~no;|+G>F^jnFX?(F/FaoFaoF/3Fh`o/F[aoF_elC$-F^jm6#%QYour~answer
~is~not~correct.~Try~the~other~choiceG>F^jnFXF\\o@$F_\oF`\o-F^jm6#%ioIndeed,~th
e~transpose~of~the~inverse~is~equal~to~the~inverse~of~the~transpose.GFghm-F^jm6
#%SSeed~of~thought!!!~Does~this~property~always~hold?G-F^jm6#%WThis~matrix~is~s
ingular.~We~cannot~check~this~propertyGF/F_pF/%+transtransGRF/6*F][oFP%(trans_A
G%-trans_transAGF^[o%#c2GF_[o%#r2GFdhmF/C8-F^jm6#%^oThe~purpose~of~this~program
~is~to~check~the~validity~of~conjecture~G-F^jm6$/))FhxFijmFijmFhx%9for~a~given~
matrix~A~.~~GF\\o@$F_\oF`\o@&F[anC&-FS6#%jnEnter~the~given~matrix.~For~example,
~A:=matrix([[1,2],[2,3]]).|+G>FhxFX?(F/FaoFaoF/F_`nC$Fi\o>FhxFX-F^jm6$F_]o/Fhx-
Fb]oF^zFa^nC$>FhxFe]o-F^jm6$Fi]oFbfo>Ffcn-Fh^oF^z>F`z-Fh^o6#FfcnFghm-F^jm6$%FTh
e~Transpose~of~the~matrix~trA~is~:~G/Fdeo-Fb]oF\goFghm-F^jm6$%MThe~Transpose~of
~the~Transpose~of~the~matrixG/Fceo-Fb]oFf\lF\\o@$F_\oF`\o>Fa^l-F^bnF\go>Fedn-F`
bnF\go>F]_l-F^bnFf\l>F\]l-F`bnFf\l@%/-F\ao6$-Fb]o6#,&FhxFaoF`zF[zF\]nF_el-F^jm6
%%=In~this~case,~the~conjectureGFbeo%'holds.G-F^jm6#%MIn~this~case,~the~conject
ure~does~not~hold~.GFghm-F^jm6#%[oThis~is~the~end~of~the~transtrans~procedure.P
lease~enter~anotherG-F^jm6#%\omatrix~and~execute~the~procedure~to~get~a~feel~fo
r~this~property.G-F^jm6#%@Try~to~give~a~proof~in~general.GF/F_pF/%)trainnetGRF/
6LFchmFd[o%#MbG%#NbGF][nF^[n%"kG%"lG%"MG%"NG%&thetaG%"bG%'bthetaG%#UMG%#UNG%#fv
G%#fuG%"uG%"vG%&deltaG%'delta1G%%averG%'deltaMG%'deltaNGFcp%#IRG%#ORGF%%#v1G%(d
eltaMbG%(deltaNbG%&fnetiG%%netiG%$UMbG%$UNbG%$IRbG%%netjG%&fnetjG%#TOG%&inputG%
+iterationsG%&alphaGFdhmF/Chn-FS6#%\o~~The~purpose~of~this~procedure~is~to~simu
late~a~neural~network.|+G-FS6#%E~~More~specifically,~this~procedure|+G-FS6#%ho~
~~1.~~simulates~and~train~a~neural~net~using~the~back~propagation~algorithm|+G-
FS6#%eo~~~2.~~checks~if~the~trained~net~recognizes~the~set~of~rules~by~producin
g|+G-FS6#%Q~~~~~~~an~output~"close~to"~the~targeted~output|+G-FS6#%hn~~For~more
~details,~please~see~Application~2.3~from~Unit~Two|+GFghm@$F[anC'-FS6#%enNeural
~net~is~trained~using~an~iterative~process.~You~may|+G-FS6#%]ochoose~the~number
~of~iterations.~How~many~iterations~do~you~want?|+G>8LFX>FenFj]p@$FgnF`\o@$Fa^n
>Fj]pFa]n-FS6#%VPlease~enter~the~input~rules~in~the~form~of~a~matrix|+G-FS6#%\o
For~example,~matrix([[1,0,1,0,0,1,0],[1,0,0,1,0,0,1]])~represent|+G-FS6#%intwo~
input~rules.~(Input~rules~(a)~and~(b)~of~Application~2.3)|+G>8KFX>FenFi^p@$FgnF
`\o?(F/FaoFaoF/4-F:6$Fi^pFeoC&-FS6#%1Matrix~expected|+G>Fi^pFX>FenFi^p@$FgnF`\o
>8G-F[_o6#Fi^p-FS6#%enPlease~enter~the~targeted~output~of~each~rule~as~a~matrix
|+G-FS6#%MFor~example,~matrix([[1,0,0,0],[0,1,0,0]]);|+G>8JFX>FenFb`p@$FgnF`\o?
(F/FaoFaoF/4-F:6$Fb`pFeoC&Fa_p>Fb`pFX>FenFb`p@$FgnF`\o>Fbz-F[_o6#Fb`p>8M.Faap-F
S6#%ZEnter~the~scaling~function~as~f:=t->1/(1+exp(-alpha*t));|+GFX>8<RF+F/6$%)o
peratorG%&arrowGF/*$,&FaoFao-%$expG6#,$*&T#FaoFKFaoF[zFaoF[zF/F/6$F]\pFaap-F^jm
6$%9You~entered~the~functionG-FgapF+-FS6#%hoChoose~alpha~appropriately~so~that~
the~net~can~be~trained~with~a~reasonable~|+G-FS6#%[pnumber~of~iterations.~In~th
is~procedure~alpha=50~is~a~good~choice.~However,~you|+G-FS6#%@may~enter~your~ch
oice~of~alpha|+G-FS6#%@Please~enter~a~value~for~alpha|+G>FaapFX@%/Faap"#]C$>Fga
pRF+F/FiapF/*$,&FaoFao-F_bp6#,$FK!#]FaoF[zF/F/F/-F^jm6$%8The~scaling~function~i
sGFhbpC$>FgapRF+F/FiapF/F\bpF/F/FdbpFbdp>Ffcn-Feo6#7&7&$F[zFdq$FaoFdq$F\]nFdqF^
ep7&F]epF^epF_ep$FgbnFdq7&F]epFaepF_epF^ep7&F]epF^epFaepF^ep>F`z-Feo6#7)7%F]ep$
FdqFdqF]ep7%F^epF_epFaep7%F^epFaepF^ep7%F^epF^epF_ep7%FaepF_epF^ep7%F_epF^epF_e
p7%F]epFaepF^ep-F^jm6$%enThe~original~weights~of~the~input~layer~is~a~random~ma
trixG/F`zFfgo-F^jm6$%fnThe~original~weights~of~the~output~layer~is~a~random~mat
rixG/FfcnFagoF\\o@$FgnF`\o-FS6#%WWe~start~training~the~net~and~updating~of~the~
weights|+G-FS6#%Fusing~the~back~propagation~algorithm|+G?(FdfnFaoFaoFj]pF\wC$?(
F]_lFaoFao-F^bnFj_pF\wC8>8H-Fgdn6$-F[_o6#-Ff]o6$-Faen6$Fh_pF]_lFeoF`z>8I-%'vect
orG6#7&Fao-Fgap6#&Fegp6$FaoFao-Fgap6#&Fegp6$FaoF\]n-Fgap6#&Fegp6$FaoFgbn>Ffel-F
gdn6$-F[_o6#-Ff]o6$F_hpFeoFfcn>Fbx-Fahp6#7&-Fgap6#&FfelFghp-Fgap6#&FfelF[ip-Fga
p6#&FfelF_ip-Fgap6#&Ffel6$Fao""%-FS6$%FThis~is~net~output~for~input~rule~%d|+GF
]_l-F^jm6#Fbx>Fh[l-Fahp6#Fhjp?(F\]lFaoFaoFhjpF\w>&Fh[lFg]l*(&FbxFg]lFao,&FaoFao
Fe[qF[zFao,&&-Faen6$FbzF]_lFg]lFaoFe[qF[zFao>8A-Feo6$FhjpFhjp?(F\]lFaoFaoFhjpF\
w?(FednFaoFaoFhjpF\w>&F\\q6$F\]lFedn,$*&Fc[qFao&F_hpFa_oFao$FgbnF[z/F^jn-Fb]o6#
-F[_o6#F\\q>8E-Fb]o6#,&F[]qFaoFfcnFao>Ffcn-Fb]o6#F^]q>F\bl-Fahp6#Fgbn?(F\]lFaoF
aoFgbnF\w>&F\blFg]l-%$sumG6$*&&Fh[lFi`lFaoF]dnFao/Fa^l;FaoFhjp>FaalFf]q?(F\]lFa
oFaoFgbnF\w>&FaalFg]l*(&F_hpFg]lFao,&FaoFaoFg^qF[zFaoFj]qFao>8B-Feo6$""(Fgbn?(F
ednFaoFaoF]_qF\w?(F\]lFaoFaoFgbnF\w>&Fj^q6$FednF\]l,$*&Fe^qFao&F\hpFa_oFaoFg\q/
Fj^q-Fb]o6#Fj^q>8F-Fb]o6#,&Fj^qFaoF`zFao>F`z-Fb]o6#Fj_q@$2FdfnFj]p-F^jm6$%8This
~is~iteration~roundG,&FdfnFaoFaoFaoF\\o@$FgnF`\o-F^jm6#%?These~are~the~trained~
matricesG-F^jm6#%QThe~adjusted~weight~matrix~of~the~input~layer~isG-F^jm6#FcfpF
ghm-F^jm6#%RThe~adjusted~weight~matrix~of~the~output~layer~isG-F^jm6#Fgfp-F^jm6
#%foDo~you~want~to~check~whether~the~above~matrices~actually~do~train~the~net?|
+G-FS6#%4Answer~yes;~or~~no|+G>FhxFX>FenFhx@$FgnF`\o@$/FhxFcaoC+Fghm?(FednFaoFa
oFbgpF\wC=-FS6$%HEnter~your~input~rule%d~as~a~1x7~matrixGFedn>FWFX>FenFW@$FgnF`
\o?(F/FaoFaoF/FboC&Fa_p>FWFX>FenFW@$FgnF`\o>Fdfn-Fgdn6$FWF`z-F^jm6$%;The~interm
ediate~output~isG/Fdfn*&Fa]oFaoFfgoFaoFghm>8?-Fahp6#7&Fao&FdfnFghp&FdfnF[ip&Fdf
nF_ip>FgapRF+F/FiapF/F\bpF/F/Fdbp>Fgcq-Fahp6#7&-F`dn6#-Fgap6#&FgcqFihm-Fgap6#&F
gcqF\^n-Fgap6#&FgcqFg]q-Fgap6#&FgcqF`[q-F^jm6$%_oThe~firing~of~the~intermediate
~output~by~the~scaling~function~yieldsGFgcqFghm>Fgcq-Ff]o6$FgcqFeo>Ffel-Fgdn6$-
F[_o6#FgcqFfcn-F^jm6#%hoMultiplying~the~fired~intermediate~output~by~the~update
d~output~matrix~yieldsG-F^jm6#/Ffel*&-Fb]o6#F[fqFaoFagoFaoFghm>Ffel-Ff]o6$FfelF
ahp>8@-Fahp6#7&-F`dn6#-Fgap6#&FfelFihm-Fgap6#&FfelF\^n-Fgap6#&FfelFg]q-Fgap6#&F
felF`[q-F^jm6$%2The~net~output~isGFjfqFghm-FS6$%FCompare~with~the~targeted~outp
ut~%d~|+GFedn-F^jm6#-Faen6$FbzFednFghmF\\o@$FgnF`\o-FS6#%goIs~the~net~output~wi
thin~a~"reasonable~tolerance"~from~the~targeted~output?|+G-FS6#%Jfor~each~input
~rule?~Answer:~yes;~or~no;|+GFghm>FhxFX>FenFhx@$FgnF`\o@%/FhxFi`o-F^jm6#%_oYou~
may~need~to~choose~more~iteration~in~trainnet()~to~train~the~netGC%-F^jm6#%FThe
~updated~input~and~output~matricesG-F^jm6$FcfpFgfp-F^jm6#%Uhave~the~proper~weig
hts~and~the~net~has~been~trainedGFghm-F^jm6#%]oYou~may~try~to~train~the~net~wit
h~different~input~and~output~rulesGF/F_pF/%)LUdecompGRF/6N%&AUGGGG%$n11GFd[oF][
oF][nF^[nF\joFgr%$numGFP%#A1G%#b1G%$AUGG%(colflagGFh[n%'norowsG%#IdG%&lowerG%$p
dtG%%tempG%%EpdtGFchm%'secmulG%%rsetG%)firstmulG%)firstrowG%'secrowG%%pdt1G%+mu
ltiplierG%'rownumG%%row2G%%leftG%%row1G%)typeflagG%'rowaddG%(tempaugG%#L1G%#u1G
%*undocountG%(leftrowG%'assrowG%(rowsaveG%"mG%%RsetGFdhmF/C2Fghm-F^jm6#%goThe~p
urpose~of~this~procedure~is~to~demonstrate~the~steps~needed~to~find~theG-F^jm6#
%[pLU~decomposition~of~a~given~matrix~and~to~interact~with~the~function~to~find
~theG-F^jm6#%]pLU~decomposition~yourself.~For~more~information~type~?LUdecomp~a
t~the~Maple~promptGFghm@%F[anC$-FS6#%hnEnter~a~matrix.~For~example~type~:~A:=ma
trix([[0,1],[3,3]]);|+G>FagnFX>FagnFa]n?(F/FaoFaoF/4-F:6$FagnFeoC&-FS6#%<Matrix
~Expected.~Try~again|+G>FagnFX>FenFagn@$Fgn-F]o6#%$ByeG>F`zF_el?(FednFaoFao-%%r
ankG6#FagnF\w@$/-Fd^o6#-%*submatrixG6%Fagn;FaoFednFb_rF_el>F`zFao@$/F`zFaoC&Fgh
m-F^jm6%%,The~matrix~GFagn%I~does~not~have~a~unique~LU~factorizationG-F^jm6#%DT
ype~?LUdecomp~for~more~informationGFghm-F^jm6#%CSelect~one~of~the~following~mod
es:G-FS6#%:~~~1.~Demonstration~mode|+G-FS6#%8~~~2.~Interactive~mode|+G-FS6#%V~~
~3.~No~Intermediate~steps.~Just~give~me~the~answer|+G>FfcnFX@(/FfcnFaoCT>FezFao
@$Fa^n>FegnFe]o@$F[an>Fegn-Ff]oF[^r>FW-Fb]o6#Fegn>F]_l-%(vectdimG6#-Faen6$FegnF
ao>FbxF]_l>F\fn-Feo6$FbxFbx?(FednFaoFaoFbxF\wC$?(F\]lFaoFaoFbxF\w>&F\fnFb_qF_el
>&F\fn6$FednFednFao>85-Fahp6#,$*&F]_lFao,&F]_lFaoFaoFaoFao#FaoF\]n>FhxF]cr?(Fed
nFaoFaoFhxF\w>&FjbrFa_oF_br>&FjbrFihmF\fn-F^jm6#%RThe~LU-Decomposition~of~the~m
atrix~is~obtained~byG-F^jm6$%Eperforming~the~gauss~elimination~on:GFegnF\\o@$55
Fhn/Fen%%ExitGFjnFc^r>F]_l-F`bnFfar>8N-F^bnFfar@$2FedrF]_l>F]_lFedr?(FednFaoFao
F]_lF\wC)>Fa^lFedn?(F/FaoFaoF/3/&Fegn6$Fa^lFednF_el2Fa^lFedr>Fa^l,&Fa^lFaoFaoFa
o>FhgnF_el?(F\]l,&FednFaoFaoFaoFaoFedrF\w@$0&FegnFc\qF_el>FhgnFao@$0&FegnFhbrF_
elC$@$/FhgnFaoC$-F^jm6$%A~Perform~elimination~on~column~:GFedn-F^jm6#%E********
****************************G@$0F^frFaoC.Fghm>Fh[lF^fr-F^jm6(%+Divide~rowGFedn%
4of~the~given~matrixGFegn%#byGF^fr>Fegn-%'mulrowG6%FegnFedn*$F^frF[zFghm-F^jm6$
%4to~get~the~matrix~:GFegn>&FjbrFa[l-Fdgr6%F\fnFedn*$Fh[lF[z-F^jm6$%RPerform~th
e~same~operation~on~the~identity~matrixG-Fb]oFbfn-F^jm6$%Lto~get~the~correspond
ing~elementary~matrix:G/&%"EGFa[l-Fb]o6#F[hr>Fez,&FezFaoFaoFaoF\\o@$F^dr-F]o6#%
*Try~LaterGFghm?(F\]lFgerFaoFedrF\w@$FierC.>Fh[l,$FjerF[z-F^jm6*%-Multiply~rowG
FednF`grFegn%$~byGFeir%4and~add~that~to~rowGF\]l>Fegn-%'addrowG6&FegnFednF\]lFe
irFghm-F^jm6$%3to~get~the~matrix:GFegn>F[hr-F]jr6&F\fnFednF\]lFh[lFghm-F^jm6$Fa
hrF\fnFchr>FezF\irF\\o@$F^drFc^r@$30&Fegn6$F]_lF]_lF_el0F\[sFaoC*-F^jm6(%(The~r
owGF]_l%.of~the~matrixGFegn%2~is~multiplied~byG*$F\[sF[z>F[hr-Fdgr6%F\fnF]_lFe[
s>Fh[lFe[s>Fegn-Fdgr6%FegnF]_lFe[sFghm-F^jm6(Fb[sF]_lFc[sF\fnFd[sFh[lFchr>FezF\
ir>FfelFfcr>Faal&FhhrFihm?(FednF\]nFao,&FezFaoF[zFaoF\wC$>Ffel-Fb]o6#-%#&*G6$Fd
crFfel>Faal*&&FhhrFa_oFaoFaalFao-F^jm6#%[pApplying~elementary~row~operations~on
~the~given~matrix~to~obtain~its~row~echelonG-F^jm6#%ioform~U,~is~equivalent~to~
pre-multiplying~the~matrix~by~the~elementary~matricesG>%"UG.Ff]s-F^jm6#/*&FaalF
ao-Fb]o6#F_fnFaoFf]sFghm-F^jm6$%>The~upper~triangular~matrix~:G/Ff]s-Fb]o6#-&Fi
^o6#%*gausselimGFfar-F^jm6#%Ois~the~row~echelon~form~of~the~original~matrixGF\\
o@$F^drFc^rFghm>%"PG.F]_s-F^jm6$%SThe~product~of~the~elementary~matrices~is~giv
en~byG/F]_sFaal-F^jm6$%4Which~is~equal~to~:G/F]_s-Fb]oFifl>F]_sFfelF\\o@$F^drFc
^r-F^jm6#%hnThe~lower~triangular~matrix~L~is~the~inverse~of~this~matrix~PG-F^jm
6#/F'-F^_oFiflFghm>F'F``sF\\o-F^jm6#%WTherefore~the~LU~decomposition~of~the~giv
en~matrix~is:G-F^jm6#/Fa]o*&-Fb]o6#F'FaoFearFaoFghm-F^jm6#%doThis~is~the~end~of
~the~~LU~decomposition~demostration~version.~Try~other~G-F^jm6#%Bexamples~to~le
arn~this~procedure.G/FfcnF\]nCao>8O-Fahp6#71%#R1G%#R2G%#R3G%#R4G%#R5G%#R6G%#R7G
%#R8G%#R9G%$R10G%$R11G%$R12G%$R13G%$R14G%$R15G>8;-Fahp6#71F_[oF[eo%#r3G%#r4G%#r
5G%#r6G%#r7G%#r8G%#r9G%$r10G%$r11G%$r12G%$r13G%$r14G%$r15G>FezFao@$Fa^n>FegnFe]
o@$F[an>FegnFcar>FWFear>F]_lFhar>FbxF]_l@$FaasC*>FbxFhar>F\fnF_br?(FednFaoFaoFb
xF\wC$?(F\]lFaoFaoFbxF\w>FebrF_el>FgbrFao>FjbrF[cr>FhxF]cr?(FednFaoFaoFhxF\w>Fd
crF_br>FfcrF\fn-%*gaussmenuGF/>F\bl%&BEGING-F^jm6$%7The~orginal~matrix~is:GFegn
Fghm>F`zFao>FezFao?(F/FaoFaoF/30F`zF_el0F`zF\]nC*-FS6#%hpPlease~enter~~at~each~
step~a~row~operation~as~in~the~above~menu~to~get~the~LU-~decomposition|+G-FS6#%
jn<enter~operations~as~Ri~<>~Rj~or~Ri~=~c*Rj~+~Ri~or~Ri~=~c*Ri~>|+G>F\blFX>FenF
\bl@$5FgnF`dr-F]o6#%&AdiosG@-5/F\blF[o/F\blFinC$>F`zF_el>Fb`pF_el5/F\bl%&rowecG
/F\bl%&ROWECGC'>FegnFd^s?(FednFaoFaoFbxF\w@$F]fr>FegnFcgr-F^jm6$%DThe~matrix~in
~row~echelon~form~is~:GFegn>F`zF\]n>Fb`pF_el5/F\bl%%undoG/F\bl%%UNDOGC%@(/F^]qF
ao>Fegn-%(swaprowG6%Fegn8DFj^q/F^]qF\]n>Fegn-Fdgr6%FegnF\\q*$FjfqF[z/F^]qFgbn?(
FednFaoFaoFcdrF\w>&Fegn6$Fj]pFedn&FaapFa_o@%/Fb`pF_elC$-F^jm6$%FBefore~your~sel
ection~the~matrix~was~GFegn>Fb`p,&Fb`pFaoFaoFao-F^jm6#%Xonly~one~level~of~undo~
is~permitted~in~the~demo~versionG>FezFd\s-F:6$-%$rhsGF`cl%)monomialGC%?(FednFao
FaoFbxF\w@$50-%&coeffG6$Ffjs&FhbsFa_oF_el0-F_[t6$Ffjs&FdasFa_oF_elC%@$F][t>Fjfq
F^[t@$Fb[t>FjfqFc[t>F\\qFedn@%/FjfqFaoC.>Fj^qF\\q>8C-%$lhsGF`cl?(FednFaoFaoFbxF
\wC$@$/Fa[tFa\t>F\isFedn@$/Fe[tFa\t>F\isFedn>FegnFihs-F^jm6&%0Interchange~rowGF
\is%$andGFj^q-F^jm6$%8This~is~the~new~matrix:GFegnFghm>F[hr-Fjhs6%F\fnF\isFj^q-
F^jm6$%CApply~the~same~row~operations~to~:GF\fn-F^jm6$%PWe~obtain~the~correspon
ding~elementary~matrix~:GFfhr>F^]qFao>FezF\irC$?(FednFaoFaoFbxF\w@$50-F_[t6$Fb\
tFa[tF_el0-F_[t6$Fb\tFe[tF_el>Fi^pFedn@%/Fi^pF\\qC+>Fegn-Fdgr6%FegnF\\qFjfq-F^j
m6&FhirF\\qFagrFjfq-F^jm6$%7This~is~the~new~matrixGFegnFghm>F[hr-Fdgr6%F\fnF\\q
Fjfq-F^jm6$%IBy~applying~the~same~row~operations~to~:GF\fn-F^jm6$%Pwe~obtain~th
e~corresponding~elementary~matrix~:GFfhr>F^]qF\]n>FezF\ir-F^jm6#%foinvalid~row~
multiplication.~Multiplied~row~must~be~assigned~to~the~same~rowG>Fb`pF_el-F:6$F
fjs%(polynomGC0>FfdlF_el>FbzF_el?(FednFaoFaoFbxF\wC$@$Fb^t@&Fc^t>Fj]pFednFf^t>F
j]pFedn@$F\[t@%/FfdlF_elC$>FfdlFedn@&F][t>FgapF^[tFb[t>FgapFc[tC$>FbzFedn@&F][t
>F^yF^[tFb[t>F^yFc[t>Faap-%$rowG6$FegnFj]p?(FednFaoFaoFcdrF\w>Feis&-Fb]o6#,&*&F
gapFao-F_bt6$FegnFfdlFaoFao*&F^yFao-F_bt6$FegnFbzFaoFaoFa_o>F[hr-F]jr6&F\fnFfdl
FbzFgap@'3/FgapFao/FfdlFj]p-F^jm6(%.multiply~row~GFbzFagrF^y%/and~add~to~rowGFf
dl3/F^yFao/FbzFj]p-F^jm6(FfctFfdlFagrFgapFgctFbz-FS6-%G~~~~~~~%s~%d~%s~%d~%s~%d
~%s~%d~%s~%d~|+G%-multiply~rowGFfdlFagrFgap%(and~rowGFbzFagrF^y%6add~and~assign
~to~rowGFj]p-F^jm6$%4The~new~matrix~is~:GFegnFghm-F^jm6$%QApply~the~same~row~op
erations~to~identity~matrixGF\fn-F^jm6$%Pto~obtain~the~corresponding~elementary
~matrix~:GFfhr>F^]qFgbn>Fb`pF_el>FezF\irC$-F^jm6#%@incorrect~selection.~Try~aga
in.G>Fb`pF_el>Fh_pFd^s@$/-F\ao6$-Fb]o6#,&Fh_pFaoFegnF[zFaoF_elC%>F\]lFfdr?(F/Fa
oFaoF//&Fegn6$F\]lF\]lF_el>F\]l,&F\]lFaoF[zFao@$3/F`ftFaoF[fsC%F\\o-F^jm6$%LYou
~obtained~the~matrix~in~row~echelon~formGFegn>F`zF\]n@$3/F`zF_elF[fs-F^jm6#%Kwa
rning.~Matrix~is~not~in~row~echelon~formGFghm-FS6#%HWhat~is~the~upper~triangula
r~matrix~U?|+G-F^jm6#%LEnter~your~answer~as~a~matrix.~For~example,G-F^jm6#%Atyp
e:~>U:=matrix([[1,2],[0,2]]);GFghm>Ff]sFX?(F/FaoFaoF/4-F:6$Ff]sFeoC&>FenFf]s@$F
gnF`\o-FS6#%1Matrix~Expected|+G>Ff]sFX@&F]htC$Fc`n>Ff]sFX50-Fiar6#-F_bt6$Ff]sFa
o-Fiar6#-FaenF_it0F\it-Fiar6#-F_btF\brC&-F^jm6#%TThe~matrix~is~not~square~or~di
mensions~incompatibleG-F^jm6#%0Enter~a~matrix|+G>Ff]sFX@$FgnF`\o@%0-F\ao6$-Fb]o
6#,&FegnFaoFf]sF[zFaoF_elC%-F^jm6#%PYour~answer~is~incorrect.~The~correct~matri
x~isG-F^jmFfar-F^jm6#%Nwhich~is~the~matrix~A~in~its~row~echelon~formG-F^jm6$%?G
ood!~The~correct~matrix~is~:~GFegnFghm>FfelFfcr>FgcqF_br>FgcqFfel?(FednF\]nFaoF
d\sF\wC$>Ffel-Fb]o6#-Fj\s6$FdcrFgcq>FgcqFfelFghm-FS6#%P~~How~would~find~the~low
er~triagular~matrix~L?|+G-FS6#%en~~~~~~1.~Take~the~inverse~of~the~upper~triangu
lar~matrix.|+G-FS6#%`o~~~~~~2.~Take~the~inverse~of~the~product~of~the~elementar
y~matrices.|+G-FS6#%A~~~~~~3.~Take~the~inverse~of~A.|+G-F^jm6#%=<choose~from~1;
~or~2;~or~3;>GFghm>FfcnFX@$FgnF`\o@%0FfcnF\]nC$-F^jm6#%XIncorrect.~The~lower~tr
iangular~matrix~L~is~obtained~byG-F^jm6#%gntaking~the~inverse~of~the~product~of
~the~elementary~matricesGC$-F^jm6#%+Very~Good!GF\\oFghm-F^jm6$%ZThe~product~of~
the~elementary~matrices~E1,~E2,~..~is~P~=~GFfel>F]_sFfelFghm-FS6#%HWhat~is~the~
lower~triangular~matrix~L?|+G-FS6#%8~~~~~~1.~inverse~of~P.|+G-FS6#%R~~~~~~2.~tr
anspose~of~the~row~echelon~form~of~P.|+G-FS6#%N~~~~~~3.~product~of~the~elementa
ry~matrices.|+GFi\uFghm>FegpF``s>FfcnFX>FenFfcn@$FgnF`\o@%0FfcnFaoC$-F^jm6#%gnI
ncorrect.~The~lower~triangular~matrix~L~is~the~inverse~of~PG-F^jm6#/)F]_s%#-1G-
Fb]o6#FegpC$Fh]uF\\o>F'Fegp>Ff]sFegn-FS6#%TTo~verify~the~answer,~multiply~the~m
atrices~L~and~UG-F^jm6$%$LU=G/*&Fi`sFao-Fb]o6#Ff]sFao-Fb]o6#-Fj\s6$F'Ff]sFghm-F
^jm6#/%OIs~this~product~the~same~as~the~given~matrix~AGFi`u-FS6#%4Answer~yes;~o
r~no;|+G>FfcnFX>FenFfcn@$F^drF`\o@%/FfcnFi`o-F^jm6#%1Redo~the~problemGC$-F^jm6#
%WYou~obtained~the~LU~decomposition~of~the~given~matrix:G-F^jm6#/Fh`sFi`uFghm-F
^jm6#%foThis~is~the~end~of~the~interactive~LUdecomp~procedure.~Please~enter~ano
therG-F^jm6#%hnmatrix~and~execute~the~procedure~to~learn~the~steps~involved.G/F
fcnFgbnC6>FezFao@$Fa^n>FegnFe]o@$F[an>FegnFcar>FWFear>F]_lFhar>FbxF]_l>F\fnF_br
?(FednFaoFaoFbxF\wC$?(F\]lFaoFaoFbxF\w>FebrF_el>FgbrFao>FjbrF[cr>FhxF]cr?(FednF
aoFaoFhxF\w>FdcrF_br>FfcrF\fn?(FednFaoFao,&F]_lFaoF[zFaoF\wC%?(Fa^lFednFaoF]_lF
\w@$0F`erF_elC%@$0Fa^lFednC(>F[hr-Fjhs6%F\fnFa^lFedn>Fegn-Fjhs6%FegnFa^lFednFgh
m>FezF\irF\\o@$F^drF^irFghm%&breakG@$F]frC%>FhgnFao?(F\]lFgerFaoF]_lF\w@$Fier>F
hgnF_el@$FjfrC%>F[hr-Fdgr6%F\fnFednFfgr>FegnFcgr>FezF\ir?(F\]lFgerFaoF]_lF\w@$F
ierC%>F[hr-F]jr6&F\fnFednF\]lFeir>FegnF\jr>FezF\ir@$FjjrC%>F[hrFg[s>FegnF[\s>Fe
zF\ir>FfelFfcr?(FednF\]nFaoFd\sF\w>FfelFg\s>F]_sFfel>F'F``s-F^jm6#%NThe~LU~deco
mposition~of~the~given~matrix~is~:GFe`sF/6&FenF]_sF'Ff]sF/%*matrixmulGRF/6'FP%"
BGF][nF^[nF^[rFdhmF/C+-F^jm6#%XThe~purpose~of~this~procedure~is~to~show~the~pro
cess~ofG-F^jm6#%enmultiplying~two~matrices~A~and~B~and~obtaining~the~entriesG-F
^jm6#%9of~the~product~matrix~ABGFghm@)Fj\nC$>FWFa]n>FhxF[^nFa^nC(>FWFa]n-F^jm6#
/%<The~matrix~you~entered~is~AGFa]o-FS6#%;Please~enter~the~matrix~B|+G>FhxFX?(F
/FaoFaoF/F_`nC$Fi\o>FhxFX-F^jm6#/%<The~matrix~you~entered~is~BGFcfoF[anC*-FS6#%
<Please~enter~the~matrix~~A|+G>FWFX?(F/FaoFaoF/FboC$Fi\o>FWFXF]]o-FS6#%BPlease~
enter~the~second~matrix~B|+G>FhxFX?(F/FaoFaoF/F_`nC$Fi\o>FhxFXF`fo-Fh`n6#%(Erro
r!!G@%/F^^oF]bnC'>Fedn-Feo6$F\^oF_bn-F^jm6#/%DThe~product~of~the~two~matrices~A
B~G*&Fa]oFaoFcfoFao-F^jm6#%7is~obtained~as~followsG?(FfcnFaoFao-F^bnFa_oF\w?(F`
zFaoFao-F`bnFa_oF\w>&Fedn6$FfcnF`z%"*G?(FfcnFaoFaoF\^oF\w?(F`zFaoFaoF_bnF\wC+F\
\o@$F_\oF`\o-FS6%%inMultiply~row~%d~of~the~matrix~A~and~column~%d~of~the~matrix
~B|+GFfcnF`z-FS6%%Pto~get~the~(%d,%d)-th~entry~of~the~product~AB~|+GFfcnF`z-F^j
m6#/*&-Fb]o6#-F_bt6$FWFfcnFao-Fb]o6#-Ff]o6$-Faen6$FhxF`zFeoFao-%*innerprodG6$F_
]vFe]vFghm>F\\vFg]v-F^jm6#%Land~write~the~entry~into~the~product~matrixG-F^jm6#
/%#ABG-Fb]oFa_o-F^jm6#%ZThe~matrix~dimensions~are~incompatible~for~multiplicati
onGFghm-F^jm6#%HEnd~of~matrix~multiplication~procedure.G-F^jm6#%KTry~your~own~e
xamples~to~learn~the~processGF/F_pF/%*trproductGRF/62F][oFPFgguFgjq%#B1GF^[oFjd
oF_[oF[eo%(mult_ABG%)trans_ABGFhdo%(trans_BG%&tA_tBGFd[o%&tB_tAGFdhmF/C3>Fhx.Fh
x>Ffcn.Ffcn-F^jm6#%YThe~purpose~of~this~program~is~to~check~the~conjecture~:G-F
^jm6#%Yif~the~transpose~of~the~product~of~two~matrices~is~equalG-F^jm6#%Lto~the
~product~of~their~transpose;~that~is,G-F^jm6#/)Fa^vFijm*&FdeoFao)FfcnFijmFaoF\\
o@$F_\oF`\o@&F[anC*-FS6#%aoPlease~Enter~the~first~matrix.~For~example,~A:=matri
x([[1,2],[2,3]]).|+G>F`zFX?(F/FaoFaoF/4-F:6$F`zFeoC$Fi\o>F`zFX-F^jm6$F_]o/FhxFf
go-FS6#%BPlease~Enter~the~second~matrix~B|+G>FednFX?(F/FaoFaoF/4-F:6$FednFeoC$F
i\o>FednFX-F^jm6$F_]o/FfcnFb^vFj\nC'>F`zFe]o>Fedn-Ff]oF_^n-F^jm6$%6The~first~ma
trix~is~:GFhavFghm-F^jm6$%=The~second~given~matrix~is:~GFebvF\\o@$F_\oF`\o>F]_l
F]ho>F\]lF_ho>F^jnFh[v>Fa^lFj[v@%3/F^jnF\]l/F]_lFa^lC=>F_fn-&Fi^o6#Fgdn6$F`zFed
n>Fagn-Fh^oF]^s>Fdfn-Fh^oFf\l>Fegn-Fh^oFa_o>Fhgn-F\dv6$FdfnFegn>Fbx-F\dv6$FegnF
dfn-F^jm6$%OThe~transpose~of~the~product~transpose(AB)~is~G/Ff`v-Fb]oFj^rF\\o@$
F_\oF`\o-F^jm6#%ZThe~product~of~the~transpose~transpose(A)transpose(B)~is~G-F^j
m6#-Fb]o6#FhgnF\\o@$F_\oF`\o-F^jm6#%ZThe~product~of~the~transpose~transpose(B)t
ranspose(A)~is~G-F^jm6#-Fb]oF][qF\\o@$F_\oF`\o-FS6#%enWhich~one~of~the~followin
g~results~will~then~hold~for~any|+G-FS6#%fnmatrices~A~and~B?~Enter~your~answer~
by~selecting:~1;~or~2;|+G-FS6#%Q~~~1.~transpose(AB)~=~transpose(A)*transpose(B)
|+G-FS6#%Q~~~2.~transpose(AB)~=~transpose(B)*transpose(A)|+G>FezFX?(F/FaoFaoF//
FezFaoC$-F^jm6#%JThis~is~not~correct.Try~the~other~choice.G>FezFX-F^jm6#%LIndee
d,~for~the~matrices~A~&~B~you~entered,G-F^jm6#%Jtranspose(AB)~=~transpose(B)*tr
anspose(A)GFghmF_do-F^jm6#%_oThe~two~matrices~are~not~compatible.~We~cannot~ver
ify~this~statementGFghmF/F_pF/F`_oRF/6PFd[oFPFgjqF][nF^[nF\jo%#i1G%#j1GFe\rFgr%
&cofacG%#IDGFijqFb[rFf\rF_[rF^jo%"CG%$AdjG%"dG%%AUG1GFchmFhjqF][oFa[rF]\rFj[rF\
\rFc[rFd[rFe[rFi[rF[\r%$eqsGFg[rF[[rFh[rF^\rFggu%#i2G%#j2G%&temp1GFa\rFb\rFc\rF
d\rFdhmF/C3Fghm-F^jm6#%ipThe~purpose~of~this~function~is~to~demonstrate~the~ste
ps~to~find~the~inverse~of~a~matrix~usingG-F^jm6#%jpeither~the~Gauss-Jordan~algo
rithm~or~the~Adjoint~method.~In~selecting~the~interactive~mode,~youG-F^jm6#%ipc
an~immediately~check~if~you~have~learned~this~algorithm.~The~nostep~mode~dispal
ys~the~inverseG-F^jm6#%^pwith~no~intermediate~steps.~For~more~information~type~
?inverse;~at~the~Maple~promptGFghm@%F[anC%-F^jm6#%gnEnter~a~matrix.~For~example
~type~:~A:=matrix([[0,1],[3,3]]);G-FS6#%7Please~enter~a~Matrix|+G>FfcnFX>FfcnFa
]n?(F/FaoFaoF/4-F:6$FfcnFeoC&-FS6#%3A~matrix~Expected|+G>FfcnFX>FenFfcn@$FgnF`\
o@&0FigoF[hoC%-F^jm6$%8The~matrix~you~entered~G/FhxFago-F^jm6#%8~is~not~a~squar
e~matrixGF`\o/-Fd^oF\goF_elC%F`[w-F^jm6#%gnhas~a~zero~determinant.~Therefore~th
e~inverse~does~not~existGF`\o-F^jm6$FaboFc[wFghmF^`rFa`rFd`rFg`r>FWFX@(/FWFaoC(
>F_fnFigo>F^jnF[ho@%/F^jnF_fnC*-FS6#%=Which~method~do~you~prefer?|+G-FS6#%>~~~~
~~1.~Gauss-Jordan~method|+G-FS6#%9~~~~~~2.~Adjoint~method|+G-FS6#%8<Select~from
~1;~or~2;>|+G>FWFX>FenFW@$FgnC$F]jmF_jm@&/FWF\]nC9Fghm-F^jm6#%ioIn~this~method~
we~find~the~cofactor~of~each~entry~and~form~the~cofactor~matrixG-F^jm6#%aoThe~c
ofactor~of~(i,j)th-entry~is~given~by~(-1)^(i+j)*det(minor(A,i,j))GFghm>Fagn-Feo
6$F_fnF_fn?(Fa^lFaoFaoF_fnF\w?(F]_lFaoFaoF_fnF\w>&Fagn6$Fa^lF]_lF^\v>FhxFfcn-F^
jm6$%9The~original~matrix~is:~GFbfoF\\oFghm?(Fa^lFaoFaoF_fnF\wC$-FS6$%R~Now~fin
d~the~cofactors~of~the~entries~in~row~%d|+GFa^l?(F]_lFaoFaoF_fnF\wC,@$FgnF`\o>F
g^w*&)F[z,&Fa^lFaoF]_lFaoFao-Fd^o6#-%&minorG6%FhxFa^lF]_lFao-F^jm6$%/In~the~mat
rix~GFbfo-FS6&%KThe~minor~of~the~(%d,%d)-th~entry~%d~is~:|+GFa^lF]_l&FhxFh^w-F^
jm6#/&F\fnFh^w-&Fi^o6#F\`wF]`wFghm-F^jm6$%7and~it's~cofactor~is:~G/&FjbrFh^wFg^
wFghm-F^jm6$%=The~updated~cofactor~matrix:G/FjbrF_evF\\o@$FgnF`\o-F^jm6#%`oThe~
adjoint~matrix~is~the~transpose~of~the~cofactor~matrix.~ThereforeG-F^jm6#/%>The
~adjoint~of~the~matrix~is:G-Fh^oFj^rF\\o@$FgnF`\o-F^jm6#%_oThe~inverse~of~the~m
atrix~A~is~(1/det(A))*Adjoint(A)~and~therefore~:GFghm>Fbx-&Fi^o6#Fd^oF^z@$0FbxF
aoC%-F^jm6$%GSince~the~determinant~of~the~matrix~isGFbx-F^jm6$%7The~inverse~of~
A~~is~:G/)FhxFi_u*&-Fb]o6#F]bwFaoFbxF[zFghm-F^jm6$%7which~is~equivalent~toG-Fb]
o6#-F^_oF^zF\\o-F^jm6$%>We~can~verify~this~by~showingG/*&FcfoFaoFgcwFao-Fb]o6#-
Fj\s6$FhxFicwFa\wC>-F^jm6#%foWe~find~the~inverse~of~the~matrix~A~by~forming~the
~augmented~matrix~[A~|gr~I]G-F^jm6#%gowhere~I~is~the~identity~matrix.~Then~use~
elementary~row~operations~to~reduceG-F^jm6#%`oit~to~a~matrix~of~the~form~[I~|gr
~B].~The~matrix~B~is~the~inverse~of~A.GFghm>FdfnFb^w?(F`zFaoFaoF_fnF\w?(FednFao
FaoF_fnF\w>&FdfnF^dvF_el?(F`zFaoFaoF_fnF\w>&Fdfn6$F`zF`zFao>FhxFfcn>Fegn-%(augm
entG6$FhxFdfn>F_fnFhar>F^jnFdit-F^jm6$%CThe~original~augmented~matrix~is~:GFegn
Fghm-F^jm6#%enApply~the~elementary~row~operations~to~[A~|gr~I]~and~convertG-F^j
m6#%Sthe~matrix~to~its~reduced~row~echelon~form~[I~|gr~B]GF\\o@$FgnF`\oFghm-F^j
m6#%NWe~employ~the~Gauss~-~Jordan~elimination~on~:GF[[uF\\o@$FgnF`\o?(F`zFaoFao
F_fnF\wC*@$/&FegnFeewF_elC(>F\]lF\]n?(F/FaoFaoF//&Fegn6$F\]lF`zF_el>F\]l,&F\]lF
aoFaoFao-F^jm6&%5Interchange~the~row~GF`z%%and~GF\]l>Fegn-Fjhs6%FegnF\]lF`z-F^j
m6$%)To~get~:GFegnF\\o@$0F_gwFaoC&-F^jm6&%,Divide~row~GF`z%$by~GF_gw>Fegn-Fdgr6
%FegnF`z*$F_gwF[z-F^jm6$%*To~get~:~GFegnF\\o>F\]l,&F`zFaoFaoFao@$1F\]lF_fn?(F/F
aoFaoF/3Fcgw2F\]lF_fn>F\]lFggw@$3/F\]lF_fn0FdgwF_el-F^jm6$%>Perform~elimination
~on~columnGF`zFghm?(FednFaiwFaoF_fnF\w@$0&Fegn6$FednF`zF_elC'-F^jm6(%.multipliy
~rowGF`zFagr,$FbjwF[zFjirFedn>Fegn-F]jr6&FegnF`zFednFhjwF_hwF\\o@$FgnF`\o?(Fedn
,&F`zFaoF[zFaoF[zFaoF\w@$FajwC'Fejw>FegnFjjwF_hwF\\o@$FgnF`\o-F^jm6#%goThe~augm
ented~matrix~[A~|gr~I]~is~now~in~the~form~[I~|gr~B],~with~B~=~inverse(A)G-F^jm6
$%<Therefore~the~inverse~of~A=GFhx-F^jm6$%$is~GFicwF\\oFjcw-F^jm6%%+The~matrixG
Fhx%Vis~not~square~.~Type~in~a~square~matrix~and~try~againGFghm-F^jm6#%foThis~i
s~the~end~of~the~inverse~procedure.~Choose~other~matrices~and~executeG-F^jm6#%c
othe~function~again~to~learn~these~two~algorithms~for~finding~the~inverseGFi]wC
>>F_fn-Fiar6#-Faen6$FfcnFao>F^jn-Fiar6#-F_btF[]x>Fh[lFh[w>FdfnFb^w?(F`zFaoFaoF_
fnF\w?(FednFaoFaoF_fnF\w>FaewF_el?(F`zFaoFaoF_fnF\w>FdewFao>FhxFfcn>FaalFhew-F^
jm6$%BThe~determinant~of~this~matrix~isGFh[lFghm-FS6#%I~~Does~the~inverse~of~th
e~matrix~exist?|+G-FS6#%8~~answer~~yes;~or~no;~|+G>FWFX?(F/FaoFaoF/45/FWFcao/FW
Fi`oC&>FenFW@$FgnF`\oFaau>FWFX@&3Fg^x0Fh[lF_el-F^jm6#%WThis~is~not~correct.~The
~inverse~of~the~matrix~exists.G3Ff^xF^_x-F^jm6$%PGood!~Proceed~to~find~the~inve
rse~of~the~matrixGFhxFghm-FS6#%Z~~Which~one~of~the~following~methods~you~prefer
~to~use~?|+G-FS6#%jn~~~~1.~Show~that~the~matrix~is~row~equivalent~to~the~identi
ty.|+G-FS6#%S~~~~2.~Use~the~Adjoint~method~to~find~the~inverse|+G-FS6#%8<select
~from~1;~or~2;>|+G>FWFX>FenFW@$FgnF`\o@&Fa\wCI-FS6#%QAugment~the~matrix~A~with~
the~identity~matrix.~|+G-FS6#%Ieg:~AUG:=matrix([[1,2,1,0],[3,4,0,1]]);|+G>FegnF
X?(F/FaoFaoF/4-F:6$FegnFeoC&>FenFegn@$FgnF`\oFcht>FegnFX?(F/FaoFaoF/50Ffdr-F^bn
6#Faal0Fcdr-F`bnFjaxC'-FS6%%9%d~x~%d~matrix~expected|+GFiaxF\bx-FS6#%+Try~again
|+G>FegnFX>FenFegn@$FgnF`\o@%/-F\ao6$-Fb]o6#,&FaalFaoFegnF[zFaoF_elC$Fghm-F^jm6
$%@You~entered~the~correct~matrix.G/FegnFear-F^jm6$%JThis~is~not~correct.The~co
rrect~matrix~isG/Fegn-Fb]oFjaxFghm-FS6#%hnWhat~is~the~best~route~you~will~selec
t~to~find~the~inverse?~|+G-FS6#%C~~~1.~Find~the~determinant~of~AUG|+G-FS6#%O~~~
2.~Use~the~Gauss-Jordan~elimination~on~AUG|+G-FS6#%Q~~~3.~Use~the~backsubstitut
ion~algorithm~on~AUG|+G-FS6#%;<select~from~1;~2;~or~3;>|+G>FWFX>FenFW@$FgnF`\o?
(F/FaoFaoF/5Fa\w/FWFgbnC&-F^jm6#%@You~can~have~a~better~selectionG>FWFX>FenFW@$
FgnF`\o-F^jm6#%KGood~choice.~Proceed~to~find~the~inverse!!G>FegnFaal>FezFeas>Fh
gnFibs>Fh_pFfdrF_es-FS6#%boPlease~enter~a~row~operation~to~reduce~the~matrix~as
~in~the~menu~above|+G>F\blFbes-F^jm6$%AThe~orginal~augmented~matrix~is:GFegnFgh
m>FhbsFao>FedrF_el?(F/FaoFaoF/30FhbsF_el0FhbsF\]nC'F`fs>F\blFX@-F[gsC$>FhbsF_el
>FedrF_elFagsC'>FegnFd^s?(F`zFaoFaoFh_pF\w@$0F_gwF_el>FegnFjhwF[hs>FhbsF\]n>Fed
rF_elF`hsC$@(/FgcqFao>Fegn-Fjhs6%FegnF\isFa\t/FgcqF\]n>Fegn-Fdgr6%FegnFegp*$Fj_
qF[z/FgcqFgbn?(F`zFaoFaoFcdrF\w>&Fegn6$8PF`z&8QFf\l@%/FedrF_elC$F[js>Fedr,&Fedr
FaoFaoFaoC%-F^jm6#%Tonly~one~level~of~undo~is~permitted~in~this~versionG-F^jm6#
%ZIf~you~wish~to~exit~the~program~type~exit;~or~to~continueGFghmFdjsC%?(F`zFaoF
aoFh_pF\w@$50-F_[t6$Ffjs&FhgnFf\lF_el0-F_[t6$Ffjs&FezFf\lF_elC%@$Fiix>Fj_qFjix@
$F]jx>Fj_qF^jx>FegpF`z@%/Fj_qFaoC*>Fa\tFegp>FbzFb\t?(F`zFaoFaoFh_pF\wC$@$/F\jxF
bz>F\isF`z@$/F`jxFbz>F\isF`z>FegnFjgx-F^jm6&F_]tF\isF`]tFa\tFa]tFghm>FgcqFaoC$?
(F`zFaoFaoFh_pF\w@$50-F_[t6$Fb\tF\jxF_el0-F_[t6$Fb\tF`jxF_el>FdasF`z@%/FdasFegp
C'>Fegn-Fdgr6%FegnFegpFj_q-F^jm6&FhirFegpFagrFj_qFb_tFghm>FgcqF\]nF``t>FedrF_el
Fd`tC,>F\\qF_el>Fj^qF_el?(F`zFaoFaoFh_pF\wC$@$F[\y@&F\\y>FfhxF`zF_\y>FfhxF`z@$F
hix@%/F\\qF_elC$>F\\qF`z@&Fiix>FjfqFjixF]jx>FjfqF^jxC$>Fj^qF`z@&Fiix>FgapFjixF]
jx>FgapF^jx>Fhhx-F_bt6$FegnFfhx?(F`zFaoFaoFcdrF\w>Fdhx&-Fb]o6#,&*&FjfqFao-F_bt6
$FegnF\\qFaoFao*&FgapFao-F_bt6$FegnFj^qFaoFaoFf\l@'3F]\t/F\\qFfhx-F^jm6(FfctFj^
qFagrFgapFgctF\\q3Fbct/Fj^qFfhx-F^jm6(FfctF\\qFagrFjfqFgctFj^q-FS6-F_dtF`dtF\\q
FagrFjfqFadtFj^qFagrFgapFbdtFfhxFcdtFghm>FgcqFgbn>FedrF_elC$F`et>FedrF_el>F_hp-
&Fi^o6#%%rrefGFfar@$/-F\ao6$-Fb]o6#,&F_hpFaoFegnF[zFaoF_elC%>FednFfdr?(F/FaoFao
F//F^frF_el>Fedn,&FednFaoF[zFao@$3/F^frFaoFgfxC$-F^jm6$%TYou~obtained~the~matri
x~in~reduced~row~echelon~formGFegn>FhbsF\]n@$/FhbsF_elC&-F^jm6$%boWarning.~You~
are~existing~too~soon.~Here~is~the~reduced~echelon~form~ofGFegnFghm>FegnFb`y-F^
jm6$%@The~reduced~row~echelon~form~isGFegnFghm-FS6#%]oFrom~this~reduced~form~ex
tract~the~inverse,B,~of~the~given~matrix|+G-FS6#%@eg~:~B:=matrix([[1,2],[3,2]])
;|+G>Fb`pFX?(F/FaoFaoF/Ff`pC&Fcht>Fb`pFX>FenFb`p@$Fgn-F]o6#%,Bye~for~nowG>Fhx-F
`_r6%Fegn;FaoF_fn;,&F_fnFaoFaoFao,$F_fnF\]n@$50-F^bnF_apF_fn0-F`bnF_apF_fnC$-F^
jm6$%SIncorrect~matrix~dimensions.~The~correct~matrix~isGFhx>Fb`pFhx@%/-F\ao6$-
Fb]o6#,&Fb`pFaoFhxF[zFaoF_el-F^jm6$%ICorrect,~the~inverse~of~the~matrix~is~:~G/
F`cw-Fb]oF_ap-F^jm6$%DIncorrect.~The~correct~inverse~is~:GF`eyFghmFi]wCJ>Fb`pFb
^w-F^jm6$%7The~original~matrix~isGFfcnF\\o-F^jm6#%hnTo~find~the~cofactor~of~the
~(i,j)~entry~use~the~Maple~commandG-F^jm6#%=>(-1)^(i+j)*det(minor(A,i,j)GFghm>F
agnFb^w?(F`zFaoFaoF_fnF\w?(FednFaoFaoF_fnF\w>&FagnF^dvF^\v>FhxFfcn?(F`zFaoFaoF_
fnF\w?(FednFaoFaoF_fnF\w>&Fb`pF^dv*&)F[z,&F`zFaoFednFaoFao-Fd^o6#-F\`w6%FhxF`zF
ednFao?(F`zFaoFaoF_fnF\w?(FednFaoFaoF_fnF\wC*Fghm-FS6%%JEnter~the~cofactor~of~t
he~entry~(%d,%d):|+GF`zFedn-F^jm6$%/of~the~matrix~GFhx>FWFX>FenFW@$Fgn-F]o6#%1T
oo~soon~to~quitG>FdfyFify@%0-%$absG6#,&FWFaoFifyF[zF_elC%-F^jm6$%EIncorrect.~Th
e~correct~cofactor~is~:GFifyFghm-F^jm6$%=The~new~cofactor~matrix~is~:GFagnC$-F^
jm6#%hnYou~entered~the~correct~cofactor.~The~new~cofactor~matrix~is:G-F^jmFj^rF
ghm-FS6#%`oHaving~the~cofactor~matrix,~what~is~the~next~step?~Enter~your~choice
|+G-FS6#%U~~~~~1.~Find~the~determinant~of~the~cofactor~matrix|+G-FS6#%K~~~~~2.~
Construct~the~adjoint~matrix~of~A|+G-FS6#%H~~~~~3.~Find~the~LU-decomposition~of
~A|+G>FWFX>FenFW@$FgnF`\o@$0FWF\]n-F^jm6#%`oThis~is~not~a~good~choice!!.~You~ne
ed~to~construct~the~adjoint~matrixGFghm>Ffel-Fh^oF_ap-FS6#%enGood.~Construct~th
e~adjoint~matrix~of~A.~Enter~the~matrix|+G>FWFX?(F/FaoFaoF/FboC&-FS6%%DA~%dx%d~
matrix~expected.~Try~again|+G-F^bnFifl-F`bnFifl>FWFX>FenFW@$Fgn-F]o6#%.See~you~
laterGFghm@%3/F\^oFb[z/F^^oFc[z@%0-F\ao6$-Fb]o6#,&FWFaoFfelF[zFaoF_elC%-F^jm6#%
EYou~did~not~enter~the~right~matrix!!G-F^jm6$%enThe~correct~matrix~is~the~trans
pose~of~the~cofactor~matrixGFagn-F^jm6$%*That~is~:G/)-%'cofactGF^zFijmFg_s-F^jm
6$%<Good.The~adjoint~matrix~is:GFhjyC%-FS6%%NIn~this~case~Adjoint~of~A~is~a~%dx
%d~matrix.|+GFb[zFc[zFi\zF\]zFghm-FS6#%]oWhich~of~the~following~will~give~the~i
nverse~of~the~given~matrix?|+G-FS6#%J~~~~1.~The~inverse~of~the~adjoint~matrix|+
G-FS6#%ao~~~~2.~The~cofactor~matrix~times~1/det(A)~where~A~is~the~given~matrix|
+G-FS6#%`o~~~~3.~The~adjoint~matrix~times~1/det(A)~where~A~is~the~given~matrix|
+GFghm>FWFX>FenFW@$Fgn-F]o6#%&AdoisG@$0FWFgbn-F^jm6#%WThis~is~not~the~correct~r
oute.~The~correct~choice~is~3GFghm-FS6#%QThe~inverse~of~the~matrix~based~on~cho
ice~3~is~:G-F^jm6#/F`cw*&-Fd^oF^zF[z-Fb]o6#-&Fi^o6#%(adjointGF^zFaoF\\oFjcw-F^j
m6#%`oThis~is~the~end~of~the~Interactive~inverse~procedure.~You~may~choose~G-F^
jm6#%[oother~examples~to~enforce~your~understanding~of~these~algorithmsGF\ex@'F
^[w-F^jm6#%CThis~matrix~is~not~a~square~matrixGFg[w-F^jm6#%IThe~inverse~of~the~
matrix~does~not~existGC$-F^jm6#%>The~inverse~of~the~matrix~is~G-F^jm6#/F`cw-Fb]
o6#-F^_oF\goF/F_pF/%&trsumGRF/6/F][oFPFgguF^[oFjdoF_[oF[eo%&addABG%,trans_addAB
GFhdoFb_v%)add_tAtBGFd[oFdhmF/C:Fghm-F^jm6#%hnThe~purpose~of~this~program~is~to
~check~the~conjecture~if~theG-F^jm6#%entranspose~of~the~sum~is~the~sum~of~the~t
ranspose;~that~is,G-F^jm6#/),&FhxFaoFfcnFaoFijm,&FdeoFaoFh`vFaoF\\o@$F_\oF`\o@&
F[anC*-FS6#%jnEnter~the~first~matrix~|frfor~example,~A:=matrix([[1,2],[2,3]])|h
r|+G>FhxFX?(F/FaoFaoF/F_`nC$-FS6#%>Matrix~is~expected.Try~again|+G>FhxFXF`fo-FS
6#%[oEnter~the~second~matrix~|frfor~example,B:=matrix([[1,-1],[4,3]])|hr|+G>Ffc
nFX?(F/FaoFaoF/FcjvC$Fbcz>FfcnFX-F^jm6$F_]oFgfpFj\nC$>FhxFe]o>FfcnFibv-F^jm6$F\
cvFbfo-F^jm6$%9~The~second~matrix~is~:~GFgfp>F\]lF]bn>F`zF_bn>Fa^lFigo>FednF[ho
@$3/F\]lFa^l/F`zFednC.>F]_l-Fb]o6#Fgbz>F^jn-Fh^o6#F]_l>F_fnFifo>FagnF[go>Fdfn-F
b]o6#,&F_fnFaoFagnFaoF\\o@$F_\oF`\oFghm-F^jm6$%?The~transpose~of~their~sum~~=~G
/Ffbz-Fb]o6#F^jnF\\o@$F_\oF`\o-F^jm6$%=The~sum~of~their~transpose=~G/Fhbz-Fb]o6
#FdfnF\\o@$F_\oF`\oFghm-FS6#%jnIs~the~sum~of~the~transpose~equal~to~the~transpo
se~of~the~sum?|+GFa`o>FegnFX@&/FegnFcao-F^jm6%%9Yes~indeed,~The~propertyGFebz%>
~holds~for~these~two~matricesG/FegnFi`oC$-F^jm6%%7Incorrect,The~propertyGFebzFd
gz>FWFaoFghm-F^jm6#%`oThe~transpose~of~the~sum~is~always~equal~to~the~sum~of~th
e~transposesG-F^jm6#%GThis~is~the~end~of~the~trsum~procedureGF/F_pF/%(commuteGR
F/61F][oFPFgguFgjqF__vF^[oFjdo%#n1G%#p1GF_[oF[eoFgr%"pGF_[rFd[oFdhmF/C8-F^jm6#%
QThe~purpose~of~this~procedure~is~to~check~if~theG-F^jm6#%Qcommutative~property
~of~multiplication~holds~forG-F^jm6#%1any~two~matricesGF\\o@$F_\oF`\o@&F[anC*-F
S6#%inEnter~the~first~matrix~|frfor~example,A:=matrix([[1,2],[2,3]])|hr|+GFX>Fh
xF\bo-F^jm6#/%8The~first~matrix~is~:~AGFcfoFfczFX>FfcnF\bo-F^jm6#/%:~The~second
~matrix~is~:~BGFagoFj\nC'>FhxFe]o>FfcnFibvFhizFghm-F^jm6#/%8The~second~matrix~i
s:~BGFago>F_fnF]bn>F\]lF_bn>FagnFigo>Fa^lF[ho@$3/F\]lFagn/Fa^lF_fnC,>Fdfn-Feo6$
F_fnFa^l>Fegn-Feo6$FagnF\]l>Fdfn-F\dvFhdnF\\o@$F_\oF`\o-F^jm6$%IThe~product~AB~
of~the~two~matrices~is~:~G/Fa^vFhfzF\\o@$F_\oF`\o>Fegn-F\dv6$FfcnFhx-F^jm6$%ITh
e~product~BA~of~the~two~matrices~is~:~G/%#BAGFearF\\o@$F_\oF`\oFghm-FS6#%YDoes~
the~commutative~property~hold~for~these~matrices~?|+GF_co>FezFX@*3/FezFcao/-F\a
o6$-Fb]o6#,&FdfnFaoFegnF[zFaoF_el-F^jm6#%\oYes~indeed,~The~commutative~property
~holds~for~these~two~matricesG3F^][l0F`][lF_elC$-F^jm6#%coIncorrect,~The~commut
ative~property~does~not~hold~for~these~two~matricesG>FWFao3/FezFi`oF_][l-F^jm6#
%gnWrong,~the~commutative~property~holds~for~these~two~matricesG3F`^[lFi][lC$-F
^jm6#%]oYes,~the~commutative~property~does~not~hold~for~these~two~matricesG>FWF
ao@$Fa\wC*Fghm-FS6#%@Choose~from~the~following~menu|+G-FS6#%G~~1.~Show~me~two~c
ommutative~matrices|+G-FS6#%]o~~2.~Give~me~another~chance.~I~will~find~two~comm
utative~matrices|+G-FS6#%E~~3.~I~am~tired.~Get~me~out~of~here|+G-F^jm6#%;<choos
e~from~1;~2;~or~3;~>G>FezFX@(F^gv-%'ABeqBAGF//FezF\]n-FahzF//FezFgbn-F^jm6#%+By
e!~Bye!!GFghm-F^jm6#%UThis~is~the~end~of~the~commute~procedure.~Try~other~G-F^j
m6#%Xexamples~to~see~whether~any~two~matrices~always~commuteGF/F_pF/F/,
I/SMARTPLOTColorR6#'F'<$FebnFc]n6&Fchv%(LocListGF][nFgrF-F/C*>Fhx7#Fi_l@$/<#-Fb
\lF^z<#-Fb\l6#-F]gmF/>FhxFddl>F`z-F]zF[b[l@$4-F56#%5SMARTPLOTColor_IndexG>Fdb[l
F_el>FW&F\b[l6#,&Fdb[lFaoFaoFao?(FfcnFaoFaoF/3-%'memberG6$FWFhx1FfcnF`zC$>Fdb[l
-%$modG6$Fib[lF`z>FWFgb[l>Fdb[lFbc[lFWF/6$F]gmFdb[lF/6",
I.CM_InertFuncs<*%$IntG%$SumG%&DESolG%&RESolG%%DiffG%&LimitG%'NormalG%(ProductG
6",
IBcontext/cfparse/splitconditionalsR6#'%%CondGF`]l6*F,%'resultGFevF^uFau%,LocAn
cestorG%2ConditionalsTableG%2ConditionalsGroupG6#FbglFiyC)>6$8&8(-Fgw6$7$Q%Test
F[[lQ)Ancestor6"9$>F`z-%'removeG6%F:Fje[l%"=G>8*Fex?&8$F`zF\wC&-F26#-F:6$Fcf[lF
`[l>8)-Fgw6$Fhe[l&Fdx6#Fcf[l@$/Fjf[l%%FAILG>Fjf[lFce[l-F"6$Fcf[l&Faf[l6#F\]l>8%
Fbr?&Fcf[l7#-%(indicesGFi`lF\wC%>Fce[l-Fb\lF^g[l>8+-Ff]o6$-Fjel6#&Faf[l6#F_h[l%
4context/StackToListG@%/Fce[lFag[l>Fhg[l6$Fhg[l-F`]l6$/Fge[lFbe[l-Fb\l6#Fah[l>F
hg[l6$Fhg[l-F`]l6%F_i[l/Fhe[lFce[lF`i[l/Fje[lFhg[lFiyFiyFiyFiy,
I&pplotR6$F'F%6/%,COLORVECTORGF_[r%%maxxG%%minxG%%maxyG%%minyGF][nF\jo%%polyGFg
rF_[n%"gG%#g5GF/F/C2>FW-Fahp6#7*Fbjn%&GREENG%&BLACKG%%PINKG%)BURGANDYG%'YELLOWG
%'ORANGEG%&BROWNGF\cn>Fhx-Fahp6#,$-F]zFj_lF\]n>Ffcn!$***>F`zFden>FednFf[\l>F\]l
Fden?(Fa^lFaoFaoFd[\lF\wC&@$2Ffcn&&FKFi`lFihm>FfcnF^\\l@$2F^\\lF`z>F`zF^\\l@$2F
edn&F_\\lF\^n>FednFf\\l@$2Ff\\lF\]l>F\]lFf\\l>F^jnF_el>F_fn-FiarFO?(F]_lFaoFaoF
_fnF\w>F^jn,&F^jnFao*&-Fjel6#&F<FeezFao)FgvF_duFaoFao?(Fa^lFaoFaoFd[\lF\wC$>&Fh
x6#,&Fa^lF\]nF[zFaoF^\\l>&Fhx6#,$Fa^lF\]nFf\\l>Fagn-Ffin6&FhxFhin/F[jn""&/%%axe
sG%'NORMALG>Fdfn-%%plotG6(F^jn/Fgv;,&F`zFaoFdqFao,&FfcnFaoF\]nFao/%"yG;,&F\]lFa
o!#IFao,&FednFao"#IFao/Ffhn&FW6#,&-Fcc[l6$F^[n"")FaoFaoFaoFe^\l/F[jnFao>Fegn-Fj
gn6&7%F^[x,&F\]lFao!"$Fao%4original~points-RedG/Fahn<$Fchn%'CENTERGFhin/%%fontG
7%%&TIMESG%%BOLDGF]`\l-F^jm6#-Fdjn6$<%FegnFdfnFagn/%&titleG-%$catG6$%?Least~Squ
are~curves~of~degree~G-Ff]o6$F]hn%'stringGF/F/F/F/,
I-ToMatrixformRF/6+FP%%varsG%%eqnsGFe\rFgrF][nF^[n%"XGFgguFdhmF/C+>FfcnFa]n>Fhx
F[^n>FW-Feo6$-FiarF\go-FiarF^z?(F\]lFaoFaoF\^oF\w?(Fa^lFaoFaoF^^oF\w>&FWF^dn-F_
[t6$-Fc\t6#&FfcnFg]l&FhxFi`l>F]_l-Feo6$Fib\lFao?(F\]lFaoFao-F^bnFeezF\w>&F]_l6$
F\]lFao&FhxFg]l>F^jn-Feo6$Fhb\lFao?(F\]lFaoFao-F^bnFbfzF\w>&F^jnF[d\l-FgjsFac\l
7%FWF]_lF^jnF/F/F/F/,
I?context/cfparse/act_LINEARFUNCR6#'%%CodeGFiclF/6#FbglF/C$@$FjvF]w-FgilFj_lF/F
/F/6",
I)lsqrsmanRF/65F][nF^[nF\joFd[oFijqFe\rFgrFgv%%prodG%*rightsideG%%xsetG%&list2G
%%solnG%+rightside1GFPFajo%$trAG%&prod1GF][oFdhmF/Cen>FednFa]n>FdfnF[^n>F\]lFh[
v>Fa^lFj[v>Fez-F`_r6%Fedn;FaoF\]l;Fao,&Fa^lFaoF[zFao>Fbx-F`_r6%FednFcf\l;Fa^lFa
^l>F]_l-Feo6$Fef\lFao?(FWFaoFaoFef\lF\w>&F]_l6$FWFao&FdfnF^p-F^jm6#%^oThe~origi
nal~system~of~equations~is~given~by~the~matrix~system~Ax=bG-F^jm6#/*&-Fb]oFa[lF
aoF]_lFaoF^fv-F^jm6%%'where~G/FezFig\l/FbxF^fv@%/-Fi^r6#-Fiew6$FezFbx-Fi^rFa[l>
FfelFao>FfelF_elF\\o@$FgnF\o-F^jm6#%XThe~augmented~matrix~of~the~original~syste
m~is~given~byG-F^jmFbh\lFghm-F^jm6$%<and~its~row~echelon~form~isG-Fe^sFbh\lF\\o
@$FgnF\o-FS6#%WInspect~the~row~echelon~form~of~the~augmented~matrix.|+G-FS6#%8W
hat~do~you~conclude~?|+G-FS6#%E~~~~~~1.~Above~system~is~consistent|+G-FS6#%G~~~
~~~2.~Above~system~is~inconsistent|+G-FS6#%2<enter~1;~or~2;>|+G>F`zFX>FenF`z@$F
gnF\oFghm@&Fe_r@%/FfelFao-FS6#%XGood~the~system~is~consistent.~What~would~you~d
o~next?|+G-FS6#%jnIncorrect.~The~system~is~inconsistent.~What~would~you~do~next
?|+G/F`zF\]n@%/FfelF_el-FS6#%ZGood~the~system~is~inconsistent.~What~would~you~d
o~next?|+G-FS6#%hnIncorrect.~The~system~is~consistent.~What~would~you~do~next?|
+GF\_[l-FS6#%gn~~~~~~~1.~Form~the~system~transpose(A)*A*x~=~transpose(A)*b|+G-F
S6#%in~~~~~~~2.~Form~the~system~A*transpose(A)*x~=~A*transpose(A)*b|+G-FS6#%D~~
~~~~~3.~Form~the~system~(Ax-b)=0|+G-FS6#%8<enter~1;~or~2;~or~3;>|+G>F`zFX>FenF`
z@$FgnC$F]jmF_jm@%Fe_rC4Fghm>F\fn-F[_oFa[l-F^jm6#%jnCorrect,~the~transpose(A)*A
~is~obtained~using~the~Maple~commandG-FS6#%Aprod:=multiply(transpose(A),A);|+G>
F^jn-Fb]o6#-Fj\s6$F\fnFez-F^jmFbfz>F_fn-Fb]o6#-Fgdn6$Fi\]lFbxF\\o@$Fgn-F]o6#Fin
-FS6#%hnAlso~the~(transpose(A)*b~is~obtained~using~the~Maple~command|+G-FS6#%G>
rightside:=multiply(transpose(A),b);|+G-F^jm6#/*&-Fb]o6#Fi\]lFaoF^fvFaoF\^s-F^j
m6$%(That~isG/*&)FezF,FaoFbxFaoF\^sF\\o@$FgnC$F]jmF_jm-F^jm6$%<Now~write~the~ne
w~system~asG/*(F__]lFaoFezFaoF]_lFaoF^_]l-F^jm6#/*&FafzFao-Fb]oFeezFaoF\^sFghmC
/-F^jm6#%JIncorrect~choice.~Now~we~do~the~followingGFghm-F^jm6#%gnMultiply~the~
above~system~Ax=b~(from~left)~by~transpose~of~AG-F^jm6$%)That~is,G/*(Fh^]lFaoFi
g\lFaoF[`]lFaoFg^]lF\\o@$FgnC$F]jmF_jm>F^jn-Fb]o6#-Fgdn6$Fi\]lFez>F_fnFg]]l>Fed
n-Fiew6$F^jnF_fnFghm-F^jm6$%HAfter~simplifying~we~get~the~new~systemGFe_]lFg_]l
FghmFghm-FS6#%YWhat~can~you~conclude~about~the~new~coefficient~matrix?|+GF\_[l-
FS6#%:~~~~~~~~1.~It~is~square.|+G-FS6#%=~~~~~~~~2.~It~is~symmetric.|+G-FS6#%>~~
~~~~~~3.~It~is~invertible.|+G-FS6#%R~~~~~~~~4.~It~is~square,~symmetric~and~sing
ular.|+G-FS6#%S~~~~~~~~5.~It~is~square,~symmetric~and~invertible|+G-FS6#%D<ente
r~1;~or~2;~or~3;~or~4;~or~5;>|+G>F`zFX>FenF`z@$FgnC$F]jmF_jm@'/F`zFd^\l@%0-Fd^o
FbfzF_el-F^jm6#%Xcorrect,~the~matrix~is~square,~symmetric~and~invertibleG-F^jm6
#%jnIncorrect,~the~matrix~is~square~and~symmetric~but~its~singular.G/F`zFhjp@%F
bc]l-F^jm6#%^oIncorrect,~the~matrix~is~square~and~symmetric~and~also~non-singul
arG-F^jm6#%incorrect,~the~matrix~is~square~and~symmetric~but~non-invertibleGC$-
F^jm6#%?Incomplete~or~incorrect~choiceG@%Fbc]l-F^jm6#%OThe~matrix~is~square,~sy
mmetric~and~invertibleG-F^jm6#%VThe~matrix~is~square~and~symmetric~but~non-inve
rtibleG>Fedn-Fe^sFbfz@%Fbc]lC,Fghm-FS6#%9What~is~the~next~step~?|+G-FS6#%]o~~~~
~~1.~Find~the~inverse~of~the~coefficient~matrix~and~multiply.|+G-FS6#%_o~~~~~~2
.~Find~the~transpose~of~the~coefficient~matrix~and~multiply.|+G-FS6#%gn~~~~~~3.
~Perform~Gauss~Elimination~on~the~augmented~matrix.|+G-FS6#%9<enter~1;~or~2;~or
~3;>~|+G>F`zFX>FenF`z@$FgnC$F]jmF_jm@'Fe_rC'-FS6#%<Correct,~the~Maple~command|+
G-FS6#%en>evalm(x):=~multiply(inverse(transpose(A)&*A),rightside);|+G-FS6#%Ryie
lds~the~solution~to~the~least~squares~problem|+GFghm-F^jm6#/F[`]l-Fb]o6#-Fj\s6$
-F^_oFbfzF_fn/F`zFgbnC&-FS6#%<Correct,~The~Maple~command|+G-FS6#%Z>backsub(gaus
selim(augment(transpose(A)&*A,rightside)));|+GF\g]l-F^jm6#/F[`]l-Ff]o6$-&Fi^o6#
%(backsubG6#-Fe^s6#Faa]lFeoC&-F^jm6#%[oIncorrect~choice.~The~solution~may~be~fo
und~using~inverse~of~theG-F^jm6#%jncoefficient~matrix~or~using~Gauss~eliminatio
n.~The~least~squareG-F^jm6#%aosolution~using~gauss~elimination~and~back~substit
ution~algorithm~is~:~GF_h]lC5FghmF\_[l-FS6#%jn~~~1.~Find~the~inverse~of~the~coe
fficient~matrix~and~multiply.|+G-FS6#%\o~~~2.~Find~the~transpose~of~the~coeffic
ient~matrix~and~multiply.|+G-FS6#%Z~~~3.~Perform~Gauss~Elimination~on~the~augme
nted~matrix.|+GF]f]l>F`zFX>FenF`z@$FgnC$F]jmF_jm@&5Fe_rF][]lC%-F^jm6$%`oIncorre
ct~choice.~Apply~the~Gauss~elimination~to~the~augmented~matrixGF^jn-F^jm6#%YThe
~reduced~row~echelon~form~of~the~augmented~matrix~is:G-F^jm6#-Fb]oFjh]lFgg]lC)-
F^jm6#%enCorrect.~Find~the~row~echelon~form~of~the~augmented~matrixG-F^jm6#%Yus
ing~the~following~Maple~command~>AUG:=gausselim(prod);G-F^jm6#%MThe~row~echelon
~form~of~the~augmented~matrixGF\[^l-F^jm6$%,is~given~byG-Fe^sF][^lF\\o@$FgnC$F]
jmF_jmF\_[l-FS6#%jn~~~1.~Find~the~solution~using~inverse~of~the~row~echelon~for
m.|+G-FS6#%hn~~~2.~Find~the~transpose~of~the~reduced~matrix~and~multiply.|+G-FS
6#%M~~~3.~Apply~the~backsubstitution~algorithm.|+G-FS6#%7enter~1;~or~2;~or~3;~|
+G>F`zFX>FenF`z@$FgnC$F]jmF_jm@%Fdj]lC&-F^jm6#%doIncorrect~choice.~Find~the~sol
ution~using~the~back~substitution~algorithmG>FegnFdh]l>Fegn-Ff]oFaax-F^jm6$%?Th
e~least~square~solution~is~:G/F[`]lFearC)-F^jm6#%^oGood.~Use~the~following~Mapl
e~command~to~apply~the~backsubstitutionG-F^jm6#%5>soln:=backsub(AUG);G>FegnFdh]
l>FegnFf]^lFg]^lF\\o@$FgnC$F]jmF_jmFghm-F^jm6#%`oThis~is~the~end~of~the~interac
tive~version~of~the~least~square~methodG-F^jm6#%7Try~your~own~examples.GF/F_pF/
F/,
I0type/CF_SUBMENURFbpF/F-F/3-F:6$FK-%)specfuncG6$%)anythingGFdal/-Fjq6$FKF\yF]r
F/F/F/6",
I)rrefautoRF/6?Fd[oFPFajoF][nF^[nF\joF][oFijqFgrF\\rFe\rFghvFchmFgjqFhjqFa[rF]\
rFj[rFc[rFd[rFe[rFb[rFi[rF[\rFg[rF[[rFh[rF^\r%%prevGFdhmF/C<>F]_lFa]n>F]_l-Ff]o
6$F]_lFeo-F^jm6$%9The~original~matrix~is~:GF]_l>F^jnFhc\l>Fagn-F`bnFeez>Fa^l-%)
chkrowecGFeez@$/Fa^lFaoC)-F^jm6%%;Note~that~the~given~matrixGF]_l%<~is~not~in~r
ow~echelon~formG-FS6#%C~~~~~Choose~one~of~the~following:|+G-FS6#%[q~~~~~1.~if~y
ou~want~to~observe~the~steps~to~display~the~row~echelon~form;~~type:~>rowec;~~o
r~1;|+G-FS6#%[q~~~~~2.~If~you~do~not~want~to~observe~the~steps~of~the~row~echel
on~form;~~~type:~>bypass;~or~2;|+G>FWFX?(F/FaoFaoF/4555/FWFcgs/FW%'bypassGFa\wF
i]wC&>FenFW@$FffsF`\o-FS6#%8Type~bypass;~or~rowec;|+G>FWFX@&55Fab^l/FWFegsFa\wC
'>%(rreflagGFao-%*gaussautoGFeez>Fac^lF_el>F]_l-Fe^sFeez?(F`zFaoFaoFhc\lF\wC%>F
ednFao?(F/FaoFaoF/3/&F]_lF^dvF_el2FednFg`^l>FednFger@$0F]d^lF_el>F]_l-Fdgr6%F]_
lF`z*$F]d^lF[z55/FW%'BYPASSGFbb^lFi]wC%>F]_lFfc^l>F^jn-Fi^rFeez?(F`zFaoFaoFhc\l
F\wC%>FednFao?(F/FaoFaoF/F[d^l>FednFger@$Fad^l>F]_lFcd^l>F`zFhc\l?(F/FaoFaoF/3-
%'iszeroG6#-F_bt6$F]_lF`z2FaoF`z>F`zF^[x>FednFao?(F/FaoFaoF/F[d^l>FednFger@$3Fa
d^l0F]d^lFao>F]_lFcd^lFghm-F^jm6#%DThe~matrix~in~ROW-ECHELON~form~is~:G-F^jmFee
z-F^jm6#%YNow~let~us~put~this~matrix~into~Reduced~Row~echelon~FormGF\\o@$FffsF^
ir>FjfqFao>F^jnFg`^l>FagnFhc\l?(FednFaoFaoF^jnF\wC%>F\]lFhc\l?(F/FaoFaoF/3/&F]_
lFc\qF_el2FaoF\]l>F\]lFcft@%/F\]lFaoF/@$2FjfqF\]lC$>FjfqF\]l?(F`zFcftF[zFaoF\wC
%-F^jm6$F^jwFedn-F^jm6#%:*************************G@$0Fgg^lF_elC(-F^jm6*FgjwF\]
lFc[sF]_lFagr,$F]d^lF[zFjirF`z>F]_l-F]jr6&F]_lF\]lF`zF\i^lFghm-F^jm6$FedtF]_lF\
\o@$FffsFgfs-F^jm6$%LThe~matrix~in~reduced~row~echelon~form~is~:GF]_lFghm-F^jm6
#%jnThis~is~the~end~of~demonstration~version~of~the~reduced~echelonG-F^jm6#%[of
orm~procedure.~You~may~try~other~examples~to~learn~the~process.GF/6$FenFac^lF/F
/,
I*obasismanRF/6;F][nF^[nFijqFfhv%&AUG11G%&AUG12G%%AUG2GFe\rFgrFhjoF\jo%"wGFgjo%
%sum1G%#eqG%%sum2GFd[o%"oGFbj[l%+setofvectsGF_[rFjhv%'temp11GF`\rF][oFdhmF/Cfq>
FfcnFa]n>F]_lF[^n>F^jn&Fb]nFg]qFghm?(FWFaoFaoF]_lF\w>&F_fnF^p-F_bt6$FfcnFW-FS6#
%]o~What~is~the~very~first~thing~you~check~for~in~the~given~vectors?|+G-FS6#%X~
~1.~Determine~if~the~vectors~are~linearly~independent|+G-FS6#%T~~2.~Determine~t
he~subspace~spanned~by~the~vectors|+G-FS6#%R~~3.~Determine~the~inner~products~o
f~the~vectors|+G-F^jm6#%;<choose~from~1;~2;~or~3;>|+GFghm>F\fnFX>FenF\fn@$Fgn-F
]o6#Fadr?(F/FaoFaoF/0F\fnFaoC&-F^jm6#%jnYou~need~to~go~over~the~demonstration~m
ode.~Try~another~Choice!G>F\fnFX>FenF\fn@$FgnFf\_l-FS6%%inGood.~Enter~the~matri
x~%dx%d~whose~rows~are~the~given~vectors|+GF]_lF^jn>Fh[l-FahpFeez>&Fh[lFihm/&F_
fnFihm-Ff]o6$-Fb]o6#Fi]_lFeo?(FWF\]nFaoF]_lF\w>&Fh[lF^p/Fa[_l-Ff]o6$-Fb]o6#Fa[_
lFeo-F^jm6#Fh[l-F^jm6#%PFor~example~type:~AUG~:=~matrix([[1,2],[4,3]]);GFghm>F\
]lFX>FenF\]l@$FgnFf\_l?(F/FaoFaoF/4-F:6$F\]lFeoC&F]^r>F\]lFX>FenF\]l@$FgnFf\_l>
Fedn-Feo6$F^jnF]_l?(FWFaoFaoF]_lF\w?(FagnFaoFaoF^jnF\w>&Fedn6$FagnFW&Fa[_lFj^r>
Fedn-F[_oFa_o@%0-F\ao6$-Fb]o6#,&F\]lFaoFednF[zFaoF_elC$-F^jm6#%FYou~did~not~ent
er~the~correct~matrix.G-F^jm6$%8The~correct~matrix~is~:GFebvC%-F^jm6$%ACorrect.
~The~augmented~matrix~isGFebvFghm-FS6#%fnHow~would~you~check~to~see~if~the~vect
ors~are~independent?|+G-FS6#%NChoose~the~next~step~from~the~following~menu|+G-F
S6#%I~~~1.~Find~the~LU-decomposition~of~AUG.|+G-FS6#%I~~~2.~Find~the~QR-decompo
sition~of~AUG.|+G-FS6#%Q~~~3.~Find~the~reduced~row~echelon~form~of~AUG.|+GF`\_l
>F\fnFX>FenF\fn@$FgnFf\_l?(F/FaoFaoF/0F\fnFgbnC&-FS6#%NYou~did~not~make~the~rig
ht~choice.~Try~again|+G>F\fnFX>FenF\fn@$FgnFf\_lFghm-FS6#%hoVery~good.~Which~Ma
ple~command~helps~you~find~the~reduced~~row~echelon~form?|+G-FS6#%in~~~~~~1.~de
t(AUG);|+~~~~~~2.~rref(AUG);|+~~~~~~3.~inverse(AUG);|+GFghmF`\_l>F\fnFX>FenF\fn
@$FgnFf\_l?(F/FaoFaoF/0F\fnF\]nC&-FS6#%UYou~did~not~make~the~appropriate~choice
.~Try~Again.|+G>F\fnFX>FenF\fn@$FgnFf\_l>F`z-Fc`yF\go-F^jm6$%UGood.~The~reduced
~row~echelon~form~of~the~matrix~is:GF`zFghm-FS6#%OBased~on~this~reduced~matrix,
~are~the~vectors|+G-FS6#%4~~~~~1.~dependent?|+G-FS6#%6~~~~~2.~independent?|+G-F
S6#%9<choose~from~1;~or~2;>~|+G>F\fnFX>FenF\fn@$FgnFf\_l?(FWFaoFao-Fi^rFf\lF\w>
Fag\l-F_bt6$F`zFW@&/F\fnFao@%2Fee_lF]_lC$-F^jm6#%(CorrectG>FgapFaoC$-F^jm6#%*In
correctG>FgapF_el/F\fnF\]n@%/Fee_lF]_lC$F^f_l>FgapF_elC$Fcf_l>FgapFao@&FbctC+-F
^jm6#%ioIn~this~case~the~set~of~independent~vectors~extracted~from~the~original
~set~isG>FaalFe]_l?(FWFaoFaoF]_lF\w>&FaalF^pFd^_l>F\bl-%*extractldGFjax>F^y-Fah
p6#-FiarF`cl?(FWFaoFaoF]h_lF\w>&F^yF^p/Fa[_l-Ff]o6$-Fb]o6#&F\blF^pFeo-F^jm6#F^y
Fghm?(FWFaoFaoFee_lF\w>Fa[_lFdh_l/FgapF_elC(-F^jm6#%\oIn~this~case~the~original
~set~of~vectors~is~linearly~independent.G-F^jm6#%?The~original~set~is~given~by~
:G>F^yFe]_l?(FWFaoFaoF]_lF\w>F`h_l/Fa[_lFd^_lFgh_lFghm-FS6#%HLet~us~construct~t
he~orthonormal~basis|+GF\\o@$FgnF`\o-FS6#%^pWe~first~construct~orthogonal~vecto
rs~from~the~extract~vectors~using~the~procedure|+G-FS6#%4~~o[1]:=evalm(v1);|+G-
FS6#%]o~~o[2]:=evalm(v2-(innerprod(v2,o[1])/innerprod(o[1],o[1]))*o[1]);|+G-FS6
#%[o~~o[3]:=evalm(v3-(innerprod(v3,o[2])/innerprod(o[2],o[2]))*o[2]|+G-FS6#%gn~
~~~~~~~~~-(innerprod(v3,o[1])/innerprod(o[1],o[1]))*o[1]);|+G-FS6#%/and~so~on~.
..|+G-FS6#%Oo[1],o[2],o[3],...~are~the~orthogonal~vectors|+GFghm>&FegnFihmFi]_l
>Fhgn-Fahp6$F^jnF_el?(FWF\]nFaoFee_lF\wC,>FhgnFc[`l>Fbx-Feo6%F^jnFee_lF_el/FhxF
ao>-Faen6$FbxF\]n-Fb]o6#*(-Fh]v6$Fa[_lFa[`lFao-Fh]v6$Fa[`lFa[`lF[z-Fb]o6#Fa[`lF
ao>Fhgn-Fb]o6#*(Fb\`lFaoFd\`lF[zFa[`lFao?(FhxF\]nFao,&FWFaoF[zFaoF\wC$>-Faen6$F
bx,&FhxFaoFaoFao,&-Faen6$FbxFhxFao*(-Fh]v6$Fa[_l&FegnF^zFao-Fh]v6$Fi]`lFi]`lF[z
-Fb]o6#Fi]`lFaoFao>Fhgn-Fb]o6#,&FhgnFao*(Fg]`lFaoFj]`lF[zFi]`lFaoFao/-Fb]o6#&Fe
gnF^p,&Fd^_lFao-Faen6$FbxFWF[z>&FezF^p/Fd^`l,&Fd^_lFaoFfevF[z>Ff^`l-Fb]o6#,&Fa[
_lFaoFhgnF[z/&Fegn6#Fa]oFd^`lFghm?(FWFaoFaoFee_lF\wC,-F^jm6#%RWe~construct~the~
orthogonal~set~using~the~vectorsGFgh_l-FS6$%KConstruct~the~orthogonal~vector~nu
mber~%d|+GFW@$Fa\w-F^jm6#%KFor~example~in~R3:~>o[1]:=vector([1,2,0]);G>Faal-Fah
pFbfz>FaalFX?(F/FaoFaoF/4-F:6$FaalFahpC&-F^jm6#%5A~vector~is~expectedG>FaalFX>F
enFaal@$FgnF`\o>FenFaal@$Fgn-F]o6#%%EXitG@%4-Fie^l6#-Fb]o6#,&FaalFaoFf^`lF[zC$-
F^jm6#%WThis~is~not~the~correct~vector.~The~correct~vector~is:G-F^jm6#/&FjbrF^p
Fd^`lC$-F^jm6#%;This~is~the~correct~vectorGF_b`lFghm?(FWFaoFaoFee_lF\w>&FhbsF^p
-Fb]o6#*&-%%sqrtG6#-Fh]v6$Ff^`lFf^`lF[zFf^`lFaoF\\o@$FgnF`\o-FS6#%RYou~have~con
structed~an~orthogonal~set~of~vectorsG>Fh[l-Fahp6#Fee_l?(FWFaoFaoFee_lF\w>F`^_l
Fab`lFf^_lFghm-FS6#%\pWhat~do~you~do~to~make~this~set~an~orthonormal~basis?~Cho
ose~from~the~following~|+G-FS6#%I~~~1.~Multiply~each~vector~by~its~norm.|+G-FS6
#%G~~~2.~Divide~each~vector~by~its~norm.|+G-FS6#%J~~~3.~Any~orthogonal~set~is~o
rthonormal.|+GF`\_l>F\fnFX>FenF\fn@$FgnF\^]l@$Fhc_lC$-F^jm6#%?Incorrect~or~inco
mplete~choiceG-F^jm6#%8The~correct~choice~is~2GFghm-F^jm6#%:Now~normalize~each~
vectorGF\\o@$FgnF`\o?(FWFaoFaoFee_lF\wC'@$Fa\wC%-F^jm6#%RA~vector~is~normalized
~by~dividing~it~by~its~normG-F^jm6#%Ri.e.~~normal(v)=~evalm(v~/sqrt(innerprod(v
,v)))~;GFghm@$/-Fcc[l6$FWF\]nF_elC$F\\o@$FgnF`\o-F^jm6&%7The~norm~of~the~vector
GFab`lF[\^lF]c`l-F^jm6$%Land~the~corresponding~orthonormal~vector~isG-Fb]o6#Fib
`lFghmF\\o@$FgnF`\o-F^jm6#%UTherefore~the~orthonormal~set~of~vectors~is~given~b
yG?(FWFaoFaoFee_lF\w-F^jm6#/&%"OGF^p-F[_o6#-Ff]o6$F\g`lFeo-F^jm6#%eoThis~is~the
~end~of~the~interactive~GramSchmidt~procedure.~Change~the~givenG-F^jm6#%joset~o
f~~vectors~and~execute~the~procedure~again~to~learn~the~steps~involved.~~~GF/F_
pF/F/,
I<convert/context/ListToStackR6#'F'Febn6#%#STGF[e\lF/C&F1>FW-&F*6#FDF/-F_r6%&F*
6#FIFKFW-FjelF^pF/F/F/6",
I-obasisnostepRF/6+F][nF^[nFijqFe\rFgrFhjoFgjo%"GG%#G1GFdhmF/C+>FfcnFa]n>F`zF[^
n>FednF^[_l-F^jm6#%WAn~orthonormal~basis~is~given~by~the~following~vectrosGFghm
?(FWFaoFaoF`zF\w>&F\]lF^pFb[_l>F]_l-&Fi^o6#%,GramSchmidtG6#7#-F[\l6$-Fb]o6#Faj`
l/FW;FaoF`z>F^jn-Ff]o6$F]_lFebn?(FWFaoFao-F]zFbfzF\w-F^jm6#/Ffg`l-Fb]o6#*&-F^c`
l6#-Fh]v6$&F^jnF^pF^\alF[zF^\alFaoF/F/F/F/,
I:context/cfparse/joinblockR6#'%'blocksG-Febn6#%+CM_IFBLOCKG6'F[v%'KeySetGFgv%(
NewCodeG%-continuationG6#F.6"C)-Ffp6#.F`]m>8(Fcjm>8'Q!6">8%<"?&8&9$F\wC&>8$-Fgw
6$Q$KeyF[[lF[^al@%-F]c[lF^c[l-Fh`n6#-Ffy6$Q1Repeated~Key:~%sF\]alF_^al>Fh]al-F]
en6$Fh]al<#F_^al>Fd]al-F`]m6%Q"~Ff]alFd]al-%<context/cfparse/ifblockcodeG6$F[^a
lFb]al>Fb]alF\w@$/Fb]alF\w>Fd]al-F`]m6%Fa_alFd]alQ$fi;6"-Ficl6#F`zF\]alF\]alF\]
alF\]al,
I(qrdgramRF/64FP%"QG%"RGFhjoFgjoFcj^lF][nF^[nF\joFe\rFgrF_[r%&tempRG%)tempvectG
Fdj^l%%sum0GFd[oF_hvFdhmF/Cco-F^jm6#%goThe~purpose~of~this~procedure~is~to~cons
truct~the~QR~decomposition~of~an~mxnG-F^jm6#%bomatrix~using~the~GramSchmidt(GS)
~process.~The~columns~of~the~orthogonalG-F^jm6#%comatrix~Q~can~be~obtained~by~p
erforming~(GS)~process~on~the~columns~of~A.GF\\o@$FgnF`\o>FWFa]n>F_fnF\^o>FagnF
^^o>Fhx-Feo6%F_fnFagnF_el>Ffcn-Feo6$FagnFagn>Fegn-Feo6%FagnFagnF_el?(Fa^lFaoFao
FagnF\w?(F]_lFa^lFaoFagnF\w>&FegnFh^wF^\v?(Fa^lFaoFaoFagnF\w?(F]_lFa^lFaoFagnF\
w>Fd`wF^\v>F`z-Fahp6$FagnF_el>FednFfbal>F\]l-Fahp6$F_fnF_el?(Fa^lFaoFaoFagnF\w>
&F`zFi`l-Faen6$FWFa^l>Fhgn-FahpFj^r?(Fa^lFaoFaoFagnF\w>&FhgnFi`l/F^cal-Ff]o6$-F
b]o6#F^calFeo-F^jm6#%?First~extract~the~columns~of~AG-F^jmFgevF\\o@$FgnF`\o-F^j
m6#%gnPerform~the~GramSchmidt~process~on~the~above~set~of~vectors.G-F^jm6#%jnTh
e~matrix~R~is~a~consequence~of~this~orthogonalization~processG>&FegnFghp-F^c`l6
#-Fh]v6$&F`zFihmF\eal>&FednFihm-Fb]o6#*&FgdalF[zF\ealFao-F^jm6#%PThe~(1,1)~entr
y~of~upper~triangular~matrix~R~isG-F^jm6#%Nthe~length~of~the~first~column~of~A.
~That~is~GFghm-F^jm6#/&FfcnFghp-F^c`l6#-%-innerproductGF[ealF\\o@$FgnF`\o-F^jm6
#%fnThe~first~orthonormal~vector~is~obtained~by~normalizing~theG-F^jm6#%enfirst
~column~of~A.~That~is~u[1]=col(A,1)/length~(col(A,1))GFghm-F^jm6#%BThe~first~or
thonormal~vector~is~:G-F^jm6#/F^eal-Ff]o6$-Fb]o6#F^ealFeoFghm?(FjbrFaoFaoF_bnF\
w>&Fhx6$FjbrFao&F^eal6#Fjbr-F^jm6#Fbfo-F^jm6#/FfcnFearFghmF\\o@$FgnF`\o-F^jm6#%
\oThe~other~entries~of~the~matrix~R~are~obtained~using~the~formulaeG>Fa^l.Fa^l>
F]_l.F]_l-F^jm6#/&FfcnFh^w-F_fal6$&F`zFeez&FednFi`l-F^jm6#%bowhere~v[j]'s~are~t
he~columns~of~A~and~u[i]'s~are~the~orthonormal~set~ofG-F^jm6#%dovectors~obtaine
d~from~v[j]'s~using~the~GramSchmidt~process~(for~i<j)~and~GFghm-F^jm6#/&Ffcn6$F
a^lFa^l-F^c`l6#-F_falFaft-F^jm6#/F\]l,&F^calFao-F\d[l6$*&&FednFeezFao-F_fal6$F^
calFajalFao/F]_lFdf\lF[zF\\o@$FgnF`\o?(Fa^lF\]nFaoFagnF\wC?-FS6$%VLet~us~find~t
he~entries~in~the~column~%d~of~R~using~|+GFa^l-FS6#%:the~following~formula(s)|+
GFghm>FbxFjbal>FezF^fv?(F]_lFaoFaoFef\lF\wC'-F^jm6#/&Ffcn6$F]_lFa^lFbjal-F^jm6#
%,This~yieldsG-F^jm6#/Fe[bl-F_fal6$Fical-Fb]o6#Fajal>Fbx-Fb]o6#,&FezFao*&-Fh]vF
cjalFaoFajalFaoFao>FezF^fvF\\o@$FgnF`\o>F]_lFchal>F\]l-Fb]o6#,&F^calFaoFbxF[z-F
^jm6#%?Since~the~vector~w~is~given~byGFjial-F^jm6#%2which~is~equal~toG-F^jm6#/F
\]l-Fb]oFg]l-F^jm6#%'we~getG-F^jm6#/Feial-F_fal6$Fg]blFg]blFghm>&FegnFfial-F^c`
l6#-Fh]vFaft?(F]_lFaoFaoFef\lF\w>&FegnFf[blFf\bl>F[ial-Fb]o6#*&Fa^blF[zF\]lFao?
(FjbrFaoFaoF_bnF\w>&Fhx6$FjbrFa^l&F[ialFfgalF\\o@$FgnF`\o-F^jm6$%?The~next~orth
onormal~vector~isG/F[ial-Ff]o6$-Fb]o6#F[ialFeoFghm-F^jm6#%Eand~the~updated~matr
ices~Q~and~R~areG-F^jm6%Fbfo%#~~GF[halF\\o@$FgnF`\o>Fa^lFahal-F^jm6#%QThe~matri
x~Q~is~formed~by~using~the~orthonormal~G-F^jm6#%Qvectors~u[1],u[2],..~u[n]~~~~a
s~the~columns~of~QG?(Fa^lFaoFaoF_fnF\w?(F]_lFaoFaoFagnF\w>Fd`w&FajalFi`l-F^jmF^
zF\\o@$FgnF`\o-F^jm6$%0The~matrix~R~isGF[halF\\o@$FgnF`\o-F^jm6$%FThe~QR-decomp
osition~of~A~is~given~byG/FW%#QRG-F^jm6$F^]z/Fa]o*&FcfoFaoFearFaoFghm-F^jm6#%^o
This~is~the~end~of~the~QR-Decomposition~process.~Try~more~examples!GF/F_pF/F/,
I0context/fullkeyR6$'%%RootG<$F)F[b\l'%(CurrentGF[b\lF/F[e\lF/C$@$5-F:6$FKF)/FK
Fe]al-F]o6#F<-Ffa\l6%FKQ"_6"F<F/F/F/Facbl,
I(trsumegRF/6&FPFgguF][nF^[nFdhmF/C2>FW-Feo6$FgbnFgbn>FhxFgcbl-F^jm6$%%Let~GF`]
oFghm-F^jm6$F[hwFbfoF\\o@$F_\oF`\o-F^jm6$F\dbl/Fchv,&FWFaoFhxFao-F^jm6$%%ThenG/
&FchvF]\v,&&FWF]\vFao&FhxF]\vFaoF\\o@$F_\oF`\o-F^jm6$%;Now~the~(i,j)th~element~
ofG/)FchvFijm%5(j,i)th~element~of~CG-F^jm6$%#SoG/)FcdblFijm,&&FW6$F`zFfcnFao&Fh
xFjeblFao-F^jm6$%<and~also~(i,j)th~element~ofG/,&)FWFijmFaoFdeoFaoFheblFghm-F^j
m6$%*ThereforeG/FgeblF`fblF/F_pF/F/,
I/type/CM_MatrixRFbpF/F[e\lF/-%&evalbG6#0-F]en6$-%'selectG6%F:-Fjq6$FK%&arrayG<
$FeoFahp-Fjq6$FK-Fb_^l6$Fd_^l<&Feo%'MATRIXGFahp%'VECTORGF]rF/F/F/6",
I3help/text/QRdecomp-F]t6'%+HELP~FOR:~G%gnA~QR~factorization~of~a~matrix~A~is~d
efined~as~A~=~QR~~whereG%hnQ~is~an~orthogonal~matrix~and~R~is~an~upper~triangul
ar~matrixG%TNote~:~A~matrix~Q~is~orthogonal~if~transpose(Q)*Q=IGF_tF/,
I<context/cfparse/conditionalR6#'F[vF[_l6%Fjd\l%,ConditionalGFevF-F/C--Ffp6#.Fg
w-Ffp6#.FiclF^]al-%(raadlibGF\]m-F26#-F56#&FdxFj_l>FhxFgibl>Ffcn-F[ibl6$Fge[lFh
x-F26#0FfcnFe]al>FW-Ffy6%Qenif~not~(~%s~)~then~stack[push](~%a~=~NULL,~Exclusio
ns)~fi;6"-F`]m6$Fa_alFfcnFK@$4-Fg\mF^p-Fh`n6$%>unable~to~parse~condition~forGFh
x-FiclF^pF/F/F/Fcjbl,
I4help/text/gausselim-F]t6DF`hblF_`bl%`o-~linsys[gausselim]~:~the~purpose~of~th
is~function~is~to~To~reduce~a~G%cq~a~matrix~OR~an~augmented~matrix~representing
~a~system~of~linear~equations~Ax=b~to~its~row~echelon~form.G%fn~Gauss~eliminati
on~is~performed~on~the~rows~of~the~matrix~AG%N~or~on~the~rows~of~the~augmented~
matrix~[A,b]G%en~There~are~THREE~versions~of~the~gauss~elimination~programGF_`b
l%]p~VERSION~1~:~~GAUSSAUTO~:~demonstrates~a~step~by~step~process~of~gauss~elim
inationG%[o~~~~~~~~~~~~~~using~the~row~operations~selected~by~the~machine.~GF_t
%K~~example~1~:~~A~:=~matrix([[1,2],[3,3]]);G%=~~~~~~~~~~~~~~~gaussauto(A);G%R~
~example~2~:~~A~:=~[[1,2,2],[2,2,2]];~b:=[3,43];G%?~~~~~~~~~~~~~~~gaussauto(A,b
);GF_t%T~~Note~that~the~matrices~can~be~entered~in~any~formG%Y~~as~~A~:=~[[2,3]
,[3,4]]~~OR~~A:=~matrix([[2,3],[3,4]]);GF_t%\o~VERSION~2~:~GAUSSMAN~:~This~proc
edure~performs~gauss~eliminationG%V~~~~~~~~~~~~~using~the~row~operations~provid
ed~by~youGF_t%J~~example~:~~A~:=~matrix([[1,2],[3,43]]);G%:~~~~~~~~~~~~~gaussma
n(A);GF_t%co~VERSION~3~:~gausselim~:~This~is~the~general~procedure.~It~allows~y
ou~toG%_o~~~~~~~~~~~~~select~the~row~operations~first.~However~you~can~switchG%
]o~~~~~~~~~~~~~to~auto~mode~any~time~and~observe~the~machine~performG%=~~~~~~~~
~~~~~row~operations.GF_t%I~~example~:~~A~:=~matrix([[1,2],[3,3]]);G%;~~~~~~~~~~
~~~gausselim(A);GF_tF_`blF/,
I;context/cfparse/recognizerR6$'FjtF\al'%%DATAGFfx6'FduF[vFiuFjd\l%'IfCodeGF[e\
lF/C0Few-Ffp6#F_il-Ffp6#Fg\m-Ffp6#F`]m-Ffp6#Ficl>Fhx-Fgw6$Fb^alFK>Ffcn-F_ilFj_l
>Fedn-F]f[l6%F:FfcnF_f[l-F26#/-F]zFa_oFao-F26#-Fg\m6#-Ficl6#-Fb\lFa_o>F`z-Ficl6
AQ(proc(f)6"QWlocal~RememberFlag,Exclusions,Vars,Nvars,Funcs,NFuncs;F__clQ`pglo
bal~`context/CM_GLOBALG`,~ContextData,~`context/Initialized`,~`context/InitProc
s`,F__clQB~~~~~_EnvContextData,~ContextKey;F__clQ/option~system;F__clQ9"Context
Menu~Procedure";F__clQ`oif~`context/Initialized`~<>~true~then~readlib('`context
/init`')()~fi:F__cl-Ffy6$Q2ContextKey~:=~%a;F__clFhx-Ffy6$Q7_EnvContextData~:=~
%a;F__clQ%DATAF__clQFContextData~:=~eval(_EnvContextData);F__cl-Ffy6$QHif~`cont
ext/InitProcs`[%a]~<>~true~thenF__clFhx-Ffy6$QJreadlib('`context/InitializeProc
s`')(%a);F__clFhx-Ffy6$QA`context/InitProcs`[%a]~:=~true;F__clFhxF[`alQ0readlib
(stack);F__clQ>Exclusions~:=~stack['new']():F__clQ6RememberFlag~:=~true;F__clFj
^clQTContext~:=~_EnvContextData[~CM_BUILD(ContextKey)~];F__clQGif~stack['depth'
](Exclusions)~>~0~thenF__clQgnExclusions~:=~convert(~Exclusions~,~`context/Stac
kToList`~);F__clQGContext~:=~subs(Exclusions~,~Context);F__clF[`alQ<if~Remember
Flag~=~true~thenF__clQhn~~~~Context~~:=~convert(Context,'CONTEXTMENU',~"Top",~a
rgs~);F__clQ?~~~~procname(args)~:=~Context;F__clQ%elseF__clQgn~~~~Context~:=~Co
nvert(Context,'CONTEXTMENU',~"Top",~args~);F__clF[`alQ)Context;F__clQ$end6"@$4-
Fg\mFf\lC&>F]flFiel>%+GlobalCodeGF`z-%(fprintfG6#-F`]m6$Q"|+F__clF`z-Fh`n6#Q;Un
able~to~parse~RecognizerF__cl>F`z-Fd]mFf\l>F`z-Fi\l6$<#/F\`clFiel-FjelFf\lF/6%F
]flF\bcl%3context/CM_GLOBALGGF/F__cl,
I8context/cfparse/topmenuR6&%3SimplificationTypeGF]vFhu'%&AListGFebn6'%(Content
G%&MenusG%+ActionListGFgv%(EntriesG6#Fbgl6"C,-Ffp6#F*-Ffp6#Fjw>8%-FfxF\dcl>&Fcd
cl6#Q%MainF\dcl-&F*6#FDF\dcl-F_r6%%9context/cfparse/pushmenuG9'Fcdcl>8$-Ff]o6$F
fdclFhh[l-F26#-F:6$FaeclFebn>Faecl-F_r6$R6#FgvF\dclF\dclF\dcl@%-F:6$9$Fcbl-Fjw6
$F`fclT#F`fclF\dclF\dcl6$FhcclFcdclFaecl>Faecl-%Acontext/cfparse/removeduplicat
esG6#Faecl-%)CM_BUILDG6&F`fcl9%9&FaeclF\dcl6#FdxF\dclF\dcl,
I>convert/CONTEXTMENU/recursiveR6#%'ActionG6,%+TypeStringGF]vFhu%+SelectionsG%+
ActionProcGFgvFhjoF_b\l%)submenusG%(varprocG6#Fbgl6"C(-Fby6%""#<#%,ContextMenuG
-F>6$F]fl-%%copyG6#Fdx>8)&9"6#;F^hcl!""-F26#3-F:6$9$Febn/-F]z6#Fbicl""%>6&8%8&8
'8(-Fb\lFeicl@-/F[jcl7#Q$VAR6"C&>8+%,_EnvCM_VarsG@$2"""-F]z6#Fejcl>Fejcl-%%sort
G6#7#-Fb\lF[[dl-F26#-F:6$Fejcl<$-Febn6#F)-Fc]nFh[dl>8,-F[\l6$7%-Ffy6$%#%aG8*Fji
cl&F\jcl6#Fb\dl/Fb\dlFejcl/F[jcl7#Q%FUNCFbjclC%>Fejcl-Ff^m6#Fghcl-F26#-F:6$Fejc
l<$-Fc]n6#%)functionG-FebnFc]dl>F[\dlF\\dl/F[jcl7#Q&PAIRSFbjclC'>Fejcl-F^[dl6#7
#-Fb\l6#-F^^mF\]dl-F26#/FjjclF^hcl>Fejcl-&%)combinatG6#.%(permuteGF[[dl-F26#-F:
6$Fejcl%)listlistG>F[\dl-F[\l6$7%-Ffy6$%&%a,%aG&Fb\dlFjgclFjicl&F\jcl6#-Fb\lFd\
dlFe\dl-F:6$F[jcl7#-Febn6#<%F)%(numericGF[b\lC$>Fejcl-Fb\l6$FijclF[jcl>F[\dlF\\
dl-F:6$F[jcl-Febn6#F[b\lC(>8--F`]m6%Q"/FjgclF^_m-Fb\l6#F`z-F26#-F:6$F^adlF)@$4-
F:6$-Fjel6#F^adl%*procedureG-Fh`n6$%CSelection~type~not~implemented~yetGF[jcl>F
ejcl-F^adlF\]dlF\_dl>F[\dl-F[\l6$7%-Faq6$F_\dl;F^hcl!"#Q.sub-selectionFjgclFi_d
lFe\dlF_bdl-F]o6#-%,CONTEXTMENUG6%FiiclFjicl7#F[\dlFjgcl6%F]flFdxFfjclFjgclFjgc
l,
I+DELETEPLOTR6$'%)positionG%'posintG'%%PlotGFd]dlF/F-F/-F]o6#-%'subsopG6$/FKFbr
F<F/F/F/6",
I+qrdecommanRF/64FP%#Q1G%"qGFc`alFhjoFgjoFcj^lF][nF^[nF\joFe\rFgrF_[rFd`alFe`al
Fdj^lFf`alFd[oFdhmF/CG>FWFa]n>FagnF\^o>FdfnF^^o>FhxFiaal>F`z-Feo6$FagnFdfn>Fhgn
-Feo6%FagnFdfnF_el?(F]_lFaoFaoFagnF\w?(F^jnFaoFaoFdfnF\w>&Fhgn6$F]_lF^jnF^\v>Fe
dn-Fahp6$FdfnF_el>F\]lFiedl>Fa^lFfbal?(F]_lFaoFaoFdfnF\w>Fajal-Faen6$FWF]_l>Fez
-FahpFifz?(F]_lFaoFaoFdfnF\w>&FezFeez/Fajal-Ff]o6$F_\blFeo-F^jm6$%/The~matrix~i
s~GF`]oFghmF\\o@$FgnF\^]l-F^jm6#%FWe~now~construct~the~matrices~Q~and~RGFghmF\\
o@$FgnF`\o-F^jm6#%ZAnswer~the~following~questions~about~the~matrices~Q~and~RGFg
hm-FS6#%>The~matrix~R~in~general~is~:|+G-FS6#%]o~~~1.~Upper~triangular~with~pos
itive~entries~on~the~main~diagonal|+G-FS6#%]o~~~2.~Lower~triangular~with~positi
ve~entries~on~the~main~diagonal|+G-FS6#%\o~~~3.~Diagonal~matrix~with~positive~e
ntries~on~the~main~diagonal|+G>FjbrFX@%0FjbrFao-F^jm6#%EIncorrect.~the~correct~
answer~is~(1)G-F^jm6#%%GoodGFghm-FS6#%CThe~columns~of~the~matrix~Q~are~:|+G-FS6
#%9~~~1.~Dependent~vectors|+G-FS6#%:~~~2.~Orthogonal~vectors|+G-FS6#%3~~~3.~Ort
honormal|+G>FjbrFX@%0FjbrFgbn-F^jm6#%EIncorrect.~the~correct~answer~is~(3)GFfhd
l@%/%'QRflagGFaoC%-F^jm6#%PThe~rotation~method~yields~the~QR-decompositionG>F`z
-&Fi^o6#%)QRdecompG6$FW/Fb`alFi_v-F^jm6#/Fa]o*&FagoFaoFfgoFaoCdp>FWFa]n>FagnF\^
o>FdfnF^^o>FhxFaedl>F`z-Feo6$FdfnFdfn>Fhgn-Feo6%FdfnFdfnF_el?(F]_lFaoFaoFdfnF\w
?(F^jnF]_lFaoFdfnF\w>FfedlF^\v?(F]_lFaoFaoFdfnF\w?(F^jnF]_lFaoFdfnF\w>&FhxFgedl
F^\v>FednFiedl>F\]lFiedl>Fa^lFfbal?(F]_lFaoFaoFdfnF\w>FajalF_fdl>FezFbfdl?(F]_l
FaoFaoFdfnF\w>FefdlFffdlF[dal-F^jmFa[lF\\o@$FgnF`\o>&FhgnFghp-F^c`l6#-Fh]v6$F^e
alF^eal>&F\]lFihm-Fb]o6#*&Fj\elF[zF^ealFaoFghm-F^jm6#%SLet~us~compute~the~entri
es~of~the~matrix~R~and~theG-F^jm6#%8columns~of~the~matrix~QGFghm-FS6#%4R[1,1]~i
s~equal~to|+G-FS6#%Y~~~~1.~The~square~root~of~the~innerproduct~of~v1~and~v1|+G-
FS6#%U~~~~2.~The~square~root~of~innerproduct~of~v1~and~v2|+G-FS6#%F~~~~3.~The~i
nnerproduct~of~v1~and~v1|+G>FjbrFX@%FbhdlC$-F^jm6#%LIncorrect~choice.~The~corre
ct~answer~is~(1)G-F^jm6$%8and~the~updated~matrix~G/F`zFfev-F^jm6$%9Good.~The~up
dated~matrixGF__elF\\o@$FgnF`\o?(F]_lFaoFaoFagnF\w>&Fhx6$F]_lFao&F`]elFeez-FS6#
%NThe~first~column~of~the~matrix~Q~is~equal~to|+G-FS6#%M~~~1.~u[1]=(1/innerprod
uct(v[1],v[2]))*v[1]|+G-FS6#%S~~~2.~u[1]=(1/sqrt(innerproduct(v[1],v[1])))*v[1]
|+G-FS6#%M~~~3.~u[1]=(1/innerproduct(v[1],v[1]))*v[1]|+G>FjbrFX@%0FjbrF\]nC$-F^
jm6#%LIncorrect~choice.~The~correct~answer~is~(2)G-F^jm6$%3and~the~new~matrixG/
Fb`alFcfo-F^jm6$%6Good.~The~new~matrix~GF_ael>F]_lFchal>F^jn.F^jn?(F]_lF\]nFaoF
dfnF\wC*>F\fnFfbal>FbxFbhr?(F^jnFaoFaoF_duF\wC$>F\fn-Fb]o6#,&FbxFao*&-Fh]v6$Faj
al&F\]lFbfzFaoFcbelFaoFao>FbxFbhr>F^jnFeael>Fa^l-Fb]o6#,&FajalFaoF\fnF[z>&FhgnF
][s-F^c`l6#-Fh]vFfial?(F^jnFaoFaoF_duF\w>&FhgnFh__lFabel>&F\]lFeez-Fb]o6#*&F[ce
lF[zFa^lFaoFghm-FS6#%6The~entry~R[1,2]~is:|+GF]^elF`^el-FS6#%F~~~~3.~The~innerp
roduct~of~v2~and~v2|+G>FjbrFX>Fegn&Fhgn6$F\]nF\]n>F_delF^\v@%Fg`elC$Fi`elF\_el-
F^jm6$%:Good.~The~updated~matrix~GF__el>F_delFegnFghm-FS6#%6The~entry~R[2,2]~is
:|+G-FS6#%W~~~~1.~The~square~root~of~the~innerproduct~of~w~and~w|+G-FS6#%T~~~~2
.~The~square~root~of~innerproduct~of~w~and~v2|+G-FS6#%D~~~~3.~The~innerproduct~
of~w~and~w|+G-FS6#%Fwhere~w=v[2]-innerproduct(v[2],u[1])|+G>FjbrFX@%FbhdlC$Fi^e
l-F^jm6$%9and~the~updated~matrix~~GF__elFddelFghm?(F]_lFaoFaoFagnF\w?(F^jnFaoFa
oFdfnF\w>F^\el&FcbelFeez-FS6#%HThe~second~column~of~the~matrix~Q~is~:|+G-FS6#%N
~~~1.~v[2]*(1/sqrt(innerproduct(v[2],v[2])))|+G-FS6#%H~~~2.~v[2]*(1/sqrt(innerp
roduct(w,w)))|+G-FS6#%N~~~3.~v[2]*(1/sqrt(innerproduct(v[1],v[2])))|+G>FjbrFX@%
Fg`elC$Fi`el-F^jm6$%4and~the~new~matrix~GF_ael-F^jm6$%4Good~the~new~matrixGF_ae
l-F^jm6#%ZRecursively~we~obtain~the~entries~of~the~matrix~R~and~theG-F^jm6#%Tco
lumns~of~the~matrix~Q~to~get~the~QR~decompositionG>F]_lFchalFb`blFe`bl?(F]_lFao
FaoFagnF\w?(F^jnFaoFaoFdfnF\w>F^\elF`felF\ablF\\o@$FgnF\^]l>F]_lFchal>F^jnFeael
?(F^jnFaoFao,&FdfnFaoF[zFaoF\w?(F]_l,&F^jnFaoFaoFaoFaoFdfnF\w>FfedlF_el-F^jm6$F
`ablF__elF\\o@$FgnF`\oFbabl-F^jm6$F^]z/-Fb]o6#-Fj\s6$FhxFhgn*&FcfoFaoFfevFaoFgh
m-F^jm6#%inThis~is~the~end~of~the~QR-Decomposition~interactive~procedure.G-F^jm
6#%3Try~more~examples!GF/6$FenF]jdlF/F/,
I<convert/CONTEXTMENU/TRIPLESRF/6#F_b\lF[e\lF/C%>FW-F^[dl6#7#-Fb\l6#--Ffp6#.F^^
m6#Fb]n-F26#/FicnFgbn>FW-&Fh^dl6#.F[_dlF^pF/F/F/6",
I0help/text/topic-F]t6&F_t%en~~~~~~~The~following~topics~are~available~in~this~
package.G%_o~~~~~~~trsum~,~commute,~trproduct,~trinverse~,~transtrans,~matrixmu
lG%N~~~~~~~type~>~?topic-name~to~get~more~detailsGF/,
I0help/text/graph-F]t67F`hblF_t%en~linsys[graph]~:~The~purpose~of~this~procedur
e~is~to~graphG%K~a~system~of~linear~equations~in~2D~or~3D~GF_t%H~An~example~on~
how~to~use~this~functionG%hnTo~graph~2~equations~of~3~variables~we~can~type~the
~followingGF_t%9~>~eq1~:=~x~+~y~+~z~=10;G%9~>~eq2~:=~2*x~-~3*y~=~z;G%4~>~graph(
eq1,eq2);~GF_t%fn~or~just~type~at~the~Maple~prompt~>graph();~and~the~programG%9
~will~prompt~for~input.~GF_t%]o~In~2D~case~the~equations~will~be~displayed~alon
g~with~the~graphs.GF_t%Z~The~graph~will~dispaly~consistency~or~inconsistency~in
foGF_t%Z~Plot~Range~:~The~program~will~select~the~best~plot~rangeG%I~necessary~
to~display~all~intersections.GF/,
I1help/text/lindep-F]t6BF`hblF_t%8~~~~~~Linear~DependenceG%:~~~~~~-------------
------GF_t%_o~The~purpose~of~this~function~is~to~go~step~by~step~over~the~proce
ssG%^o~of~checking~whether~a~given~set~of~vectors~is~linearly~independentG%.~or
~dependentG%\o~To~invoke~the~function~first~enter~the~vectors~and~then~call~the
G%8~function.~For~example:GF_t%L~>~u1:=vector([a1,a2]);u2:=vector([b1,b2]);G%8~
>~u3:=vector([c1,c2]);G%5~>~lindep(u1,u2,u3);G%)~~~~~or~G%,~>~lindep()GF_t%D~an
d~enter~the~vectors~upon~requestGF_t%9~Background~Information:GF_t%fo~Definitio
n~:~A~set~of~vectors~|frv1,v2,v3,......vn|hr~is~said~to~be~a~linearlyG%co~~~~~~
~~~~~~~~independent~set~of~vectors~|frv1,v2,v3,......vn|hr~if~wheneverG%[o~~~~~
~~~~~~~~~c1v1+c2v2+c3v3+.....cnvn~=~0~implies~c1=c2=..cn=0.G%jo~~~~~~~~~~~~~~if
~at~least~one~ci~is~not~zero~then~the~set~is~linearly~dependentG%+~Examples:GF_
t%en~1.~Is~the~set~of~vectors~|fru1,u2,u3|hr~linearly~independent?G%K~~~~where~
~u1=(2,4)~,u2=(4,8)~,~u3~=(7,8)?G%M~~~~Ans~:~no~,~since~the~vector~u2=2*u1+0*u3
G%L~~~~Therefore~the~set~is~linearly~dependentGF_tF/,
I3help/text/trainnet-F]t6.F_t%;HELP~SCREEN~for~:~trainnetGF_t%1PACKAGE~:~linmat
G%W~Purpose~:~This~function~trains~a~neural~network~whoseG%]ooutput~represents~
the~adjusted~weights~of~the~output~and~the~inputG%'layersG%=~HOW~TO~CALL~THE~FU
NCTION~?~G%hn~~~~~~~~~~~To~invoke~the~procedure~type~at~the~Maple~prompt~>G%8~~
~~~~~~~~~>trainnet();GF_tF_tF/,
I4type/CF_CONDITIONALRFbpF/F-F/-F:6$FK-Fb_^l6$Fd_^lF_]lF/F/F/6",
I3help/text/solveqns-F]t64F`hblF_`bl%ao-~solvepack[solveqns]~:~the~purpose~of~t
his~function~is~to~To~solve~a~G%^o~a~system~of~linear~equations.~The~set~of~equ
ations~must~be~enteredG%T~as~a~list~or~set.~Example~:~>S:=[x-y=3,2*x+3*y=9];GF_
`bl%\o~The~set~of~variables~you~are~solving~for~must~also~be~entered~asG%I~a~se
t~or~list.~Example~:~>~vars:=[x,y];GF_t%jn~If~no~arguments~are~entered~the~prog
ram~will~prompt~for~input.G%9~Then~follow~directions.GF_t%E~~example~1~:~~S:=[x
-y=3,2*x+3*y=9];G%<~~~~~~~~~~~~~~~vars:=[x,y];G%A~~~~~~~~~~~~~~~solveqns(S,vars
);GF_t%<~~example~2~:~~solveqns();~GF_`blF/,
I2type/CF_SEPARATORRFbpF/F-F/-F:6$FK-Fb_^l6$Fd_^l.F[dlF/F/F/6",
I2help/text/backsub-F]t6-F`hblF_t%ao~linsys[backsub]~:~The~purpose~of~this~func
tion~is~to~perform~the~backG%gn~~~~~~~~~~~substitution~algorithm~in~solving~a~l
inear~systemG%in~~~~~~~~~~~The~procedure~will~check~the~row~reduced~matrix~forG
%W~~~~~~~~~~~consistency~and~apply~the~back~substitutionG%]o~~~~~~~~~~~Matrix~m
ust~be~in~row~echelon~form~as~shown~by~examplesG%1~~~~~~~~~~~belowGF_t%K~~examp
le~1~:~~A~:=~matrix([[1,2],[0,1]]);G%;~~~~~~~~~~~~~~~backsub(A);GF/,
I9type/context/Unbounded@@RFbpF/F[e\lF/3-F:6$FKFd]dl53-F:6$-Fb\l6$F_elFKFd]dl-F
:6$-Fb\l6$F_elF^bfl-.%#@@G6$Fd_^lF)-F:6$F^bflFdbflF/F/F/6",
I.qrdecomnostepRF/6%FPFc`alFgddlF/F/C&>FWFa]n>FhxFcjdl-F^jm6#%FThe~QR-Decomposi
tion~of~the~matrix~isG-F^jm6#/Fa]o*&FagoFaoFcfoFaoF/F/F/F/,
I(rrefmanRF/6@Fd[oFPFajoF][nF^[nF\joF][oFijqFgrF\\rFe\rFghvFchmFgjqFhjqFa[rF]\r
Fj[rFc[rFd[rFe[rFb[rFf\rFi[rF[\rFg[rF[[rFh[rF^\rF\`^lFdhmF/C2>F^yFeas>F\blFibs>
F]_lFa]n>FbzFhc\l-%,rdgaussmenuGF/>FegnFbes-F^jm6$%BThe~original~augmented~matr
ix~is:GF]_lFghm>Fa^lFao?(F/FaoFaoF/30Fa^lF_el0Fa^lF\]nC)-FS6#%^pPlease~enter~an
~appropriate~row~operation~from~the~above~menu~to~reduce~the~matrix|+G>FegnFX>F
enFegn@$FffsFc^r@/5/FegnF[o/FegnFin>Fa^lF_el5/FegnFcgs/FegnFegsC%>F]_lFfc^l?(F`
zFaoFaoFhc\lF\wC%>FednFao?(F/FaoFaoF/F[d^l>FednFger@$Fad^l>F]_lFcd^l-F^jm6$F]hs
F]_l5/Fegn%$recG/Fegn%$RECGC$>F]_l-Fc`yFeez>Fa^lF\]n5/FegnFbhs/FegnFdhsC$@(/F_f
nFao>F]_l-Fjhs6%F]_lFgapFhbs/F_fnF\]n>F]_l-Fdgr6%F]_lFgcq*$FfdlF[z/F_fnFgbn>F]_
l-F]jr6&F]_lFh[lF\fn,$FfelF[z-F^jm6$F]jsF]_l-F:6$-FgjsFfarFhjsC$?(F`zFaoFaoFbzF
\w@$50-F_[t6$Fehfl&F\blFf\lF_el0-F_[t6$Fehfl&F^yFf\lF_elC%@$Fjhfl>FfdlF[ifl@$F^
ifl>FfdlF_ifl>FgcqF`z@%/FfdlFaoC*>FhbsFgcq>Fjbr-Fc\tFfar?(F`zFaoFaoFbzF\wC$@$/F
]iflFjbr>FgapF`z@$/FaiflFjbr>FgapF`z>F]_lFegfl-F^jm6&F_]tFgapF`]tFhbs-F^jm6$Fc]
tF]_lFghm>F_fnFaoC'>F]_l-Fdgr6%F]_lFgcqFfdl-F^jm6&FhirFgcqFagrFfdl-F^jm6$Fd_tF]
_lFghm>F_fnF\]n-F:6$FehflFf`tC+>Fh[lF_el>FaalF_el?(F`zFaoFaoFbzF\wC$@$50-F_[t6$
F]jflF]iflF_el0-F_[t6$F]jflFaiflF_el>F\fnF`z@$Fihfl@%/Fh[lF_elC$>Fh[lF`z@&Fjhfl
>FfelF[iflF^ifl>FfelF_iflC$>FaalF`z@&Fjhfl>FbxF[iflF^ifl>FbxF_ifl@$0FfelFaoC$>F
]_l-F]jr6&F]_lFh[lF\fnFfel-F^jm6(FhirFh[lFagrFfelFgctF\fn@$FgbwC$>F]_l-F]jr6&F]
_lFaalF\fnFbx-F^jm6(FhirFaalFagrFbxFgctF\fn@$3Ffj\l/FbxFaoC$>F]_lFf]glFh]glF`i^
lFghm>F_fnFgbn-F^jm6#%BChoose~a~selection~from~the~menu.G>FjfqF\gfl@$/-F\ao6$-F
b]o6#,&FjfqFaoF]_lF[zFaoF_elC$-F^jm6$%KThe~matrix~in~reduced~row~echelon~form~i
s:GF]_l>Fa^lF\]n@$3/Fa^lF_elFjdfl-F^jm6$%Swarning.~Matrix~is~not~in~reduced~row
~echelon~formGF]_lFghm@$Fa\w-F^jm6#%[oTo~solve~the~system~apply~backsubstitutio
n~to~the~reduced~matrixGFghm-F^jm6#%doThis~is~the~end~of~the~interactive~Gauss-
Jordan~procedure.~You~may~selectG-F^jm6#%coother~examples~to~enforce~your~under
standing~of~this~important~algorithmGF/F_pF/F/,
I*lasterror%OIEEE~Math:~SignalNaN,~nan,~0,~000000000000FC7FGF/,
I2type/CM_SEPARATORRFbpF/F-F/-F:6$FK-Fb_^l6$Fd_^l.F\dlF/F/F/6",
I+APPENDPLOTR6$Fbj[l'%*dimensionGFicdl6+%,CoordinatesG%)NewPlotsGFcp%&PlistGF]_
s%&PlotsG%/ExcludedColorsG%)newcolorG%(OptionsGF-F/C-@$Fjv--Ffp6#FawF/@'/FarFee
m>F`z7#FK/FarFiem>F`zFK>F`zFfbgl-F26#-F:6$F`z-Febn6#<%Febn%*algebraicG/FacglFac
gl-F26#5/F<F\]n/F<Fgbn>F\]l-F_gbl6$R6#FgvF/F/F/-F:6$FKFebnF/F/F/7#&Fb]n6#;FgbnF
[]n>F^jn-F]f[l6$RF\dglF/F/F/F]dglF/F/F/F_dgl>Fdb[lF_el>Fa^l--Ffp6#FagmFg]l>FhxF
br@&FfcglC%>FW--Ffp6#FegmFg]l>%6_EnvSMARTPLOTCoords2DGFW?&FfcnF`zF\wC$@'-F:6$Ff
cn<$F_f[lFacglC'@$53-F:6$FfcnFacgl2Fao-F]z6#-F^^mF\go3-F:6$FfcnF_f[l2F\]nFbfgl-
Fh`n6#%Eattempting~to~add~3D-Plot~to~2D-PlotG>F]_l-FifmFi`l>Fa^l7$-Fb\lFi`lF]_l
>Fedn-Fb\l6#-%*smartplotGF\go>Fedn-Fefm6$Fedn/.FfhnF]_l-F:6$FfcnFebnC%@$53-F:6$
&FfcnFihmFacgl2Fao-F]z6#-F^^m6#Fchgl3-F:6$FchglF_f[l2F\]nFehglFifgl>FednFfcn@%4
-%$hasG6$Fedn<$Fjggl.%'colourGC%>F]_lF]ggl>Fa^lF_ggl>FednFgggl>Fa^l7$F`ggl-Fb\l
6#-Fagm6#7#Fedn-Fh`n6#%2unknown~structureG>Fhx6$FhxFednFgcglC%>FW--Ffp6#FigmFg]
l>%6_EnvSMARTPLOTCoords3DGFW?&FfcnF`zF\wC$@'FheglC$@$53F_fglFhfgl3Fffgl2FgbnFbf
gl-Fh`n6#%3too~many~variablesG>Fedn-Fb\l6#-%,smartplot3dGF\goF[hglC$@$53FahglF\
igl3Fjhgl2FgbnFehglFf[hl>FednFfcnFajgl>FhxFejgl6%-Fb\lFg]lFhx-Fb\lFbfzF/Fcb[lF/
6",
I0CM_GetArrayVarsR6#'FcpFcgbl6$%%valsGF,F[e\lF/C%>FWF]r?&Fhx<#Fb]nF\w@%-F:6$Fhx
Fcgbl>FW-F]en6$FW-F_r6$Fb\l<#-Fe\lF^z>FW<$FWFhx-F^^mF^pF/F/F/6",
I)continueRF/6#%(defaultGFdhmF/C'@%Fa^n>FWFa]n>FWFao-FhhmF^p-FS6#%ao<~type~;~an
d~press~Enter~to~continue~?topic~for~help~~exit;~to~quit~>|+G>FenFXFj^hlF/F_pF/
F/,
I4help/text/trproduct-F]t6-F_t%KHELP~SCREEN~for~:~Transpose~of~the~ProductGF_tF
at%foCONTENTS~:~This~function~checks~if~the~product~of~the~transpose~is~equal~t
oG%Z~~~~~~~~~~~transpose~of~the~product~for~~matrices~A~and~BGF_t%U~~~~~~~~~~~~
To~Call~:~~>~A~:=~matrix([[1,2],[3,3]]);G%U~~~~~~~~~~~~~~~~~~~~~~~>~B~:=~matrix
([[1,0],[7,7]]);G%I~~~~~~~~~~~~~~~~~~~~~~~>~trproduct(A,B);GF_tF/,
I1context/cmglobalR6$'%#wwG<$%'symbolGF[b\l'FcpFd_^l6'Fcj^l%,NotAssignedG%%expr
G%'newvarGF][nF-F/C.>FWFK-F26#-F:6$FWF]`hl>FhxRF\dglF/F[e\lF/C$@$4Fibbl-F]o6#Fc
jm-Fifbl6#/FK-FjelFj_lF/F/F/@$-F:6$Far%(indexedGC$>&%7context/cmglobal/namesGF^
p,&-Fb\l6$FaoFarFaoF[zFao-F26#3-F:6$F[bhlFcs1F[zF[bhl@$2FaoF[]n>FfcnF<@$-F:6$FW
F[b\lC&@$4-F56#F[bhl@%4-Faigl6$-Fjq6$FfcnF[b\lFWC$>F[bhlF[zF]p>F[bhlF[z?(Fedn,&
F[bhlFaoFaoFaoFaoF/F\wC$>F`z-Ff]o6$-Ffa\l6%%!GFWFednF[b\l@$4-Faigl6$FechlF`zF`e
u>F[bhlFedn-F]oFf\l-F26#-F:6$FWF^`hl>FW-Ff]o6$FKF[b\l@&455-F5Fj_lF_chl-Faigl6$-
Fjq6$FfcnF^`hlFW>F[bhlF[zF^chl>F[bhlF[z@%-F:6$FfcnF^bdl?(FednF[dhlFaoF/F\wC$>F`
zF`dhl@$3-FhxFf\l4-Faigl6$-FjelF\goF`zF`eu?(FednF[dhlFaoF/F\wC$>F`zF`dhl@$3Fcfh
l4-Faigl6$FgehlF`zF`eu>F[bhlFednFhdhlF/6#F\bhlF/6",
I(cleanupRF/6(%#s2GF^[n%#chG%#s1GF\`^l%)prevprevGFdhmF/C(>FWF_t>F`zFa]n>FhxFgbn
?(F/FaoFaoF/1Fhx-%'lengthGFf\lC&>F\]l-Faq6$F`z;,&FhxFaoFdqFaoFghhl>Fedn-Faq6$F`
z;,&FhxFaoF[zFaoF\ihl>Ffcn-Faq6$F`z;FhxFhx@%330F\]l%")G0F\]l%"(G0F\]lF^\v@)33/F
\]l%"+G/Fedn%"-G/Ffcn%"1GC$>FW-Ffa\l6$FW%#~-G>Fhx,&FhxFaoFgbnFao3/F\]lF_jhl/Fed
nFajhlC$>FW-Ffa\l6$FWF_jhl>Fhx,&FhxFaoF\]nFaoF[jhlC$>FWF^[il>FhxFa[ilC$>FW-Ffa\
l6$FWF\]l>FhxFb]`l>FhxFb]`l>FW-Ffa\l6%FWFednFfcnF]pF/F/F/F/,
I4help/text/trinverse-F]t6,F_t%GHELP~SCREEN~for~:~Transpose~of~inverseGF_tFat%f
oCONTENTS~:~This~function~checks~if~the~transpose~of~the~inverse~is~equal~toG%Q
~~~~~~~~~~~inverse~of~the~transpose~of~matrix~A~GF_t%U~~~~~~~~~~~~To~Call~:~>~~
A~:=~matrix([[1,2],[3,3]]);G%F~~~~~~~~~~~~~~~~~~~~~~>~trinverse(A);GF_tF/,
I.type/CM_CLASSRFbpF/F-F/-F:6$FK-Fb_^l6$Fd_^l.F\alF/F/F/6",
I(getListRF/Fj`sF/F/C(-FS6#%hnPlease~enter~a~set~of~2D~points.~eg:~L:=|fr[1,2],
[2,-1],[3,1]|hr|+G>FWFX>FenFgr@$FgnF\o?(F/FaoFaoF/F]_nC&-FS6#%PPlease~try~again
:~eg:~L:=|fr[1,2],[2,-1],[3,1]|hr;|+G>FWFX>FenFgr@$FgnF\oF]pF/F_pF/F/,
I)chkvalidR6$'F%Feo'FajoFahp6#F][nF/F/C$?(FWFaoFao-F^bnFj_lF\w@$/-F\ao6$-Fb]o6#
,&F<Fao-F_bt6$FKFWF[zF\]nF_el>%'answerG%#okGF]oF/6#Fb_ilF/F/,
I*constructRF/6-%&eqsetGFej^l%(varlistGF][nF^[nF\joFe\rFgrF,F_[rFijqFdhmF/C+>FW
Fa]n>FfcnF[^n>Fa^lFicn?(F`zFaoFaoFa^lF\w>&FhxFf\l&FWFf\l>F]_l-F]zF\go>Fagn-Feo6
%Fa^lF_crF_el?(F`zFaoFaoFa^lF\wC$>F_fn<#-Fb\l6#-Fc\t6#F``il?(FednFaoFao-F]zF]^s
F\w?(F\]lFaoFaoF]_lF\w@$0-F_[t6$&F_fnFa_oFbc\lF_el>&Fagn6$F`zF\]lFdail?(F`zFaoF
aoFa^lF\w>&Fagn6$F`zF_cr-FgjsF^ail-F]oFj^rF/F/F/F/,
I&blankRF+6$F][nFe^hlF/F/C$@%Fa^n>FhxFa]n>FhxFao?(FWFaoFaoFhxF\w-F^jmF`jmF/F/F/
F/,
I0type/CM_NOPRINTRF\dglF/F[e\lF/-F:6$FK-Fb_^l6$Fd_^l<$%&PADICG%&CFRACGF/F/F/6",
I0type/CM_SUBMENURFbpF/F-F/-F:6$FK-Fb_^l6$F[b\l.FcblF/F/F/6",
I'bksub1RF/6?F][nF_hvF^[nF\joFe\rFgr%&soln1GFgjoFijq%(charsetG%)solnvarsG%)soln
vectG%(tempsumGFPFajoFgvF_[r%)solneqnsGFej^lF`b\l%&eqns1GFge\l%&rightG%)tyepfla
gGFghvF][o%&countGF_b\l%)templistGFdhmF/C5>F_fn-Fahp6#7.&%#_tGFihm&F_eilF\^n&F_
eilFg]q&F_eilF`[q&F_eil6#Fd^\l&F_eil6#""'&F_eil6#F]_q&F_eil6#F]`\l&F_eil6#""*&F
_eil6#F\jn&F_eil6#"#6&F_eil6#"#7>F^jnFa]n>FgcqF[^n>Fedn-Fiar6#-Faen6$F^jnFao>F\
]l,&-Fiar6#-F_btF]gilFaoF[zFao>Fbx.Fbx>Fbx-FahpFg]l?(FWFaoFao-FiarF\fqF\w>&FbxF
^p&FgcqF^p>Fh[l-Fb]o6#-Fj\s6$-F`_r6%F^jnFb_rFcf\l-Ff]o6$FbxFeo>FaalF]hil>F^y-F`
_r6%F^jnFb_r;FggwFggw>F^jnF^e]l?(FWFaoFaoFednF\w@$0&F^jn6$FWFWF_el>F^jn-Fdgr6%F
^jnFW*$F^iilF[z>Fh[lF]hil>FhgnFahil>FdfnFfgil?(FWFaoFaoF\]lF\w>Fag\lFjgil>Fjbr-
Fgdn6$FhgnFdfn@%/-Fi^rFgev-Fi^rFbfzC4>FWFedn?(F/FaoFaoF/-Fie^l6#-F_bt6$F^jnFW>F
WF]]`l>FednFjfil>F\]lF_gil@$2FWFedn>Fh[lF]hil@%/FednF\]lF/C$-F^jm6#%JThis~syste
m~has~infinitely~many~solutionsGFghm>F\blFfgil?(FWFaoFaoF\]lF\w>Ffh_lFjgil>Ffdl
-Fahp6$F\]lF_el>FbzF\]l?(F`zF_jilF[zFaoF\wC,>FWFao?(F/FaoFaoF//&F^jnFhe_lF_el>F
W,&FWFaoFaoFao>&FfdlF^pFao?(FfcnFa\jlFaoF\]lF\w@$0&FfdlF\goFaoC%>&F\blF\go&F_fn
6#,(F\]lFaoFbzF[zFaoFao>Fg\jlFao>Fbz,&FbzFaoF[zFao>FegnF_el?(FfcnFa\jlFaoF\]lF\
w>Fegn,&FegnFao*&&F^jnFjeblFaoFj\jlFaoFao>Ffh_l*&,&&F^jn6$F`zFggwFaoFegnF[zFaoF
_\jlF[z@%0FWF\]lC(@%/F`zF_jil-F^jm6#%KChoose~the~free~variable(s)~and~substitut
eG-F^jm6#%-substitute~:G?(FfcnFa\jlFaoF\]lF\w-F^jm6#/&FbxF\goFj\jlFghm-F^jm6$%/
in~equation~:~G/&FjbrFf\lFj]jl-F^jm6$%5to~find~the~variableG/FjgilFfh_lFghmC$-F
^jm6#%FBy~using~the~last~equation~we~can~getG-F^jm6#/Fe[qFj]qF\\o@$FgnF`\o-F^jm
6#%STherefore~the~values~of~the~variables~are~given~byG>FjfqFfgil?(FWFaoFaoF\]l
F\w>&FjfqF^pFd_jl>F\fnFfgil?(FWFaoFaoF\]lF\w?(FhxFWFaoF\]lF\w@$/&FgcqF^z-Fc\t6#
&FjfqF^z>&F\fnF^z/Fi`jl-FgjsF[ajl-F^jmFbfnFghmC%Fghm-F^jm6#%coIn~the~above~syst
em~we~observe~that~one~or~more~of~the~statement(s)~leadG-F^jm6#%foto~inconsiste
nt~equation(s).~Therefore~the~original~system~is~Inconsistent.GF/F_pF/F/,
I'bksub2RF/6?F][nF_hvF^[nF\joFe\rFgrF^dilFgjoFijqF_dilF`dilFadilFbdilF_[rFPFajo
FgvFcdilFej^lF`b\lFddilFge\lFedilF\\rFghvF][oFgdilF_b\lFhdilFdhmF/C6>F_fnF[eil>
F^jnFa]n>FgcqF[^n>FednFjfil>F\]lF_gil?(FWFaoFaoFad\lF\w?(FfcnFaoFao-F`bnFbfzF\w
F/>F\fn.F\fn>F\fnFfgil?(FWFaoFaoFhgilF\w>&F\fnF^pF[hil>Fh[l-Fb]o6#-Fj\s6$Fahil-
Ff]o6$F\fnFeo>FaalF\cjl>F^yFghil>F^jnF^e]l?(FWFaoFaoFednF\w@$F]iil>F^jnFaiil>Fh
[lF\cjl>FezFahil>FdfnFfgil?(FWFaoFaoF\]lF\w>Fag\lFjbjl>Fjbr-Fgdn6$FezFdfn@%/Feh
\lF_jilC7>FWFedn?(F/FaoFaoF/Fcjil>FWF]]`l>F^jn-F`_r6%F^jn;FaoFW;FaoFggw>FednFjf
il>F\]lF_gil@$F[[jl>Fh[lF\cjl@%F^[jlF/F/Fghm>F\blFfgil?(FWFaoFaoF\]lF\w>Ffh_lFj
bjl>FfdlFg[jl>FbzF\]l?(F`zF_jilF[zFaoF\wC)>FWFao?(F/FaoFaoF/F^\jl>FWFa\jl>Fc\jl
Fao?(FfcnFa\jlFaoF\]lF\w@$Ff\jlC%>Fj\jlF[]jl>Fg\jlFao>FbzF`]jl>FegnF_el?(FfcnFa
\jlFaoF\]lF\w>FegnFd]jl>Ffh_lFh]jl>FjfqFfgil?(FWFaoFaoF\]lF\w>Fc`jl/FjbjlFfh_l-
F^jm6#%IThe~values~of~the~variables~are~given~byG>FjfqFfgil?(FWFaoFaoF\]lF\w>Fc
`jlFhfjl>FhgnFfgil?(FWFaoFaoF\]lF\w?(FhxFWFaoF\]lF\w@$Fh`jl>&FhgnF^zF_ajlF^dalC
%FghmFcajlFfajlF/F_pF/F/,
I3help/text/LUdecomp-F]t6UF_t%CHELP~SCREEN~for~:~LU-DecompositionGF_tFf^fl%NDEF
INITION~:~A~=~LU,~where~L=lower~triangularG%N~~~~~~~~~~~~~and~U~is~upper~triang
ular~~~~~~~G%doEXISTENCE~:~The~LU~factorization~of~a~given~matrix~always~does~n
ot~exist.G%ao~~~~~~~~~~~~Even~if~the~LU~factorization~exists,~it~may~not~be~uni
que.G%bo~~~~~~~~~~~Given~below~are~some~sufficient~conditions~for~existence~and
G%7~~~~~~~~~~~uniqueness.G%in~~~~~~~~~~~1.~If~rank(A)=k~then~A~can~be~factored~
as~A~=~LU~ifG%\o~~~~~~~~~~~~~the~jth~principle~submatrix~of~A~is~non-singular~f
orG%6~~~~~~~~~~~~~j~=~1..kG%`o~~~~~~~~~~~~~jth~principle~submatrix~of~A~is~defi
ned~as~the~submatrixG%T~~~~~~~~~~~~~consists~of~rows~1..j~and~columns~1..jGF_t%
eo~~~~~~~~~~~2.~If~A~is~non-singular~then~A~=~LU~has~a~unique~factorization.G%e
o~~~~~~~~~~~3.~If~A~is~any~nxn~matrix~then~there~exist~permutation~matricesG%L~
~~~~~~~~~~~~~P~and~Q~such~that~A~=~P*LU*Q.GF_t%F~METHOD~OF~FINDING~LU~FACTORIZA
TION~:G%eo~~~~~~~~~~~If~a~given~matrix~A~can~be~written~as~a~product~of~two~mat
ricesG%Z~~~~~~~~~~~A~=~LU~~where~L~is~a~lower~triangular~matrix~~G%jn~~~~~~~~~~
~and~U~is~upper~triangular,~then~U~can~be~obtained~byG%[o~~~~~~~~~~~performing~
Gauss~Elimination~on~A~to~convert~A~to~itsG%`o~~~~~~~~~~~row~echelon~form.~The~
matrix~L~is~obtained~by~applying~theG%co~~~~~~~~~~~same~operations~to~the~ident
ity~matrix~and~taking~its~inverseGF_t%ho~~~~~~~~~~~Each~elementary~row~operatio
n,~when~applied~to~the~identity~matrixG%co~~~~~~~~~~~produces~an~elementary~mat
rix~Mi.~Therefore~in~the~process~ofG%ao~~~~~~~~~~~converting~A~to~its~row~echel
on~form,~we~produce~a~sequenceG%hn~~~~~~~~~~~of~elementary~matrices~M1,M2,M3,..
.etc~as~follows.G%Z~~~~~~~~~~~Mi*....*M3*M2*M1*A~=~row~echelon~form~of~A~=~UG%Y
~~~~~~~~~~~Let~L~=~inverse(Mi*....*M3*M2*M1)~then~A~=~LUGF_t%0~APPLICATIONS~:G%
[o~~~~~~~~~~~LU~decomposition~of~a~matrix~can~be~used~to~solve~theG%M~~~~~~~~~~
~linear~system~~Ax~=~b~as~follows.G%jn~~~~~~~~~~~Consider~Ax~=~b~==>~LUx~=~b~==
>~Ly~=~b~where~Ux~=~y~G%Y~~~~~~~~~~~solve~Ly~=~b~for~y~using~forward~substituti
onG%fn~~~~~~~~~~~Then~solve~Ux~=~y~~for~x~using~back~substitutionGF_`blFj^flF[_
fl%G~~~~~~~~~~~>A:=~matrix([[1,2],[4,4]]);G%9~~~~~~~~~~~>LUdecomp(A);GF_t%hn~~~
~~~~~~~~The~procedure~can~also~be~called~without~argumentsG%8~~~~~~~~~~~>LUdeco
mp();GF_tF_tF/,
I8context/InitializeProcsR6#'F[vF[b\l6$Fgv%%KeysGF[e\lF/C&>Fhx-F_gbl6%F:7#-F\h[
lF[fl7#F^`hl>Fhx-F_r6$Fb\lFhx?&FWFhxF\wC$-F26#-F:6$&FdxF^pF^bdl-F>6$FW-Fjel6#Fe
[[m>&%2Context/InitProcsGFj_lF\wF/6$FdxF\\[mF/6",
I4help/text/leastsqrs-F]t6A%4HELP~FOR:~leastsqrsG%[oGiven~a~system~of~linear~eq
uations~Ax=b,~one~can~find~a~solutionG%[ousing~Gauss~Elimination~if~the~system~
is~consistent.~However~forG%_oinconsistent~systems~we~obtain~a~solution~x~that~
minimize~the~2-normG%boof~the~vector~|grAx-b|gr.~This~solution~is~called~the~le
ast~square~solutionG%5of~the~system~Ax=b.~GF_t%bo~The~procedure~takes~a~set~of~
equations,~an~augmented~matrix,~or~set~ofG%]o~2D~points~and~the~degree~of~the~l
east~square~polynomial~as~input.G%bo~If~no~arguments~are~entered,~the~procedure
~will~prompt~for~input.~ThisG%eo~procedure~can~be~used~in~3~modes,~Demonstratio
n,~Interactive,~and~No-StepG%\o~modes.~No-Step~mode~is~equivalent~to~linalg[lea
stsqrs]~function.GF_t%Q~USAGE~:~>~A:=matrix([[1,2,1],[2,3,1],[2,4,0]]);G%9~~~~~
~~~~>~leastsqrs(A);GF_t%0~~~~~~~~~~~~~ORGF_t%;~~~~~~~~~>~eq1:=x1+2*x2=1;G%=~~~~
~~~~~>~eq2:=2*x1+3*x2=1;G%=~~~~~~~~~>~eq3:=2*x1+4*x2=0;G%C~~~~~~~~~>~leastsqrs(
eq1,eq2,eq3);GF_tF_][m%L~~~~~~~~~>~pointList:=|fr[1,2],[3,4],[3,-1]|hr;G%<~~~~~
~~~~>~degreeOfPoly:=2;G%M~~~~~~~~~>~leastsqrs(pointList,degreeOfPoly)GF_][mF_t%
8~~~~~~~~~>~leastsqrs();GF_tF/,
I1help/text/linsys-F]t6<F`hbl%'~~~~~~G%9~LINEAR~SYSTEMS~package.G%%~~~~G%Q~Writ
ten~by~:~Elias~Deeba~and~Ananda~GunawardenaG%:~DATE~~~~~~~:~Spring~1994G%co~The
~purpose~of~this~package~is~to~introduce~INTERACTIVE~procedures~intoG%in~MAPLE~
enviornment.~These~packages~are~specifically~written~toG%]o~display~the~linear~
algebra~topics~in~a~highly~interactive~seetingGF_t%gnCALLING~SEQUENCE:~You~can~
call~any~function~BY~the~statementG%5<function>(args)~~ORG%9linsys[<function>](
args)GF_t%*SYNOPSIS:G%P-~to~use~a~linsys~function,~invoke~the~functionG%D~using
~the~form~linsys[<function>].G%$~~~G%@-~The~functions~available~are~:GFi^[m%C~~
~~gausselim~~gaussauto~gaussman~GFi^[m%hn-~for~more~information~on~a~particular
~function~refer~to~highG%E~~level~function~manual~or~just~typeG%3>~?<function~n
ame>G%2~eg:~>~?gausselimGF/,
I=context/cfparse/act_completeRFhd\lF/F[e\lF/C%-Ffp6#.Fd]mF^]al-Fd]m6#-Ficl6ZQg
nproc(s1::uneval,s2::uneval,s3::uneval,s4::uneval,s5::uneval)6"Q_o~~~~local~Eva
l,~EvalLabel,Parameters,AutoAssignFlag,ParseFlag,Nargs,F[`[mQO~~~~~~~~F,Labels,
Params,i,ProcBody,CodeString;F[`[mQ;~~~~global~none,CM_Global;F[`[mQ@~~~~"Compl
ete~Action~Template";F[`[mQD~~~~readlib(~'`context/cmglobal`');F[`[mQB~~~~readl
ib(~'`context/testeq`');F[`[mF``[mQQ~~~~Parameters~:=~select(~type~,~[args]~,~`
=`~);F[`[mQjn~~~~AutoAssignFlag~:=~subs(~op(Parameters),~"autoassign"=FAIL~,F[`
[mQ6~~~~~~"autoassign"~);F[`[mQX~~~~ParseFlag~:=~~subs(~op(Parameters),~"parse"
=false~,F[`[mQ1~~~~~~"parse"~);F[`[mQG~~~~Nargs~:=~nargs~-~nops(Parameters);F[`
[mQap~~~~if~not~type([args[1]]~,~list(list)~)~then~ERROR(`list~of~lists~expecte
d`,args)~fi;F[`[mQD~~~~Eval~:=~()~->~op(~eval(~subs(~|frF[`[mQM~~~~~~~~~~~PIECE
WISE=`context/fixpiecewise`,F[`[mQ:~~~~~~~~~~~MATRIX=matrix,F[`[mQI~~~~~~~~~~~V
ECTOR=vector|hr~,~[args]~)));~F[`[mQ\p~~~~Labels~:=~vector(~map(~convert~,~map2
(~op~,~1~,~[args[1..Nargs]]~)~,~name~));F[`[mQ6~~~~for~i~to~Nargs~doF[`[mQ<~~~~
~~~~if~Labels[i]~=~noneF[`[mQK~~~~~~~~or~(not~type(Labels[i],name))~thenF[`[mQS
~~~~~~~~~~~~~Labels[i]~:=~Eval(op(2..-1,args[i]));F[`[mQ\p~~~~~~~~elif~not~`con
text/testeq`(eval(Labels[i]),~eval(op(2..-1,args[i]))~)~thenF[`[mFaa[mQ,~~~~~~~
~fi;F[`[mQ(~~~~od;F[`[mQG~~~~if~type(~procname~,~indexed~)~thenF[`[mQdo~~~~~~~~
Params~:=~seq(~CF_ARG.i~=~op(i,procname)~,~i=1..nops(procname)~);F[`[mQ)~~~~els
eF[`[mQ8~~~~~~~~Params~:=~NULL;F[`[mQ(~~~~fi;F[`[mQ8~~~~ProcBody~:=~subs(~|frF[
`[mQ]p~~~~~~~'f'=~~~`if`(~type([Labels[1]],[indexed])~and~evalb(op(0,Labels[1])
=Labels),F[`[mQN~~~~~~~~~~~~~~~~eval(Labels[1],1),Labels[1]),F[`[mQar~~~~~~~'AR
GS'~=~(seq(`if`(~type([Labels[i]],[indexed])~and~evalb(op(0,Labels[i])=Labels),
eval(Labels[i],1),Labels[i]),F[`[mQ:~~~~~~~~~~~i=1..Nargs~)),F[`[mQ/~~~~~~~Para
ms,F[`[mQS~~~~~~~'`CM_Global`(a)'~=~`context/cmglobal`('a'),F[`[mQS~~~~~~~'`CM_
Global`(b)'~=~`context/cmglobal`('b'),F[`[mQS~~~~~~~'`CM_Global`(x)'~=~`context
/cmglobal`('x'),F[`[mQL~~~~~~~'CM_GLOBALG'~=~`context/CM_GLOBALG`,F[`[mQ1~~~~~~
~~NULL~|hr,~F[`[m-F`]m6&Fa_alQ9proc()~global~CM_ASSIGN;F[`[mFi_lFdaclQ'~~~~);F[
`[mQ]o~~~~CodeString~:=~readlib('`context/BuildInputString`')(ProcBody);F[`[mQh
o~~~~CodeString~:=~readlib('`context/autoassign`')(CodeString,AutoAssignFlag);F
[`[mQ>~~~~if~ParseFlag~=~true~then~F[`[mQP~~~~~~~~CodeString~:=~`context/jointe
xt`(~"~"~,F[`[mQR~~~~~~~~~~~~"if~true~then"~,~CodeString~,~"fi"~);F[`[mQJ~~~~~~
~~~parse(~CodeString~,~statement~);F[`[mFga[mQ4~~~~~~~~CodeString;F[`[mFia[mFda
clF/F/F/F[`[m,
I6SMARTPLOTFindCoords2DR6#'FjaglFebn6)F][nFfhz%+candidatesG%(nextposG%*newcoord
sGFab\l%"YGF[e\lF/C)>F\]l.%%_NoXG>Fa^l.%%_NoYG-FfpF6>FednF@>F`z-&F*6#.%&depthGF
a_o?&FhxFK2F`zF\]nC(@$1FgbnFcd[mF`eu>Ffcn-F]f[l6%F]c[l&Fhx6#;F\]nFgbn<$F\]lFa^l
@$/FfcnFddl%%nextG-F_r6%FFFfcnFedn?(FWFaoFao-%$minG6$F\]nFcd[mF\w@&Fa\w>F\]l&Fe
dnF^pFi]w>Fa^lF_f[m>F`zFcd[m7$F\]lFa^lF/F/F/6",
I,qrdecomautoRF/6AF][nF_hvF`hvFhhvFihvF^[nF\joFgrFfjqF][oFgjqFhjqFijqFfhvFbj^l%
&RmatsGFjjqFh[nF[[rF\[rF][rF^[rF]_sF'FfddlFb`al%'nocolsGFPFc`alFd[o%,processtil
lGFdhmF/C=>FjbrFao@$F[anC&-F^jm6#%XPlease~enter~a~matrix~A.~e.g.~A:=matrix([[1,
2],[2,3]]);GFX>FagnFfeq>Fegn-FiewFj^r@$Fa^n>FegnFe]o>F]_lFhar>FbzFdit>FfelF]_l>
Fh[l-FeoF][s?(FWFaoFaoFfelF\wC$?(F\]lFaoFaoFfelF\w>&Fh[lFh[ilF_el>&Fh[lF_iilFao
>Fgcq-Feo6$FfelFbz?(FWFaoFaoFfelF\w?(F\]lFaoFaoFbzF\w>&FgcqFh[il&FegnFh[il>Fbx-
Fahp6#*&F_duFaoF]_lFao?(Fa^lFaoFaoF]i[mF\w>&FbxFi`lFjg[m>Fa^lFao>FhgnFegn>Fjfq.
Fjfq-F^jm6$%<The~matrix~you~entered~is~:G/FgcqFearF\\o@$FgnF`\o@&/F]jdlF\]nC$-F
_`alF\fqF`\o/F]jdlF_elC*-F^jm6#%hnThere~are~two~popular~methods~of~finding~the~
QR~decompositionG-F^jm6#%Rof~a~matrix.~Please~select~one~from~the~followingGFgh
m-FS6#%W~~~~~~1.~Using~Gram~Schmidt~Orthogonalization~Process|+G-FS6#%Q~~~~~~2.
~Using~the~orthogonal~rotation~matrices|+G-F^jm6#%3<type~~1;~~or~2;~>G>F\\qFX@$
/F\\qFaoC$F]j[mF`\oFghm>Fh[l-Feo6%FfdrFfdrF_el?(FWFaoFaoFfdrF\w>Fah[mFao>F_fnF_
el?(F\]lFaoFaoFfelF\w?(FWFggwFaoFfdrF\w@$0Fih[mF_elC$>F_fnFaoF`eu@%/F_fnF_elC$-
F^jm6#%gnA~is~upper~triangular~and~a~trivial~QR~Decomposition~of~A~isG-F^jm6#/F
ear*&-Fb]oFg^_lFaoFearFaoCS-F^jm6#%boThe~QR~decomposition~of~A~can~be~obtained~
by~applying~rotation~matricesG-F^jm6#%coR(i,j)~to~A~that~will~convert~the~matri
x~A~to~an~upper~triangular~matrixGF\\o@$FgnFf\_l>FjbrFao>F]_lFhar>FbzFdit>FfelF
]_l>Fh[l-Feo6%F]_lF]_lF_el?(FWFaoFaoF]_lF\w>Fah[mFao>FagnFch[m?(FWFaoFaoFfelF\w
?(F\]lFaoFaoFbzF\w>&FagnFh[ilFih[m>FbxF[i[m?(Fa^lFaoFaoF]i[mF\w>F`i[mFg]\m>Fa^l
Fao>FezFch[m?(FWFaoFaoFfelF\w?(F\]lFaoFaoFbzF\w>&FezFh[ilFih[m>FjfqFdi[m@%1Fcdr
Ffdr>Fj^q,&FcdrFaoF[zFao>Fj^q,&FfdrFaoF[zFao?(F\]lFaoFaoFj^qF\wC'>FhxF\]l?(F/Fa
oFaoF/3/&Fegn6$FhxF\]lF_el1FhxFfel>FhxFb]`l@'/FhxF\]l>F_fnF_el2FfelFhx>F_fnFao>
F_fnF\]n@$Fggfl>Fegn-Fjhs6%FegnFhxF\]l@$0F_fnFaoC)-FS6$%F~~~~~Working~on~column
~%d~of~A......|+GF\]l?(FWFggwFaoF]_lF\w@$5F\\\mFggflC2?(F`zFaoFaoFfelF\w>&F`i[m
FeewFao>&F`i[mFaft-%$cosG6#-%'arctanG6#*&Fih[mFaoF`ftF[z>&F`i[mF_iilFca\m>&F`i[
mFh[il,$-%$sinGFea\mF[z>&F`i[m6$F\]lFWF_b\m@$Fggfl>F`i[m-Fjhs6%F`i[mFhxF\]l-F^j
m6$%;Apply~the~rotation~matrix~G/&FjfqFh[il-Fb]o6#F`i[mFghm@$0F_fnF\]n>Fegn-Fb]
o6#-Fj\s6$F`i[mFhgn-F^jm6#%:from~left~of~A~and~we~getG-F^jm6#/*&F\c\mFaoFgcqFao
FearFghmF\\o@$FgnF\^]l>FhgnFegn>Fa^lFderFghm-F^jm6#%fnMultiply~the~above~rotati
on~matrix(matrices)~from~left~of~AG-F^jm6%%Wto~make~the~entries~of~A~below~the~
diagonal~of~column~GF\]l%&~zeroG-F^jm6$%5The~new~matrix~is~:~GFegnFghm>FfdlFjg[
m>Fgap&FbxFihm@%1FgbnFa^l?(FWF\]nFaoFef\lF\wC$>Ffdl-Fb]o6#-Fj\s6$FjgilFgap>Fgap
Ffdl>FfdlF\e\m>FjbrFef\l>Faal-FahpFfgal?(FWFaoFaoFjbrF\w>Ffg_lFjgil-F^jm6#%ZDis
playing~matrices~in~the~form~Rn*R(n-1)*...*R2*R1*A~=~RG-F^jm6#%=where~each~Ri~i
s~orthogonal.GF\\o@$FgnF`\oFghm@$/FjbrFao-F^jm6#/*&-Fb]o6#&FaalFihmFao-Fb]oF\fq
FaoFear@$/FjbrF\]n-F^jm6#/*(-Fb]o6#&FaalF\^nFaoFjf\mFaoF]g\mFaoFear@$/FjbrFgbn-
F^jm6#/**-Fb]o6#&FaalFg]qFaoFdg\mFaoFjf\mFaoF]g\mFaoFear@$/FjbrFhjp-F^jm6#/*,-F
b]o6#&FaalF`[qFaoF]h\mFaoFdg\mFaoFjf\mFaoF]g\mFaoFear@$/FjbrFd^\l-F^jm6#/*.-Fb]
o6#&FaalFdeilFaoFfh\mFaoF]h\mFaoFdg\mFaoFjf\mFaoF]g\mFaoFear@$/FjbrFgeil-F^jm6#
/*0-Fb]o6#&FaalFfeilFaoF_i\mFaoFfh\mFaoF]h\mFaoFdg\mFaoFjf\mFaoF]g\mFaoFear@$/F
jbrF]_q-F^jm6#/*2-Fb]o6#&FaalFieilFaoFhi\mFaoF_i\mFaoFfh\mFaoF]h\mFaoFdg\mFaoFj
f\mFaoF]g\mFaoFear>Ffdl-F[_o6#Ffdl-F^jm6#%gnQ~is~obtained~by~taking~the~transpo
se~of~Rn*R(n-1)*...*R2*R1GFghm-F^jm6$%<Therefore~the~matrix~Q~is~:GFfdl-F^jm6$%
2The~matrix~R~is~:GFegn>F\bl-Fb]o6#-Fj\s6$FfdlFegnF\\o@$FgnF`\o-F^jm6$Fdabl/Fgc
qFfabl-F^jm6$F^]z/Fig\l*&-Fb]oFfj\mFaoFearFaoFghm-F^jm6#%jnThis~is~the~end~of~t
he~~QR~decomposition~demonstration~version.G-F^jm6#%2Try~more~examplesGF/FiielF
/F/,
I.EDITSMARTPLOTRF/6(%1characterizationGF][n%*newoptionGFjaglF]bglF]_sF-F/C%@$Fj
vF`bgl>Fdb[lF_el@%-F:6$Fa]nFcsC)>FhxFa]n>FfcnF[^n>F`z-F_gbl6$RF\dglF/F/F/F]dglF
/F/F/F_dgl>Fedn-F]f[l6$RF\dglF/F/F/F]dglF/F/F/F_dgl>F\]l-Fefm6$&F`zF^zFfcn>F`z-
F_ddl6$F[`\mF`z-F]o6$-Fb\lFf\lFj^clC(>FfcnFa]n>F`z-F_gbl6$RF\dglF/F/F/F]dglF/F/
F/7#&Fb]n6#;F\]nF[]n>Fedn-F]f[l6$RF\dglF/F/F/F]dglF/F/F/Fj^]m@%-F:6$Ffcn/F)Fd_^
l>FW-Fc\tF\go>FWFfcn>Fedn-F]f[l6%R6$FgvFa_\lF/F/F/-FaiglFJF/F/F/FednFW-F]o6%Fc^
]mFj^clFfcnF/Fcb[lF/6",
I4SMARTPLOTColor_ListRF/F/F[e\lF/7*.Fghn.%&greenG.Fcin.%(magentaG.%'orangeG.%%p
lumG.%&coralG.%$tanGF/F/F/6",
I'ABeqBARF/6%FbhvFPFgguFdhmF/C9-F]cn6#Fi^oFghm>FW-Feo6%F\]nF\]nF_el>&FWFghpFao>
&FWF`delFao-FS6#%enIn~general~matrices~do~not~commute.~However,~following~is|+G
-FS6#%hna~brief~list~of~conditions~on~A~and~B~,~so~that~they~commute|+GFghm-FS6
#%Z~~1.~A~and~B~commute~if~B~is~a~power~of~A~and~vice~versa|+G-F^jm6$%-~Proof~:
~LetG/Ffcn)FhxFgr-F^jm6%%&Then~G/Fa^v)Fhx,&FgrFaoFaoFao/Ff\[lFcc]mFghmF\\o@$F_\
oF`\o-FS6#%jn~~2.~More~generally~A~and~B~commute,~if~B~is~a~polynomial~of~A|+G>
Fhx-Feo6#7$7$FaoF\]n7$FgbnFao-F^jm6$%$LetGFbfo>Ffcn-Fb]o6#,(-Fj\s6$,$FhxF\]nFhx
FaoFhxFd`\lFWFhjp-F^jm6$F\dbl/Ffcn,(*$FcfoF\]nF\]nFcfoFd`\lFa]oFhjp-F^jm6$%+Tha
t~is~:~GFgfpFghm-F^jm6$Ffdbl/Fa^v-Fb]o6#-Fj\sFhdn-F^jm6$F[hw/Ff\[l-Fb]o6#-Fj\sF
a\[lF/F_pF/F/,
I0help/text/trsum-F]t6-F_t%GHELP~SCREEN~for~:~Transpose~of~the~sumGF_tFat%boCON
TENTS~:~This~function~checks~if~the~sum~of~the~transpose~is~equal~toG%T~~~~~~~~
~~~transpose~of~the~sum~of~matrices~A~and~BGF_tFdt%T~~~~~~~~~~~~~~~~~~~~~~>~B~:
=~matrix([[1,0],[7,7]]);G%D~~~~~~~~~~~~~~~~~~~~~~>~trsum(A,B);GF_tF/,
I)linspace=F/FcjmE\[l(%-graphvectaddGRF/65F]joFd[oFhjo%%sizeG%)arrowsetGF][nF^[
nF\jo%)colorsetG%'txtsetG%$txtG%'sumsetG%%sumxG%%sumyG%$ln1G%$ln2G%*sumstringG%
&psumxG%&psumyGF/F/C8>F^jn-Fahp6#7;%+aquamarineG%&blackGFcin%%navyGF`a]m%%cyanG
%&brownG%%goldGFg`]m%%grayG%%greyG%&khakiGFj`]m%'maroonGF\a]m%%pinkGF^a]mFghn%'
siennaGFba]m%*turquoiseG%'violetG%&wheatG%&whiteG%'yellowG-F]cn6#%*plottoolsGF\
cn-F^jm6#%VThe~purpose~of~this~procedure~is~to~help~us~visualizeG-F^jm6#%Ythe~c
oncept~of~vector~addition.~The~program~will~displayG-F^jm6#%Xthe~original~vecto
rs~and~their~sums.The~sum~vectors~areG-F^jm6#%6drawn~with~red~arrowsGF\\o>Ffcn-
Fahp6#"#?@%F[anC'-FS6#%LHow~many~vectors~would~you~like~to~enter?~|+G>FhxFX>F`z
Fhx>F\]lFao?(F/FaoFaoF/1F\]lF`zC%-FS6#%[oPlease~enter~a~vector~with~two~compone
nts.~eg:~>~vector([1,2]);|+G>Fbc\lFX@%4-F:6$Fbc\lFahp-F^jm6#%GERROR!~a~vector~i
s~expected.~Try~againG>F\]lFggwC$?(F\]lFaoFaoF[]nF\w@%-F:6$&Fb]nFg]lFahp>Fbc\lF
^\^m-Fh`n6#%:Vector~arguments~expectedG>F`zF[]n?(F\]lFaoFaoF`zF\w@$0-FiarFac\lF
\]n-Fh`n6#%Fvectors~must~have~only~two~componentsG>FW-FahpFf\l>FagnF\]^m?(F\]lF
aoFaoF`zF\wC$>Fien-F[bp6(7$F_elF_el7$&Fbc\lFihm&Fbc\lF\^nF^ep$F\]nF[z$F^filFdq/
Ffhn&F^jnFg]l>&FagnFg]l-Fjgn6%7%Fe]^mFf]^m-Ff]o6$Fbc\lF[b\l/Fahn<$%&RIGHTGFchn/
Fj`\l7%F\a\lF]a\lF\jn>FednF]r>F_fnF]r?(F\]lFaoFaoF`zF\w>Fedn-F]en6$Fedn<$FienF\
^^m>FdfnF]r>Fjbr&FchglFihm>Ffel&FchglF\^n?(F\]lFaoFaoF^[xF\wC.>FegnF_el>FhgnF_e
l>F\fnFbdhl?(F]_lF\]nFaoF\]lF\w>F\fn-Ffa\l6$F\fnFgihl?(Fa^lFaoFaoFggwF\wC%>Fegn
,&FegnFao&&FfcnFi`lFihmFao>Fhgn,&FhgnFao&Fa`^mF\^nFao@%0Fa^lFggw@%1F\]nFa^l>F\f
n-Ffa\l6%F\fn-Ff]o6$Fa`^mF[b\l%#)+G>F\fn-Ffa\l6%F\fnF\a^mF]jhl>F\fn-Ffa\l6$F\fn
F\a^m>Fien-F[bp6(Fc]^m7$FegnFhgnF^epFg]^m$FaoF[zFehn>F\^^m-Fjgn6%7%FegnFhgnF\fn
F`hnFi`\l>&FezFg]l-%%lineG6&7$FjbrFfelFha^mFbin/%*linestyleGF\]n>Fe[q-Fab^m6&7$
&&Ffcn6#FggwFihm&F[c^mF\^nFha^mFbinFdb^m>Fdfn-F]en6$Fdfn<&FienFe[qF\^^mF_b^m>Fj
brFegn>FfelFhgn-&F_cn6#Fdjn6$-F]en6$FednFdfn/Ff^\l%'normalGF/F/F/%&basisGRF/6HF
][nF^[nF_hvF`hvFe\rFgrFhjoFcj^lFijq%'chkansGFd[oFchmFfhvFjhvF]_sFfhz%"sGFhjmF]j
o%)basissetG%)basisaugG%'bassetG%'orgsetGF\joFf`alFdj^lFge\lF_[r%'vecsetGFP%$au
gGFgjo%%mat1G%%aug1GF][o%(tempsetGFgdil%#m1GFdhmF/C0Fghm-F^jm6#%]pThe~purpose~o
f~this~procedure~is~to~learn,~how~to~verify~if~a~set~of~vectors~formsG-F^jm6#%\
pa~basis.~Recall~that~a~basis~for~a~linear~space~is~a~minimal~linearly~independ
entG-F^jm6#%]psubset~of~the~given~vectors~that~spans~the~space.~For~more~inform
ation~type~?basisGFghm-FS6#%DSelect~one~of~the~following~modes:|+G-FS6#%<~~~~~1
.~Demonstration~mode|+G-FS6#%:~~~~~2.~Interactive~mode|+G-FS6#%6~~~~~3.~No-Step
~mode|+G>FagnFX>FenFagn@$FgnF`\oFghm@(/FagnFaoCP@%0F[]nF_elC'?(FWFaoFaoF[]nF\wC
$>&Fa^lF^p&Fb]nF^p@$4-F:6$F]g^mFahp-F]o6#%EError!~All~arguments~must~be~vectors
G>F\]lF[]n>Fedn-Fiar6#&Fa^lFihm>F^jn-FeoFb_q?(FWFaoFaoFednF\w?(FhxFaoFaoF\]lF\w
>&F^jnF^c[l&&Fa^lF^zF^pC.-FS6#%gnPlease~Enter~the~set~of~vectors.~All~vectors~m
ust~be~of~the|+G-FS6#%Asame~dimension.~To~end~type~>0;|+G-FS6#%WExample~:~>~v1:
=vector([3,3,2]);~v2:=vector([0,1,1]);|+G>FdfnFX>FenFdfn@$FgnF`\o>FWFao>FenFdfn
@$FgnF`\o?(F/FaoFaoF/0FdfnF_el@%-F:6$FdfnFahpC+>FenFdfn@$FgnF`\o>F]g^mFdfn@%Fa\
w>F^jn-Ff]o6$Fjg^mFeo>F^jn-Fiew6$F^jnF]g^m>FWFa\jl-FS6#%LEnter~the~next~vector;
or~type~0~~when~done|+G>FdfnFX>FenFdfn@$FgnF`\oC&-FS6#%AA~vector~is~expected.~T
ry~again|+G>FdfnFX>FenFdfn@$FgnF`\o>F\]lF]]`l@$1FaoF\]l>FednFad\l?(FWFaoFaoF\]l
F\wC$>F]g^m-FahpFa_o?(FhxFaoFaoFednF\w>&F]g^mF^z&F^jn6$FhxFW>F^yF]r?(FWFaoFaoF\
]lF\w>F^y-F]en6$F^y<#-Ff]o6$-Fb]o6#F]g^mFeo-FS6#%]qA~set~of~vectors~S~is~a~basi
s~for~a~linear~space~V|+if~the~following~two~conditions~are~satisfied,|+G-FS6#%
E~~~~~1.~~S~is~linearly~independent.|+G-FS6#%F~~~~~2.~~S~spans~the~linear~space
~V.|+GFghm-F^jm6%%.The~given~setG/Fjbr-FjelFhh_l%5may~form~a~basis~forG-F^jm6%)
Fc`alFedn%6or~for~a~subspace~of~GFi]_mF\\o@$FgnF`\o-F^jm6#%_oFirst~we~check~to~
see~if~the~given~vectors~are~linearly~independent.G-F^jm6#%fnConstruct~the~matr
ix~AUG,~whose~rows~are~the~given~vectors.G-F^jm6#/%/The~matrix~AUGG-Fb]o6#-F[_o
FbfzF\\o@$FgnF`\o-F^jm6#%MApply~Gauss-Jordan~Elimination~to~matrix~AUGG-F^jm6#%
Hto~obtain~its~reduced~row~echelon~form.G>Fegn-Fc`yFg^_m-F^jm6#/%DThe~reduced~r
ow~echelon~form~of~AUGGFearFghm-F^jm6#%hnInspect~the~reduced~matrix~and~determi
ne~how~many~independentG-F^jm6#%envectors~can~be~drawn~from~the~rows~of~the~red
uced~matrix~?GF\\o@$FgnF`\o@*32-Fi^rFfarF\]l/F``_mFednC'-F^jm6#%epThe~reduced~r
ow~echelon~form~of~the~matrix~shows~that~at~least~one~of~the~original~vectorsG-
F^jm6#%epis~a~linear~combination~of~the~others.~Therefore~the~set~of~vectors~is
~linearly~dependent.G-F^jm6&%DHowever,~we~can~extract~a~subset~ofGF``_m%Jvector
s~that~forms~a~basis~for~the~space~GFi]_mFghm>Fj_qFao3F_`_m2F``_mFednC(Fc`_mFf`
_m-F^jm6%F[a_mF``_m%=linearly~independent~vectorsG-F^jm6&%Fthat~forms~a~basis~f
or~a~subspace~of~GFi]_m%-of~dimensionGF``_mFghm>Fj_qF\]n3/F``_mF\]lF_a_mC/-F^jm
6#%aoThe~reduced~row~echelon~form~of~the~matrix~shows~that~no~vector~of~theG-F^
jm6#%^ooriginal~set~of~vectors~can~be~written~as~linear~combination~of~theG-F^j
m6#%aoothers.~Therefore~the~original~set~of~vectors~is~linearly~independent.GF\
\o@$FgnC$-F^jm6#%(Bye~nowGF_jm-F^jm6#/%=The~original~set~of~vectors~GFe]_m-F^jm
6%%.does~not~haveGFedn%0vectors~to~spanG-F^jm6%Fi]_m%Hbut~they~form~a~basis~for
~a~subspace~ofGFi]_m-F^jm6$Fga_mF``_mFghm>Fj_qFgbn-F^jm6#%;End~of~the~basis~pro
cedureGF`\o3Fja_mFa`_mC1F\b_m-F^jm6#%`ooriginal~set~of~vectors~can~be~written~a
s~a~linear~combination~of~theGFbb_mF\\o@$FgnC$Fgb_mF_jmFjb_m-F^jm6&%*does~haveG
Fedn%>independent~vectors~that~spanGFi]_mF\\o-F^jm6$%RTherefore~the~set~is~a~ba
sis~for~the~linear~spaceGFi]_mFec_mFghm>Fj_qFhjp-F^jm6#%aoThis~is~the~end~of~th
e~demonstration~version.~You~may~change~the~givenG-F^jm6#%_ovectors~and~execute
~the~procedure~again~to~learn~the~steps~involved.GF`\oF\\o@$FgnF`\o@&FhjxC'-F^j
m6#%hnSince~the~given~set~of~vectors~is~linearly~dependent,~we~mustG-F^jm6%%4ex
tract~a~basis~forGFi]_m%>from~the~given~set~of~vectorsG-F^jm6%%+Choose~anyGF``_
m%5linearly~independentG-F^jm6#%>vectors~from~the~original~setGFghm/Fj_qF\]nC(F
ce_m-F^jm6&%Bextract~a~basis~for~a~subspace~ofGFi]_mFga_mF``_m-F^jm6%F\f_mF``_m
%Slinearly~independent~vectors~from~the~original~setGFghmF\\o@$FgnF`\o-F^jm6#/%
FRecall~the~original~set~of~vectors~isGFe]_mFghm>Fh[l<#-Fb]oFig^m?(FhbsF\]nFaoF
fdrF\wC&>Faal-Ff]o6$-Fb]o6#Fg]_lFeo?(FWF\]nFao-F]zFg^_lF\w>Faal-Fiew6$Faal-Ff]o
6$F`^_lFeo>Faal-Fiew6$Faal-Ff]o6$-Fb]o6#&Fa^l6#FhbsFeo@$/-Fi^rFjaxF\bx>Fh[l-F]e
n6$Fh[l<#Fdh_m>F\blF]r?(FWFaoFaoFig_mF\w>F\bl-F]en6$F\bl<#-Ff]o6$-Fb]o6#F`^_lFe
o-FS6$%YExtract~%d~linearly~independent~vectors~to~form~a~basis|+GFig_m-F^jm6#/
Fggu-FjelF`clFghmF\\o@$FgnF`\o@$5Fhjx/Fj_qFhjpC6-F^jm6$%inLet~us~verify~that~th
e~above~set~is~basis~for~the~linear~spaceGFi]_mFghm-F^jm6#%hn~~1.~Clearly~the~a
bove~set~of~vectors~is~linearly~independentG-F^jm6$%Z~~2.~Therefore~it~is~suffi
cies~to~show~that~the~set~spansGFi]_mF\\o@$FgnC$Fgb_mF_jm>F]_l-F`_r6%-Feo6#7,7#
F%7#Fajo7#%"cG7#Fehv7#%"eG7#Fcp7#Fbj[l7#%"hG7#FW7#FhxFb_r;FaoFao-F^jm6&%1Given~
any~vectorGF]_l%+belongs~toGFi]_m-F^jm6%%/We~shall~writeGF]_l%Xas~a~linear~comb
ination~of~the~basis~vectors~as~followsGFghm>Fgcq&F\blFihm?(FWF\]nFao-F]zF`clF\
w>Fgcq-Fiew6$FgcqFfh_l>Fgcq-Fiew6$FgcqF]_l>Fbz-Feh]l6#-Fe^sF\fq>Ffdl-Fahp6$Fedn
F_el?(FWFaoFaoF_]`mF\w>Ffdl,&FfdlFao*&&FbzF^pFaoFdh_lFaoFao>FgapFfdl>Ffdl.Ffdl>
Ffdl,&FgapFaoFfdlF[z-F^jm6#/F[`]lFfdl@$5Faf_m/Fj_qFgbnC$-F^jm6#%inThe~above~set
~of~vectors~is~a~minimal~linearly~independent~setG-F^jm6&%9that~spans~a~subspac
e~ofGFi]_mFga_mF``_mFghmFjd_mF]e_m/FagnF\]nCU@%Fhf^mC'?(FWFaoFaoF[]nF\wC$>F]g^m
F^g^m@$F`g^mFcg^m>F\]lF[]n>FednFhg^m>F^jnF\h^m?(FWFaoFaoFednF\w?(FhxFaoFaoF\]lF
\w>F`h^mFah^mC.Fdh^mFgh^mFjh^m>FdfnFX>FenFdfn@$FgnF`\o>FWFao>FenFdfn@$FgnF`\o?(
F/FaoFaoF/Fdi^m@%Ffi^mC+>FenFdfn@$FgnF`\o>F]g^mFdfn@%Fa\w>F^jnF^j^m>F^jnFaj^m>F
WFa\jl-FS6#%LEnter~the~next~vector;~or~type~0~when~done|+G>FdfnFX>FenFdfn@$FgnF
`\oC&F[[_m>FdfnFX>FenFdfn@$FgnF`\o>F\]lF]]`l@$Fc[_m>FednFad\l?(FWFaoFaoF\]lF\wC
$>F]g^mFh[_m?(FhxFaoFaoFednF\w>F[\_mF\\_m-FS6#%in~A~set~of~vectors~S~is~said~to
~be~a~basis~for~a~linear~space,|+G-FS6#%M~if~the~following~conditions~are~satis
fied:|+GF[]_m-FS6#%=~~~~~2.~~S~spans~the~space.|+GF\\o@$FgnF`\o>F^yFfgil>Fh_pF]
r?(FWFaoFaoF\]lF\wC$>F`h_lFd\_m>Fh_p-F]en6$Fh_pFc\_m-FS6%%ZEnter~the~%dx%d~matr
ix~whose~rows~are~the~given~vectors.|+GF\]lFedn-FS6#%gnExample:~To~enter~a~matr
ix~type:~matrix([[1,2,3],[1,1,1]]);|+G-F^jm6$%AThe~original~set~of~vectors~is~:
GF^y>FagnFX?(F/FaoFaoF/Fi]rC&>FenFagn@$FgnF`\oFa_p>FagnFX?(F/FaoFaoF/50-F^bnFj^
rF\]l0-F`bnFj^rFednC'-FS6%%jnERROR!!~Incorrect~Dimensions.~%dx%d~matrix~expecte
d.~Try~again|+GF\]lFedn-FSF\jt>FagnFX>FenFagn@$FgnF`\o>FegpFao>F^jnFh^_m?(F/Fao
FaoF/0-F\ao6$-Fb]o6#,&FagnFaoF^jnF[zF\]nF_elC(-F^jm6#%DIncorrect~answer.~please
~try~again.G>FagnFX>FenFagn@$FgnF`\o@$/FegpFgbnF`eu>Fegp,&FegpFaoFaoFao@%Fef`mC
%-F^jm6#%EYou~attempted~3~times~unsuccessfullyG-F^jm6#%KThe~matrix~formed~by~th
e~set~of~vectors~isGFe]]l-F^jm6$%>Good,~the~correct~matrix~is~:GFagn>F\\qFagnFg
hm-FS6#%gnSelect~the~most~appropriate~method~from~the~following~menu:|+G-FS6#%T
~~1.~Find~the~LU-decomposition~of~the~above~matrix|+G-FS6#%Y~~2.~Find~reduced~r
ow~echelon~form~of~the~above~matrix~|+G-FS6#%N~~3.~Find~the~transpose~of~the~ab
ove~matrix.|+G-FS6#%:~<Type~:~1;~or~2;~or~3;>|+GFghm>FdfnFX>FenFdfn@$FgnF`\o?(F
/FaoFaoF/0FdfnF\]nC&-F^jm6#%;Incorret~method.~Try~againG>FdfnFX>FenFdfn@$FgnC$F
]jmF_jm>Fj^q-Fc`yFbfz-F^jm6#%OThe~reduced~row~echelon~form~of~the~matrix~is:G-F
^jmFh_q>F_hp-Fi^rFh_qF\\o@$FgnC$Fgb_mF_jm-FS6#%foHow~many~linearly~independent~
rows~can~be~extracted~from~the~above~matrix?|+G-FS6#%IEnter~the~number~followed
~by~semicolon;|+G>FagnFX>FenFagn@$FgnF`\o?(F/FaoFaoF/0FagnF_hpC&-FS6#%6Incorrec
t.~Try~again|+G>FagnFX>FenFagn@$FgnF`\oFghm>Fedn-F^bnFh_q>F\]l-F`bnFh_q@%/Fgi`m
FednC.-F^jm6%%-Correct,~allGFedn%Lvectors~of~the~set~are~linearly~independentGF
ghm-FS6#%SDoes~the~original~set~of~vectors~form~a~basis~for|+G-FS6$%;~~~1.~a~su
bspace~of~R%d~?|+GF\]l-FS6$%+~~~2.~R%d|+GF\]l-FS6#%1Answer~1;~or~2;|+G>FagnFX>F
enFagn@$FgnF`\o@*3Fef^mF^[jl-F^jm6$%NIncorrect.~The~original~set~forms~a~basis~
forG)Fc`alF\]l3Fef^m0FednF\]l-F^jm6$%enCorrect.~The~original~set~forms~a~basis~
for~a~subspace~of~GF[]am3Fd_`mF]]am-F^jm6$%gnIncorrect.~The~original~set~forms~
a~basis~for~a~subspace~of~GF[]am3Fd_`mF^[jl-F^jm6$%MCorrect.~The~original~set~f
orms~a~basis~for~GF[]amFghm-F^jm6$%-A~basis~is~:GFh_pC/-F^jm6%%=Correct.~We~nee
d~to~extract~GFgi`mFhf_mFghm-FS6$%inExtract~%d~Linearly~Independent~vectors~fro
m~the~original~set|+GFgi`m-FS6#%JTo~enter~a~vector,~type~:~vector([1,2]);|+G>Fh
bsFao>FhxFao>Fh[lF]r?(F/FaoFaoF/31FhbsF\]l2Fig_mFgi`mC/@$/FhbsF\]n>Fj_qFao-F^jm
6$%?The~original~set~of~vectors~isGF^yFghm-FS6#%SEnter~an~independent~vector~fr
om~the~original~set|+G>&Fa\tFgh_mFX?(F/FaoFaoF/4-F:6$Fh_amFahpC&-FS6#%6A~vector
~is~expected|+G>Fh_amFX>FenFh_am@$FgnF\^]l@$/-Fcc[l6$FhbsF\]nFaoF]d`m>Fb_il%&no
tokGFghm?(F/FaoFaoF//Fb_ilFi`amC%-F_^il6$F^jn-Fb]o6#Fh_am@%F[aamC(-F^jm6$%+The~
vectorG-Ff]o6$F_aamFeo-F^jm6#%endoes~not~belong~to~the~original~set~of~vectors.
~Try~again.G-FS6#%MPlease~enter~a~vector~from~the~original~set|+G>Fh_amFX>FenFh
_am@$FgnF`\o-F^jm6$%MYou~entered~a~vector~from~the~original~set~:GFfaamFghm@$F[
\`lC(-F^jm6#%^oSince~a~set~consisting~of~a~non-zero~vector~is~linearly~independ
entG-F^jm6%FeaamFfaam%0is~in~the~basisG>F\is-Ff]o6$-Fb]o6#&Fa\tFihmFeo>Fh[l-F]e
n6$Fh[l<#F]cam>FhxFb]`l>Fj_qF_el@$32FaoFhx0Fj_qF_elC9>Fgcq-Feo6$-F^bn6#F\is,&-F
`bnFadamFaoFaoFao?(FfcnFaoFaoF`damF\w?(F`zFaoFaoFbdamF\w@%0F`zFbdam>&FgcqF]\v&F
\isF]\v>Fidam&Fh_amF\go>F\is.F\is>F\is-Feo6$-F^bnF\fq-F`bnF\fq?(FfcnFaoFaoFbeam
F\w?(F`zFaoFaoFceamF\w>FjdamFidam>Fgcq.Fgcq>F^]q-Fc`yFadam@$0-Fi^rFd]qFhxC(>Fbz
-Feh]l6#-Fe^sFadam>FfdlF[^`m?(FWFaoFao-F]zF^[lF\w>Ffdl,&FfdlFao*&Fa^`mFao-Fb]o6
#-Ff]o6$-Faen6$F\isFWFeoFaoFao>FgapFfdl>FfdlFd^`m>FgapFf^`mF\\o@$FgnF`\o-F^jm6#
%inLet~us~form~the~matrix~whose~rows~consist~of~the~current~basisG-F^jm6$%6and~
the~new~vector~:~GFfaam-F^jm6#-F[_oFadamF\\o@$FgnC$Fgb_mF_jm-F^jm6$%EFind~its~r
educed~row~echelon~form~:~G-F[_oFd]qFghm-FS6#%hoBased~on~the~reduced~row~echelo
n~form,~are~the~vectors~linearly~independent?|+G-FS6#%8<~answer~yes;~or~no;~>|+
G>FagnFX>FenFagn@$FgnF`\o@&/FagnFcao@%/F]famFhxC&-F^jm6#%aoGood.~We~can~add~the
~new~vector~to~form~a~new~linearly~independent~setG>Fh[l-F]en6$Fh[l<#FfaamFf^_l
>FhxFb]`lC(-F^jm6#%[oIncorrect.~The~new~vector~can~be~written~as~a~linear~combi
nationG-F^jm6#%Jof~the~current~basis~vectors~as~follows~:G-F^jm6#/-Fb]o6#-Ff]o6
$Fh_amFeoFgapFghm-F^jm6#%VTherefore~the~new~vector~does~not~belong~to~the~basis
G>F\is-F`_r6%F\isFcf\l;FaoF\ihl/FagnFi`o@%F^iamC&-F^jm6#%foincorrect.~We~can~ad
d~the~new~vector~to~form~a~new~linearly~independent~setG>Fh[lFdiamFf^_l>FhxFb]`
l@$F\famC(-F^jmFd]q-F^jm6#%incorrect.~The~new~vector~can~be~written~as~a~linear
~combinationGF\jamFghmFfjam>F\isFjjam>Fhbs,&FhbsFaoFaoFaoF\\o-F^jm6%%2We~have~e
xtractedGFgi`mFhf_mFghm-F^jm6$%WTherefore,~the~basis~we~extracted~from~the~give
n~set~:GF^y-F^jm6$%%~is~GFh[lFghm-F^jm6#%`pThis~is~the~end~of~the~interactive~v
ersion~of~the~basis~procedure.~Try~other~examplesG/FagnFgbnC5@%Fhf^mC'?(FWFaoFa
oF[]nF\wC$>F]g^mF^g^m@$F`g^mFcg^m>F\]lF[]n>FednFhg^m>F^jnF\h^m?(FWFaoFaoFednF\w
?(FhxFaoFaoF\]lF\w>F`h^mFah^mC.Fdh^mFgh^mFjh^m>FdfnFX>FenFdfn@$FgnF`\o>FWFao>Fe
nFdfn@$FgnF`\o?(F/FaoFaoF/Fdi^m@%Ffi^mC+>FenFdfn@$FgnF`\o>F]g^mFdfn@%Fa\w>F^jnF
^j^m>F^jnFaj^m>FWFa\jlFdj^m>FdfnFX>FenFdfn@$FgnF`\oC&F[[_m>FdfnFX>FenFdfn@$FgnF
`\o>F\]lF]]`l@$Fc[_m>FednFad\l?(FWFaoFaoF\]lF\wC$>F]g^mFh[_m?(FhxFaoFaoFednF\w>
F[\_mF\\_m>F^yF]r?(FWFaoFaoF\]lF\w>F^yFa\_m>FegnF^e]l@*F^`_m>Fj_qFaoF^a_m>Fj_qF
\]nFia_m>Fj_qFgbnF[d_m>Fj_qFhjp-F^jm6#/%@The~original~set~of~vectors~is~GFe]_mF
ghm>FegnFa__m@*F^`_m>Fj_qFaoF^a_m>Fj_qF\]nFia_mC(Fjb_mF^c_mFbc_mFec_m>Fj_qFgbnF
`\oF[d_mC(Fbd_mFfd_mFec_mFghm>Fj_qFhjpF`\oFghm>Fh[lF_g_m?(FhbsF\]nFaoFfdrF\wC&>
FaalFdg_m?(FWF\]nFaoFig_mF\w>FaalF[h_m>FaalF`h_m@$Fih_m>Fh[lF\i_m>F\blF]r?(FWFa
oFaoFig_mF\w>F\blFbi_m-FS6$%UA~basis~consists~of~%d~linearly~independent~vector
s|+GFig_mF\j_m@$Fbj_m-F^jm6$%NThe~above~set~is~a~basis~for~the~linear~spaceGFi]
_m@$F[_`m-F^jm6&%KThe~above~set~is~a~basis~for~a~subspace~ofGFi]_mFga_mF``_mF/6
$FenFb_ilF/%-graphlincombGRF/6=F]joFd[oFhjoF\g]mF]g]mF][nF^[nF^g]mF_g]mF`g]mFag
]mFbg]mFcg]m%'framesG%&frameGFe\r%#clG%%ctxtG%#slG%%stxtGFfg]mF\joFi[`mFdg]mFeg
]mFgg]mFhg]mF/F/C6>F]_lF[h]mF_i]mF\cn-F^jm6#%ZThe~purpose~of~this~procedure~is~
to~help~us~visualize~theG-F^jm6#%Yconcept~of~linear~combination.~The~program~wi
ll~take~twoG-F^jm6#%envectors~and~create~linear~combinations~of~the~two~vectors
.G-F^jm6#%gnThe~optional~third~argument~of~the~function~is~the~number~ofG-F^jm6
#%inlinear~combinations~you~want~to~view.~Click~on~the~play~buttonG-F^jm6#%\oto
~see~the~animations.~You~can~control~the~speed~of~the~animationG-F^jm6#%Iand~yo
u~may~continuously~play~the~movie.GF\\o@$FgnC$Fgb_mF_jm@%/F[]nFgbn>FhgnF^[_l>Fh
gnFgbn@'2FgbnF[]n-Fh`n6#%gnThis~program~display~linear~combinations~of~only~two
~vectorsGF[anC%>F\]lFao?(F/FaoFaoF/1F\]lF\]nC%F][^m>Fbc\lFX@%4-F:6$FchglFahp-FS
6#%IERROR!!~A~vector~is~expected.~Try~again|+G>F\]lFggw>F`zF\]nC$?(F\]lFaoFaoF\
]nF\w@%F\\^m>Fbc\lF^\^mF`\^m>F`zF\]n?(F\]lFaoFaoF`zF\w@$Ff\^mFh\^m>FWF\]^m>F_fn
F\]^m?(F\]lFaoFaoF`zF\wC$>Fien-F[bp6(Fc]^mFd]^mF^epFg]^mFh]^m/Ffhn&F]_lFg]l>&F_
fnFg]lF]^^m?(F\blFaoFaoFhgnF\wC3@'/-Fcc[l6$F\blFgbnF_elC$>&F^yFihm,&-Fcc[l6$-%%
randGF/FhjpFaoFaoFao>&F^yF\^nFjgbm/FegbmFaoC$>FigbmFjgbm>F`hbm,&F[hbmF[zF[zFaoC
$>FigbmFjgbm>F`hbmFjgbm?(F\]lFaoFaoF`zF\wC$>&F\fnFg]l-F[bp6(Fc]^m7$*&&F^yFg]lFa
oFe]^mFao*&FaibmFaoFf]^mFaoF^epFg]^mFh]^m/Ffhn&F]_l6#,&F\]lFaoF\]nFao>&FjbrFg]l
-Fjgn6%7%F`ibmFbibm-Ffa\l6%-Ff]o6$FaibmF[b\lF^\vF`^^mFb^^mFe^^m>FdfnF_el>FegnF_
el>FaalFbdhl?(Fa^lFaoFaoF\]nF\wC%>Fdfn,&FdfnFao*&&F^yFi`lFaoF``^mFaoFao>Fegn,&F
egnFao*&FhjbmFaoFd`^mFaoFao@%F\a^l>Faal-Ffa\l6'Faal-Ff]o6$FhjbmF[b\lF^\vF\a^mF]
jhl>Faal-Ffa\l6&FaalF`[cmF^\vF\a^m>Ffel-F[bp6(Fc]^m7$FdfnFegnF^epFg]^mFia^mFehn
>Fh[l-Fjgn6%7%FdfnFegnFaalFb^^mFi`\l>Fhbs-Fab^m6&7$*&FigbmFaoF`_^mFao*&FigbmFao
Fb_^mFaoFh[cmFbinFdb^m>Fgap-Fab^m6&7$*&F`hbmFao&&FfcnF\^nFihmFao*&F`hbmFao&Fi\c
mF\^nFaoFh[cmFbinFdb^m>&Fez6#,&F\blF]_q!"'Fao-Fec^m6$<$FdbnFi]_lFjc^m>&Fez6#,&F
\blF]_q!"&Fao-Fec^m6$<&FdbnFi]_l&F_fnF\^n&FWF\^nFjc^m>&Fez6#,&F\blF]_q!"%Fao-Fe
c^m6$<(FdbnFfcrFi]_lF\^cmF]^cm&F\fnFihmFjc^m>&Fez6#,&F\blF]_qFd`\lFao-Fec^m6$<*
FdbnFfcrFi]_lF\^cmF]^cmFf^cm&F\fnF\^n&FjbrF\^nFjc^m>&Fez6#,&F\blF]_qFdqFao-Fec^
m6$<+FdbnFfcrFhbsFi]_lF\^cmF]^cmFf^cmF^_cmF__cmFjc^m>&Fez6#,&F\blF]_qF[zFao-Fec
^m6$<,FdbnFfcrFhbsFgapFi]_lF\^cmF]^cmFf^cmF^_cmF__cmFjc^m>&Fez6#,$F\blF]_q-Fec^
m6$<.FdbnFh[lFfelFfcrFhbsFgapFi]_lF\^cmF]^cmFf^cmF^_cmF__cmFjc^m-F^jm6#-Fdjn6%7
#-F[\l6$&FezF`cl/F\bl;Fao,$FhgnF]_q/%+insequenceGF\w/%(scalingG%,constrainedGF/
F/F/%'lindepGRF/6JF][nF^[nF\joFe\rFgrFhjoFijqFd[oFchmFdj^lFf`al%(tempAUGGFfhv%%
AUG3G%*setofvarsG%+solnvectorG%*eqnvectorGFcj^lFge\l%%AUG0G%&soln0G%(homosumGF`
b\l%+consisflagGF][o%(zerovecGF_[rFbj^lF\^q%(chkans1G%(chkans2G%(chkans3GFgdil%
%res1G%&res11G%%AUGGG%)solutionG%(youransGFbdil%)smallsumGFdhmF/C.Fghm-F^jm6#%h
oThe~purpose~of~this~procedure~is~to~determine~if~a~set~of~vectors~is~linearlyG
-F^jm6#%hodependent~or~not.~It~includes~three~versions:~demonstration,~interact
ive,~andG-F^jm6#%jnno-step.~For~more~information~type:~?lindep~at~the~Maple~pro
mptGFghmFghmFee^mFhe^mF[f^mF^f^m>F]_lFX@(/F]_lFaoCin@%F[anC+-FS6#%VPlease~Enter
~the~set~of~vectors.~All~vectors~must~be|+G-FS6#%Lof~the~same~dimension~as~w.~T
o~end~type~0;|+G>F^jnFao>FWFao-FS6#%@Please~enter~the~first~vector.|+G>Faj`lFX?
(F/FaoFaoF/0F^jnF_elC'>FenFaj`l@$FgnF`\o?(F/FaoFaoF/30Faj`lF_el4-F:6$Faj`lFahpC
&>FenFaj`l@$FgnF`\o-FS6#%4A~vector~expected.|+G>Faj`lFX@$F^ecmC%>FWFa\jl-FS6#%O
Please~enter~the~next~vector.~To~end~type:~0;|+G>Faj`lFX>F^jnFaj`l>FednF]]`l>F`
z-Fiar6#F`]elC%?(FWFaoFaoF[]nF\wC$>Faj`lF^g^m@$F_ecm-Fh`n6#%>All~arguments~must
~be~vectorsG>FednF[]n>F`zFcfcm>Fa^l-FeoF^dv?(FWFaoFaoF`zF\w?(FhxFaoFaoFednF\w>&
Fa^lF^c[l&&F\]lF^zF^p?(FWFaoFao-F`bnFi`lF\w?(FhxFa\jlFaoFhgcmF\w@$/-F\ao6$,&-Fa
en6$Fa^lFWFao-Faen6$Fa^lFhxF[zF\]nF_elC%-F^jm6#%cpSince~among~the~vectors~enter
ed,~two~vectors~are~the~same,~the~set~is~linearly~dependentG-F^jm6#%hnEnter~dis
tinct~vectors~and~run~the~procedure~again.~Good~LuckG-F]o6#Fbdhl>Fez-Fahp6#7,F^
[oFjdo%#c3G%#c4G%#c5G%#c6G%#c7G%#c8G%#c9G%$c10G>Ffdl-Ff]o6$-Fahp6$F`zF_elFeo>Fe
z-F`_r6%-Ff]o6$FezFeoFb_rFc\`m>Fh[lFa^l?(FWFaoFaoFednF\w?(FhxFaoFaoF`zF\w>&Faj`
lF^z&Fa^lF]\_m>F\blF_el?(FWFaoFaoFednF\w>F\bl,&F\blFao*&&FezF`g\lFao-Fb]o6#-Ff]
o6$F_hcmFeoFaoFao-F^jm6#%hnIn~this~particular~example,~we~begin~with~the~vector
~equationG-F^jm6#/F\blF]\]mFghm-F^jm6#%_oThe~problem~reduces~to~solving~a~homog
eneous~system~of~equations.~IfG-F^jm6#%`othe~homogeneous~system~has~a~unique~so
lution~(trivial~solution),~thenG-F^jm6#%fothe~vectors~are~linearly~independent;
~otherwise,~the~vectors~are~dependent.GF\\o@$FgnC$F]jmF_jm>Fa^lF`gcm?(FWFaoFaoF
`zF\w?(FhxFaoFaoFednF\w>FdgcmFegcm>FbxFh[_m?(FWFaoFaoFednF\w>FjgilF][dm>F\fn-Fg
dn6$Fa^lFbx-F^jm6#%]oThe~above~vector~equation~yields~the~following~homogeneous
~system:G?(FWFaoFaoF`zF\w-F^jm6#/FjbjlF_elF\\o@$FgnC$F]jmF_jm-F^jm6#%^oNow~cons
truct~the~augmented~matrix~of~the~above~homogeneous~system.G>FjbrF[jcm>Fa^l-Fie
w6$Fa^lFjbr-F^jm6$%0The~matrix~is~:G/Fa^l-Fb]oFi`lF\\o@$FgnC$F]jmF_jm>Fegn-Fe^s
Fi`l-F^jm6#%coApply~the~Gauss~Elimination~to~obtain~the~row~echelon~form~of~the
~matrixG-F^jm6#/%<The~row~echelon~form~of~AUGGFear>Fhgn-Feo6$F``_mFcdr>FWFfdr?(
F/FaoFaoF//-F\ao6$-F_bt6$FegnFWF\]nF_el>FWF]]`l?(FhxFaoFaoFWF\w?(FfcnFaoFaoFcdr
F\w>&FhgnFhdn&FegnFhdn@%3-F:6$&Fhgn6$FWFedn%(complexG0Fa`dmF_elC(>FW-F^bnFgev?(
F/FaoFaoF/3-Fie^l6#-F_bt6$FhgnFW2FaoFW>FWF]]`l>FhxFao?(F/FaoFaoF/3/&FhgnF^c[lF_
el2Fhx-F`bnFgev>FhxFb]`l@$30FdadmF_el0FdadmFao>Fhgn-Fdgr6%FhgnFW*$FdadmF[z>Fegn
FfevC%>F\]l-Fahp6#Fbay?(FfcnFaoFaoFbayF\w>&F\]lF\go&FhgnF`]v@$34F_`dm/-F\ao6$F\
]lF\]nF_elC*-F^jm6%%.Assuming~thatGFd`dm%Lthe~echelon~form~of~the~augmented~mat
rix~isG>FWFg`dm?(F/FaoFaoF/Fi`dm>FWF]]`l>FhxFao?(F/FaoFaoF/Fbadm>FhxFb]`l@$Fiad
m>FhgnF]bdm>FegnFfevF[[uF\\o@$FgnC$F]jmF_jm-F^jm6#%jnThis~row~echelon~form~of~t
he~augmented~matrix~yields~the~systemGFghm>F^y-Fb]o6#-Fj\s6$-F`_r6%Fegn;FaoF``_
mFb_rFez?(FWFaoFaoF``_mF\w-F^jm6#/&F^yF`g\lF_el-F^jm6#%\oCan~you~tell~from~the~
reduced~system~or~from~the~row~echelon~formG-F^jm6#%gnof~the~augmented~matrix,~
if~it~has~the~unique~zero~solution?G-F^jm6#%fnOr~if~the~system~has~infinitely~m
any~non-trivial~solutions?GF\\o@$FgnC$F]jmF_jm>Ffel-Feh]lFc^s?(FWFaoFaoF]_\mF\w
>&FfelF^p-Fi\l6(/F^eilF[z/F`eilF[z/FaeilF[z/FbeilF[z/FceilF[zF^fdm>F_fnF_el?(FW
FaoFaoFednF\w>F_fn,&F_fnFao-Fdhy6#F^fdmFao@%F`\\mC$-F^jm6#%boThe~trivial~soluti
on~is~the~only~solution~to~the~homogeneous~system~andG-F^jm6#%aotherefore~we~co
nclude~that~the~set~of~vectors~is~linearly~independent.GC4-F^jm6#%VThe~homogene
ous~system~has~a~non-trivial~solution~andG-F^jm6#%Wa~non-trivial~solution~to~th
e~above~system~is~given~byG-FS6#%7~~~~~~~~~~~~~~~~~~~~~~GFghm?(FWFaoFaoFednF\wC
$-FS6%%0~~~~~~c%d~=~%a,GFWF^fdm@$/-Fcc[l6$FWFhjpF_el-FS6#%"|+GF\\o-F^jm6#%[oThi
s~implies~that~some~ci's~are~non-zero~in~the~zero~combinationG-F^jm6#%hnc1*v1+c
2*v2+...+cn*vn~=~0~and~therefore~the~set~of~vectors~isG-F^jm6#%4linearly~depend
ent.GFghmF\\o-F^jm6#%goSome~vectors~can~be~written~as~a~linear~combination~of~t
he~others~as~followsG>Fdfn-F`_r6%Fh[l;Fao-F^bnFg^_l;Fao,&-F`bnFg^_lFaoF[zFao>Fe
dn-F`bnFifz?(FhxFaoFaoFbayF\wC'>Fh[lFdfn>Fh[l-%(swapcolG6%Fh[lFhxFedn>Fhbs-%+co
nsistentGFg^_l@%/FhbsFaoC*>Faal-Feh]l6#-Fe^sFg^_l?(FWFaoFaoFbayF\w>Ffg_l-Fi\l6(
FafdmFbfdmFcfdmFdfdmFefdmFfg_l>FagnF[jcm>F_fnF[jcm?(FWFaoFaoFbayF\wC$>F_fn,&Fag
nFao*&Ffg_lFao-Fb]o6#-Ff]o6$-Faen6$Fh[lFWFeoFaoFao>FagnF_fn>Fagn.Fagn>F_fn,&F_f
nFaoFagnF[z@%F`\\mC)>FbzF_fn>FagnF[jcm>F_fnF[jcm?(FWFaoFaoFbayF\w@$4-Fie^l6#F`\
emC$>F_fn,&FagnFao*&%"0GFaoF\\emFaoFao>FagnF_fn>FagnFd\em>F_fnFf\em-F^jm6#/-Fb]
o6#FiicmF_fn-F^jm6#/-Fb]o6#-Ff]o6$-Faen6$Fh[lFednFeoF_fn-F^jm6$Fa^em%incannot~b
e~written~as~a~linear~combination~of~the~other~vectorsG@$0FbzF_elC$F\\o@$FgnF`\
o>Fh[lFdfn>FhbsFhjdm@%F[[emC*>FaalF^[em?(FWFaoFaoFbayF\w>Ffg_lFc[em>FagnF[jcm>F
_fnF[jcm?(FWFaoFaoFbayF\wC$>F_fnFj[em>FagnF_fn>FagnFd\em>F_fnFf\em@%F`\\mC(>Fag
nF[jcm>F_fnF[jcm?(FWFaoFaoFbayF\w@$F^]emC$>F_fnFc]em>FagnF_fn>FagnFd\em>F_fnFf\
emFi]emF^^emFg^emFghm-F^jm6#%foThis~is~the~end~of~the~linear~dependence~procedu
re.~Please~change~the~givenG-F^jm6#%invectors~and~execute~the~procedure~again~t
o~learn~this~process~G/F]_lF\]nC^o@%F[anC+F[dcmF^dcm>F^jnFao>FWFaoFcdcm>Faj`lFX
?(F/FaoFaoF/FhdcmC'>FenFaj`l@$FgnF`\o?(F/FaoFaoF/F]ecmC&>FenFaj`l@$FgnF`\oFeecm
>Faj`lFX@$F^ecmC%>FWFa\jlF\fcm>Faj`lFX>F^jnFaj`l>FednF]]`l>F`zFcfcmC%?(FWFaoFao
F[]nF\wC$>Faj`lF^g^m@$F_ecmFjfcm>FednF[]n>F`zFcfcm>Fa^lF`gcm?(FWFaoFaoF`zF\w?(F
hxFaoFaoFednF\w>FdgcmFegcm>FezF]icm>Fh[lFa^l?(FWFaoFaoFednF\w?(FhxFaoFaoF`zF\w>
FfjcmFgjcm>FjbrF[jcm>FegnF[^dm>Fb`p*&&FezFihmFao-Ff]o6$-Fb]o6#-Faen6$Fa^lFaoFeo
Fao?(FWF\]nFaoFednF\w>Fb`p,&Fb`pFao*&F[_`lFao-Ff]o6$-Fb]o6#F_hcmFeoFaoFao-FS6#%
^pHow~do~you~check~if~the~given~set~of~vectors~is~linearly~independent~or~depen
dent?|+G-FS6#%X~~1.~Form~the~zero~combination~c1*v1+c2*v2+...+cn*vn=0|+G-FS6#%f
n~~2.~Visually~inspect~vectors~to~find~a~dependency,~if~any|+G-FS6#%9<~choose~f
rom~1;~or~2;>|+G>F]_lFX>FenF]_l@$FgnFf\_l@'Fgccm-F^jm6#%XVery~good.~This~is~the
~best~way~to~check~the~dependencyGF^aemC$-F^jm6#%`oThis~is~not~the~best~choice.
~In~most~cases~it~would~be~very~difficultG-F^jm6#%\oto~detect~a~linear~dependen
cy~of~vectors~by~inspection.~Choose~1;G-F^jm6#%EInvalid~choice.~The~best~choice
~is~1G-F^jm6#%doThe~zero~combination~of~the~vectors~yields~the~following~vector
~equation:G-F^jm6#/Fb`p-Ff]o6$-Fb]o6#-Faen6$FegnFgerFeoFghmFghm-FS6#%]oThe~prob
lem~reduces~to~solving~a~homogeneous~system~of~equations:|+G-FS6#%hnIf~the~syst
em~has~only~the~trivial~solution,~the~vectors~are|+G-FS6#%;~~~1.~linearly~depen
dent~|+G-FS6#%=~~~2.~linearly~independent.|+GFaeem>F]_lFX>FenF]_l@$FgnFf\_l@'Fg
ccm-F^jm6#%\pNo.~A~trivial~solution~to~the~system~implies~the~vectors~are~linea
rly~independentGF^aem-F^jm6#%^pGood.~A~trivial~solution~to~the~system~implies~t
he~vectors~are~linearly~independentG-F^jm6#%hpInvalid~choice.~A~trivial~solutio
n~to~the~system~implies~the~vectors~are~linearly~independentGFghm-FS6#%WNow~fro
m~the~zero~combination~of~the~vectors,~extract|+G-FS6#%Mthe~homogeneous~system~
of~linear~equations.|+G?(FWFaoFaoF`zF\wC)-FS6$%CPlease~enter~equation~number~(%
d)|+GFW>&F_hpF^pFX?(F/FaoFaoF/4-F:6$Ffiem%)equationGC&>FenFfiem@$FgnFf\_l-FS6#%
DAn~equation~is~expected.~Try~again|+G>FfiemFX>Fi^pF_el?(FhxFaoFaoFednF\w>Fi^p,
&Fi^pFao*&&FezF^zFaoFdgcmFaoFao>&F]_lF^p/Fi^p&Fegn6$FWFger@$0FjjemFfiem-F^jm6$%
DIncorrect.~The~correct~equation~is:GFjjemF\\o@$FgnFf\_l-FS6#%TThe~system~of~li
near~equations~resulting~from~the~|+G-FS6#%Mzero~combination~of~the~vector~equa
tion~is:|+G?(FWFaoFaoF`zF\w-F^jm6#FjjemFghm-FS6%%hnEnter~the~%dx%d~associated~a
ugmented~matrix~in~Maple~format;|+GF`zFger-F^jm6#%VFor~example~type:~AUG:=matri
x([[1,2,2,0],[9,3,0,0]]);G>F]_lFX?(F/FaoFaoF/4-F:Fa`^lC&>FenF]_l@$FgnFf\_l-FS6#
%6A~matrix~is~expected|+G>F]_lFX?(F/FaoFaoF/50Fhc\lF`z0Fg`^lFgerC'-F^jm6#%=Inco
rrect~matrix~dimensions.G-FS6%%=Please~enter~a~%dx%d~matrix|+GF`zFger>F]_lFX>Fe
nF]_l@$FgnF`\o>F\isFao?(F/FaoFaoF/30-F\ao6$-Fb]o6#,&F]_lFaoFegnF[zF\]nF_el1F\is
FgbnC'-FS6#%KThis~is~not~the~correct~matrix.~Try~again|+G>F]_lFX>F\is,&F\isFaoF
aoFao>FenF]_l@$FgnC$F]jmF_jm@%Fe^fmC%-F^jm6#%AVery~good.~The~correct~matrix~isG
F[[uFghmC$-F^jm6#%BIncorrect.~The~correct~answer~is:GF[[u-FS6#%9What~would~you~
do~next?|+G>%(resflagGF_el?(F/FaoFaoF//F]`fmF_elC%>F^jn-F`hmF/@*30-F^bnFi`lFhgc
m5/F^jnFao/F^jnF\]n-F^jm6#%>Please~make~another~selectionGFh`fm-%&menu1G6%Fa^lF
`zFednFi`fm-%&menu2GF_afm/F^jnFgbn-%&menu3GF_afm@$FgnC$F]jmF_jm-FS6#%8Is~the~se
t~of~vectors:|+G-FS6#%=~~~1.~Linearly~independent?|+G-FS6#%;~~~2.~Linearly~depe
ndent?|+G-FS6#%3<~Type~1;~or~2;~>|+G>F]_lFX>FenF]_l@$FgnC$F]jmF_jm>FfelF[fdm@&F
gccm@%-Fie^lFifl-F^jm6#%TCorrect.~The~set~of~vectors~is~linearly~independentGC$
-F^jm6#%^oIncorrect.~The~set~of~vectors~is~linearly~dependent~as~shown~below.G>
FgapFaoF^aem@%Fjbfm-F^jm6#%VIncorrect.~The~set~of~vectors~is~linearly~independe
ntGC$-F^jm6#%\oCorrect.~The~set~of~vectors~is~linearly~dependent~as~shown~below
.G>FgapFaoFghm?(FWFaoFaoFhgcmF\w?(FhxFa\jlFaoFhgcmF\w@$F[hcmC$-F^jm6#%_oNote~tw
o~vectors~are~the~same~and~thus~the~set~is~linearly~dependentGFjhcm@$FbctC)Fcid
m>Fdfn-F`_r6%Fh[lFiidm;FaoF]jdm>FednF_jdm?(FhxFaoFaoFbayF\wC'>Fh[lFdfn>Fh[lFdjd
m>FhbsFhjdm@%F[[emC*>FaalF^[em?(FWFaoFaoFbayF\w>Ffg_lFc[em>FagnF[jcm>F_fnF[jcm?
(FWFaoFaoFbayF\wC$>F_fnFj[em>FagnF_fn>FagnFd\em>F_fnFf\em@%F`\\mC)>FbzF_fn>Fagn
F[jcm>F_fnF[jcm?(FWFaoFaoFbayF\w@$F^]emC$>F_fnFc]em>FagnF_fn>FagnFd\em>F_fnFf\e
mFi]emF^^emFg^em@$F[_emC$F\\o@$FgnF`\o>Fh[lFdfn>FhbsFhjdm@%F[[emC*>FaalF^[em?(F
WFaoFaoFbayF\w>Ffg_lFc[em>FagnF[jcm>F_fnF[jcm?(FWFaoFaoFbayF\wC$>F_fnFj[em>Fagn
F_fn>FagnFd\em>F_fnFf\em@%F`\\mC(>FagnF[jcm>F_fnF[jcm?(FWFaoFaoFbayF\w@$F^]emC$
>F_fnFc]em>FagnF_fn>FagnFd\em>F_fnFf\em-F^jm6#/F\^em-FjelF]^s-F^jm6#/Fa^emFhhfm
Fg^em-F^jm6#%boThis~is~the~end~of~the~interactive~linear~dependence~procedure.~
You~mayG-F^jm6#%hochange~the~given~vectors~and~execute~the~procedure~again~to~l
earn~the~processG/F]_lFgbnCE@%F[anC+F[dcmF^dcm>F^jnFao>FWFaoFcdcm>Faj`lFX?(F/Fa
oFaoF/FhdcmC'>FenFaj`l@$FgnF`\o?(F/FaoFaoF/F]ecmC&>FenFaj`l@$FgnF`\oFeecm>Faj`l
FX@$F^ecmC%>FWFa\jlF\fcm>Faj`lFX>F^jnFaj`l>FednF]]`l>F`zFcfcmC%?(FWFaoFaoF[]nF\
wC$>Faj`lF^g^m@$F_ecmFjfcm>FednF[]n>F`zFcfcm>Fa^lF`gcm?(FWFaoFaoF`zF\w?(FhxFaoF
aoFednF\w>FdgcmFegcm?(FWFaoFaoFhgcmF\w?(FhxFa\jlFaoFhgcmF\w@$F[hcmC%FdhcmFghcmF
jhcm>FezF]icm>FfdlFiicm>FezF^jcm>Fh[lFa^l?(FWFaoFaoFednF\w?(FhxFaoFaoF`zF\w>Ffj
cmFgjcm>F\blF_el?(FWFaoFaoFednF\w>F\blF[[dm>Fa^lF`gcm?(FWFaoFaoF`zF\w?(FhxFaoFa
oFednF\w>FdgcmFegcm>FbxFh[_m?(FWFaoFaoFednF\w>FjgilF][dm>F\fnF[]dm>FjbrF[jcm>Fa
^lF[^dm>FegnFe^dm-F^jm6#%\oSetting~the~linear~dependence~problem~yields~the~aug
mented~matrixG-F^jm6#Fa^dmF\\o-F^jm6#%:whose~row~echelon~form~isGF[[uF\\o>FhgnF
^_dm>FWF``_m?(F/FaoFaoF/Fb_dm>FWF]]`l?(FhxFaoFaoFWF\w?(FfcnFaoFaoFcdrF\w>F[`dmF
\`dm@%F^`dmC(>FWF^jil?(F/FaoFaoF/Fi`dm>FWF]]`l>FhxFao?(F/FaoFaoF/Fbadm>FhxFb]`l
@$Fiadm>FhgnF]bdm>FegnFfevC%>F\]lFcbdm?(FfcnFaoFaoFbayF\w>FgbdmFhbdm@$FjbdmC*-F
^jm6%FbcdmFd`dm%Ithe~above~echelon~form~yields~the~matrixG>FWFg`dm?(F/FaoFaoF/F
i`dm>FWF]]`l>FhxFao?(F/FaoFaoF/Fbadm>FhxFb]`l@$Fiadm>FhgnF]bdm>FegnFfevF[[u>F^y
Fcddm>FfelF[fdm>F_fnF_el?(FWFaoFaoFednF\w>F_fnFifdm@%F`\\m-F^jm6#%_oFrom~this~w
e~deduce~that~the~set~of~vectors~is~linearly~independent.GC*-F^jm6#%YThe~set~of
~vectors~is~linearly~dependent~and~some~can~beG-F^jm6#%Ywritten~as~a~linear~com
bination~of~the~others~as~followsG>FdfnFgidm>FednF_jdm?(FhxFaoFaoFbayF\wC'>Fh[l
Fdfn>Fh[lFdjdm>FhbsFhjdm@%F[[emC*>FaalF^[em?(FWFaoFaoFbayF\w>Ffg_lFc[em>FagnF[j
cm>F_fnF[jcm?(FWFaoFaoFbayF\wC$>F_fnFj[em>FagnF_fn>FagnFd\em>F_fnFf\em@%F`\\mC)
>FbzF_fn>FagnF[jcm>F_fnF[jcm?(FWFaoFaoFbayF\w@$F^]emC$>F_fnFc]em>FagnF_fn>FagnF
d\em>F_fnFf\emFehfmFihfmFg^em@$F[_em@$FgnF`\o>Fh[lFdfn>FhbsFhjdm@%F[[emC*>FaalF
^[em?(FWFaoFaoFbayF\w>Ffg_lFc[em>FagnF[jcm>F_fnF[jcm?(FWFaoFaoFbayF\wC$>F_fnFj[
em>FagnF_fn>FagnFd\em>F_fnFf\em@%F`\\mC(>FagnF[jcm>F_fnF[jcm?(FWFaoFaoFbayF\w@$
F^]emC$>F_fnFc]em>FagnF_fn>FagnFd\em>F_fnFf\emFi]emF^^emFg^emF/6$FenF]`fmF/%)su
bspaceGRF/6HFd[oFhjmF][oF][nF^[nF\jo%#k1G%)alphasetGFee\l%,relationsetG%*vector
setGFgvFe\r%)subsflagGF_[rFgjoFhjoFid^m%%tsetG%*relation1G%*relation2G%'sumvecG
%*relation3G%*subspflagG%#nuG%*relation4GFijqF_b\lFhghlFfghlFgdil%+inrelationG%
#kuG%$ku1G%%ans2G%#s3GFge\l%+scalarflagGFdhmF/C/-F^jm6#%joThe~purpose~of~this~p
rocedure~is~to~demonstrate~the~steps~needed~to~verify~if~aG-F^jm6#%`pgiven~set~
is~a~subspace~or~not.~It~also~includes~an~interactive~version~that~requiresG-F^
jm6#%\presponses~to~questions~posed~and~a~no-step~version~that~yields~the~resul
t~withoutG-F^jm6#%ioany~intermediate~steps.For~more~information~type~?subspace~
at~the~Maple~promptGFghmFee^m-FS6#%;~~~~1.~Demonstration~mode|+G-FS6#%9~~~~2.~I
nteractive~mode|+G-FS6#%5~~~~3.~No-Step~mode|+G>FWFX>FenFW@$FgnF`\o@(Fa\wCN@'F[
anC(-FS6#%8Please~enter~the~set~S|+G-FS6#%EFor~example:~S:=|fr[x,y,z],~x+y-z=0|
hr;|+G>FhxFX>FenFhx@$FgnF`\o@$4-F:6$FhxFc]n-F]o6#%:ERROR!!~A~set~is~expectedGFa
^nC$>FhxFa]n@$FchgmFfhgm-F]o6#%DERROR~!!~Only~one~argument~expectedG>F]_l-Fahp6
#7;F%FajoFi[`mFehvF\\`mFcpFbj[lF`\`mFednF\]lF]joFegnFgrFgj^lFfhzFgddl%"rGF`d^mF
,FbxF\fnFcj^lFdfnFa_\l%"zG>F^jn-Fahp6#7,Fa[nFb[n%#x3G%#x4G%#x5G%#x6G%#x7G%#x8G%
#x9G%$x10GFghm>Ffel-Fahp6#7,Fe[nFf[n%#t3G%#t4G%#t5G%#t6G%#t7G%#t8G%#t9G%$t10G>F
_fnF]r>FagnF]r>F\]lFao>FfcnFao?(F`zFaoFaoF\zF\w@(-F:6$F``il%)relationG>F_fn-F]e
n6$F_fn<#F``il-F:6$F``ilFebnC$>FgcqF``il>FfcnF_el0FfcnF_elC$>&FgcqFg]lF``il>F\]
lFggw>Fgcq-Ff]o6$FgcqFebn>FfcnF_el?(F`zFaoFaoF`ailF\w@$4-F:6$&F_fnFf\l%'linearG
>FfcnFao>Fjfq-Ffa\l6%%(~S~=~|fr~G-Ff]o6$FgcqF[b\l%$~|gr~G?(F`zFaoFaoF`ailF\w>Fj
fq-Ffa\l6%Fjfq-Ff]o6$F]]hmF[b\l%#~,G>Fjfq-Ffa\l6$Fjfq%$~|hr~G>Fjfq-Fcghl6#Fjfq>
F\\qFjfq-F^jm6$%8The~set~you~entered~is~GFjfq>Fbz-Fe_il6$F_fnFgcq>Fez-FijdmF^[l
@&/FezF_elC$-F^jm6$%CThe~set~is~the~trivial~subspace~ofG)Fc`al-F]zF\fqF_jmF^gv>
Fegp-Feh]l6#-Fe^sF^[l>FjbrF]r?(F`zFaoFaoFe_hmF\w>Fjbr-F]en6$Fjbr<#/&FfelFf\l&Fg
cqFf\l?(F`zFaoFaoFe_hmF\w>&FegpFf\l-Fi\l6$FjbrFe`hm>Fegp-Ff]o6$FegpFebn>Fjfq-Ff
a\l6%%'S~=~|fr~G-Ff]o6$FegpF[b\l%>~|gr~where~x~,~y,~z..etc~in~R~|hrG>FjfqFc^hmF
ghm@$/FfcnF_elC$-F^jm6#%@This~set~can~also~be~written~asG-F^jmFd^hm>Fdfn-Ff]o6$
FgcqFahp@$1-FiarFif