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",
I5CM_RemoveAssumptionsR6#%"fGF+F-F/C&-%(readlibG6#%'assumeG@$-F:6$FK%.HasAssump
tionGC$-F>6$-%*substringG6$FK;"""!"#.Fgn-%'RETURNG6#Fgn>8$-%'indetsGFX@%0Fbo<"-
%$mapG6$9!Fbo%%NULLGF/F/F/6",
I5help/text/transtrans-%%TEXTG6,%"~G%MHELP~SCREEN~for~:~Transpose~of~the~transp
oseGFbp%3PACKAGE~:~MatricesG%hoCONTENTS~:~This~function~checks~if~the~transpose
~of~the~transpose~is~equal~toG%8~~~~~~~~~~~the~matrix~AGFbp%T~~~~~~~~~~~~To~Cal
l~:~>~A~:=~matrix([[1,2],[3,3]]);G%G~~~~~~~~~~~~~~~~~~~~~~>~transtrans(A);GFbpF
/,
I0context/cfparseR6#'%#CFG%)CF_CLASSG6=%(arglistG%(ActionsG%+ActionNameG%+Actio
nKeysG%)AncestorG%-ConditionalsG%0ConditionalKeysG%%DataG%,DescriptionG%%fileG%
%headG%+HelpStringG%(IfBlockG%%MenuG%$KeyG%&LocCFG%+MenuStringG%,MenuEntriesG%+
RecognizerG%-ReplacementsG%)SubMenusG%-SubMenusKeysG%#t0G%%tailG%%TestG%)warnin
gsG%"xGF-F/CK@$0%4context/InitializedG%%trueG--FS6#.%-context/initGF/-FS6#%1con
text/cmglobalG-FS6#%.context/cfgetG-FS6#%8context/cfparse/contentG-FS6#%6contex
t/cfparse/menusG-FS6#%;context/cfparse/recognizerG>83FK>%0_EnvContextDataG-%&ta
bleGF/>8%-Fdo6$Fet.%*CF_ACTIONG>8:-%%timeGF/-%)userinfoG6%F[o<#Fip-%(sprintfG6%
QF%7.3f~:~Replacing~%d~Actions~by~Keys.6",&FbuF[oFau!""-%%nopsG6#F[u>8'Fht?&8>F
[uF_sC&>82-Fgs6#-Fjs6$Q+MenuString6"Fev>&Fcv6#Fev-%*CM_ACTIONG6#Fhv>&Fgt6#FbwFe
v-F]t6$FbwFht>87<#-%$seqG6$/&FgtFawFev/Fev-Fio6$%#opG7#-%(entriesG6#Fcv>Fet-%%s
ubsG6$F[xFet>8)-Fdo6$Fet.%/CF_CONDITIONALG-Feu6%F[oFgu-Fiu6%QK%7.3f~:~Replacing
~%d~Conditionals~by~Keys.F\vF]v-F`v6#F_y>F[x-Fio6$%Bcontext/cfparse/splitcondit
ionalsGF_y>F_y-F\y6$F[xF_y>FetF[y>8*Fht?&FevF_yF_sC'>Fhv-Fgs6#Q*ConditionF\v>&F
dzFaw-%/CM_CONDITIONALGFdw>8+-Fio6$RF+F/F/F/@$-F:6$FKFcw&Fgt6#-%/CM_ACTION_DATA
G6#-Fex6#FKF/F/F/Fev>&Fgt6#F][lFev-F]t6$F][lFht>F[x<#-F^x6$F`x/Fev-Fio6$Fex7#-F
hx6#Fdz>Fet-%)CM_CLASSG6#-Fex6#F[y>88-Fdo6$Fet.%+CF_SUBMENUG-Feu6%F[oFgu-Fiu6$Q
D%7.3f~:~Replacing~Submenus~by~Keys.F\vF]v>89Fht?&FevFd]lF_sC%>FhvF[w>&F_^lFaw-
%+CM_SUBMENUGFdw>&Fgt6#Fd^lFev>F[x<#-F^x6$F`x/Fev-Fio6$Fex7#-Fhx6#F_^l>FetF[y>F
et-F\y6$<%/F^qF_]l/%(CF_CODEG%(CM_CODEG/%-CF_SEPARATORG%-CM_SEPARATORGFet-Feu6%
F[oFgu-Fiu6$QD%7.3f~:~Building~Menu~Templates~...F\vF]v-F`t6%Fet7"Fg`l>8=-%*int
erfaceG6#.%*warnlevelG-F[al6#/F]al""!-Feu6%F[oFgu-Fiu6$Q@%7.3f~:~Building~Recog
nizer~...F\vF]v>86-Fct6$Fet-%%evalG6#Fgt>%,ContextDataGF\bl-F[al6#/F^alFi`l-Feu
6%F[oFgu-Fiu6$Q>%7.3f~:~CONTEXTMENU~CompletedF\vF]v-F_o6#-F]bl6#FialF/6$FgtF`bl
F/F\v,
I-context/initR6"6#%.ReadlibDefineG6$%FCopyRight~1997~by~Waterloo~Maple~Inc.G%f
nCopyright~(c)~1997~Waterloo~Maple~Inc.~All~rights~reserved.GF`clCco@$/F^sF_s-F
_o6#F\p>8$FS-F\dl6#.%3context/autoassignG-F\dl6#.%9context/BuildInputStringG-F\
dl6#.%8context/InitializeProcsG-F\dl6#.Fjs-F\dl6#.Fip-F\dl6#.%?context/cfparse/
classtoifblockG-F\dl6#.%<context/cfparse/act_TRIPLESG-F\dl6#.%=context/cfparse/
act_completeG-F\dl6#.%?context/cfparse/act_LINEARFUNCG-F\dl6#.%<context/cfparse
/conditionalG-F\dl6#.F]t-F\dl6#.F`t-F\dl6#.Fct-F\dl6#.F">%*CM_GlobalG-FS6#.Fgs-
F\dl6#.%<convert/CONTEXTMENU/TRIPLESG-F\dl6#.%?convert/CONTEXTMENU/LINEARFUNCG-
F\dl6#%6context/defaultconfigG-F\dl6#.%5context/fixpiecewiseG-F\dl6#.%0context/
fullkeyG-F\dl6#.%3context/isparsableG>%.CM_IsParsableG.-FS6#Fihl-F\dl6#.%1conte
xt/jointextG-F\dl6#.%3context/multiparseG-F\dl6#.F\`l-F\dl6#.FM-F\dl6#.%+CM_Get
VarsG-F\dl6#.%0CM_GetArrayVarsG-F\dl6#.%,CM_GetFuncsG-F\dl6#.%.CM_InertFuncsG-F
\dl6#.%4convert/CONTEXTMENUG-F\dl6#.%<convert/context/ListToStackG-F\dl6#.%<con
vert/context/StackToListG-F\dl6#.%/type/CF_ACTIONG-F\dl6#.%.type/CF_CLASSG-F\dl
6#.%-type/CF_CODEG-F\dl6#.%1type/CF_CODETYPEG-F\dl6#.%4type/CF_CONDITIONALG-F\d
l6#.%2type/CF_SEPARATORG-F\dl6#.%0type/CF_SUBMENUG-F\dl6#.%/type/CM_ACTIONG-F\d
l6#.%.type/CM_BUILDG-F\dl6#.%.type/CM_CLASSG-F\dl6#.%-type/CM_CODEG-F\dl6#.%4ty
pe/CM_CONDITIONALG-F\dl6#.%0type/CM_IFBLOCKG-F\dl6#.%3type/CM_InertFuncsG-F\dl6
#.%2type/CM_SEPARATORG-F\dl6#.%0type/CM_SUBMENUG-F\dl6#.%1type/CM_MathFuncG-F\d
l6#.%/type/CM_MatrixG-F\dl6#.%0type/CM_NOPRINTG-F\dl6#.%2type/CM_PIECEWISEG-F\d
l6#.%3type/HasAssumptionG-F\dl6#.%9type/context/Unbounded@@G-FS6#.F*-F\dl6#.%+A
PPENDPLOTG-F\dl6#.%,APPENDPLOTSG-F\dl6#.%+DELETEPLOTG-F\dl6#.%.EDITSMARTPLOTG-F
\dl6#.%4SMARTPLOTADD_OptionG-F\dl6#.%/SMARTPLOTColorG-F\dl6#.%4SMARTPLOTColor_L
istG-F\dl6#.%3SMARTPLOTGetColorsG-F\dl6#.%6SMARTPLOTFindCoords2DG-F\dl6#.%6SMAR
TPLOTFindCoords3DG-F\dl6#.%-SMARTPLOTFixG>F^sF_sF\pF`cl6%F`glF\ilF^sF`clF`cl,
I)mainmenuRF/6#%)responseG6#%SCopyright~1996~by~Ananda~Gunawardena~&~Elias~Deeb
aGF/C,-%&blankG6#F[o-%'printfG6#%aoChoose,~from~the~following~options,~the~appr
opriate~route~to~proceed.|+G-F^em6#%]oAt~the~prompt~sign,~enter~your~selection~
followed~by~a~semi-colon|+G-F^em6#%S~~~1.~Find~the~inverse~of~the~coefficient~m
atrix.|+G-F^em6#%W~~~2.~Find~the~determinant~of~the~coefficient~matrix.|+G-F^em
6#%jo~~~3.~Find~the~row~echelon~form~of~the~matrix~to~find~a~solution~to~the~sy
stem|+G>Fbo-%)readstatGF/>%'finishGFbo@$5/Fafm%%exitG/Fafm%%EXITGC$-%&printG6#%
4Execution~SuspendedG-F_o6#Fbp-F_o6#FboF/6#FafmF/F/,
I'linmat=F/%&falseGE\[l+%)GeometryGRF/63%"SG%"TG%#S1G%#S2G%#T1G%"iG%"jG%#g1G%#g
2G%#x1G%#x2G%#y1G%#y2G%#t1G%#t2G%%min1G%&indexGFgdmF/C*-Fjfm6#%]pThe~purpose~of
~this~function~is~to~demonstrate~the~effect~of~matrix~multiplicationG-Fjfm6#%`p
on~a~2D~image.~The~image~must~be~entered~as~a~set~of~n~elements~where~n~is~the~
numberG-Fjfm6#%^pof~points~that~make~up~the~image.~The~matrix~is~a~2x2~matrix.~
The~function~displaysG-Fjfm6#%jothe~original~image~and~the~transformed~image.~T
his~function~can~be~used~to~viewG-Fjfm6#%Dreflections,~rotations,~and~scalingGF
jdm@)/9#""#C$@&-F:6$&9"F\em%$setG>FboFbjm-F:6$Fbjm%'matrixG>F[uFbjm@&-F:6$&Fcjm
6#F]jmFdjm>FboF][n-F:6$F][nFhjm>F[uF][n/F\jmF[o@'FfjmC'>F[uFbjm-F^em6#%Aplease~
enter~the~image~as~a~set|+G-Fjfm6#%FFor~example:~~S:=|fr[1,2],[3,2],[2,1]|hr;G>
FboF^fm?(F/F[oF[oF/4-F:6$FboFdjmC$-F^em6#%>A~set~is~expected.~Try~again|+G>FboF
^fmF`jmC'>FboFbjm-F^em6#%;please~enter~the~matrix~T|+G-Fjfm6#%HFor~example:~A:=
matrix([[1,1],[-1,1]]);G>F[uF^fm?(F/F[oF[oF/4-F:6$F[uFhjmC$-F^em6#%<Matrix~expe
cted.~Try~again|+G>F[uF^fm-%&ERRORG6#%MIncorrect~arguments.~Type~?Geometry~for~
helpG/F\jmFbalC*Fg[nFj[n>FboF^fm?(F/F[oF[oF/F_\nC$Fc\n>FboF^fmFi\nF\]n>F[uF^fm?
(F/F[oF[oF/Fa]nC$Fe]n>F[uF^fm-Fj]n6#%Ltoo~many~arguments.~Type~?Geometry~for~he
lpG@%5550-%'rowdimGFav-%'coldimGFav4-F:6$&FboF\em%%listG2""$-F`v6#Ff_n2Fi_nF__n
Fi]nC$-%%withG6#%&plotsG@&3/F__nF]jm/Fj_nF]jmC4>8&-Fhjm6$F]jm-F`vF`gm?(F_yF[oF[
oF]jmF_s?(FdzF[oF[oF[anF_s>&Fh`n6$F_yFdz-%&evalfG6$&&FboF\]lFjyF]jm>8(-%)multip
lyG6$F[uFh`n>FcvFgo?(FdzF[oF[oF[anF_s>Fcv-%&unionG6$Fcv<#-%$colG6$FganFdz>Fet"$
***?(F_yF[oF[oF[anF_s@$2&&FboFjyF\emFetC$>84F_y>FetFjbn>8-&&Fbo6#F^cnF\em>8/&Fc
cnF^[n>FetFfbn?(F_yF[oF[o-F`vFixF_s@$2&&FcvFjyF\emFetC$>F^cnF_y>FetF]dn>8.&&Fcv
FdcnF\em>80&FednF^[n>81-%)textplotG6%7%,&FacnF[oF^vF[o,&FfcnF[oF]jmF[o%/origina
l~imageG/%&alignG<$%&ABOVEG%%LEFTG/%&colorG%$redG>Fhv-F\en6%7%,&FcdnF[oF^vF[o,&
FgdnF[oF\oF[o%2transformed~imageG/Fcen<$Ffen%&BELOWG/Fhen%%blueG>F`[l-%*pointpl
otG6%Fbo/Fhen%$REDG/%*thicknessG"#5>8,-Fhfn6%Fcv/Fhen%%BLUEGF\gn-%(displayG6#<&
F`[lF`gnFhvFjdn3/F__nFi_n/Fj_nFi_nF/F/FagmF/%*trinverseGRF/6+%"AG%%flagG%#c1G%#
r1G%'transAG%&invtAG%%invAG%(transiAG%$ansGFgdmF/C-Fjdm-Fjfm6#%boThe~purpose~of
~this~program~is~to~check~if~the~inverse~of~the~transposeG-Fjfm6#%Wof~a~matrix~
A~is~the~transpose~of~the~inverse;~that~isG-%)continueGF/@$5FffmFdfm-F_oF/@&F]^
nC&-F^em6#%ZEnter~the~matrix.~|frfor~example,~A:=matrix([[1,2],[2,3]])|+G>FboF^
fm?(F/F[oF[oF/4-F:6$FboFhjmC$-F^em6#%?Matrix~is~Expected.~Try~Again|+G>FboF^fm-
Fjfm6$%;The~matrix~you~entered~is~G/Fbo-%&evalmGF`gmFc[nC$>Fbo-%(convertGFgjm-F
jfm6$%6The~given~matrix~is~:GFfjnF_in@$FbinFcin>Fcv-F`_nF`gm>Fh`n-Fb_nF`gm@%3/F
cvFh`n0-%$detGF`gmFbalC6>Fgan-&%'linalgG6#%*transposeGF`gm>F_y-&F_\o6#%(inverse
G6#Fgan>Fdz-Fd\oF`gm>F`[l-F^\oF\]l-Fjfm6$%IThe~inverse~of~the~transpose~inv(trA
)~=~GF_yF_in@$FbinFcin-Fjfm6$%KThe~transpose~of~the~inverse~tr(inv(A))~=~GF`[lF
_in@$FbinFcin-F^em6#%^oIs~the~transpose~of~the~inverse~equal~to~the~inverse~of~
transpose?|+G-F^em6#%CEnter~your~answer~as:~yes;~or~no;|+G>F`gnF^fm?(F/F[oF[oF/
53/F`gn%#noG0-%%normG6$-Fhjn6#,&F_yF[oF`[lF^vF]jmFbal3/F`gn%$yesGF`^oC1-F^em6#%
TTry~to~find~a~matrix~for~which~this~property~holds|+GFjdm-F^em6#%2Enter~the~ma
trix|+GF^fm>Fbo-F\[o6$%"%GFhjm-Fjfm6$%7You~entered~the~matrixGFbo>FcvFb[o>Fh`nF
d[o@$Ff[oC*>FganF]\o>F_yFc\o>FdzFi\o>F`[lF[]oF\]oF_in@$FbinFcinF`]oF_in@$FbinFc
inFjdm-F^em6#%JDoes~this~property~hold~for~this~matrix?|+G-F^em6#%CType~your~an
swer~as~:~yes;~or~no;|+G>F`gnF^fm?(F/F[oF[oF/3F^^o/Fa^oFbalC$-Fjfm6#%QYour~answ
er~is~not~correct.~Try~the~other~choiceG>F`gnF^fmF_in@$FbinFcin-Fjfm6#%ioIndeed
,~the~transpose~of~the~inverse~is~equal~to~the~inverse~of~the~transpose.GFjdm-F
jfm6#%SSeed~of~thought!!!~Does~this~property~always~hold?G-Fjfm6#%WThis~matrix~
is~singular.~We~cannot~check~this~propertyGF/FagmF/%+transtransGRF/6*F`hnF_hn%(
trans_AG%-trans_transAGFahn%#c2GFbhn%#r2GFgdmF/C8-Fjfm6#%^oThe~purpose~of~this~
program~is~to~check~the~validity~of~conjecture~G-Fjfm6$/))F[uFjgmFjgmF[u%9for~a
~given~matrix~A~.~~GF_in@$FbinFcin@&F]^nC&-F^em6#%jnEnter~the~given~matrix.~For
~example,~A:=matrix([[1,2],[2,3]]).|+G>F[uF^fm?(F/F[oF[oF/Fa]nC$F_jn>F[uF^fm-Fj
fm6$Fejn/F[u-FhjnFavFc[nC$>F[uF[[o-Fjfm6$F_[oFhco>Fh`n-F^\oFav>Fcv-F^\o6#Fh`nFj
dm-Fjfm6$%FThe~Transpose~of~the~matrix~trA~is~:~G/Fjbo-FhjnFbdoFjdm-Fjfm6$%MThe
~Transpose~of~the~Transpose~of~the~matrixG/Fibo-FhjnFixF_in@$FbinFcin>Fdz-F`_nF
bdo>Fgan-Fb_nFbdo>F`[l-F`_nFix>F_y-Fb_nFix@%/-Fb^o6$-Fhjn6#,&F[uF[oFcvF^vF]jmFb
al-Fjfm6%%=In~this~case,~the~conjectureGFhbo%'holds.G-Fjfm6#%MIn~this~case,~the
~conjecture~does~not~hold~.GFjdm-Fjfm6#%[oThis~is~the~end~of~the~transtrans~pro
cedure.Please~enter~anotherG-Fjfm6#%\omatrix~and~execute~the~procedure~to~get~a
~feel~for~this~property.G-Fjfm6#%@Try~to~give~a~proof~in~general.GF/FagmF/%)tra
innetGRF/6LFfdmFghn%#MbG%#NbGF^hmF_hm%"kG%"lG%"MG%"NG%&thetaG%"bG%'bthetaG%#UMG
%#UNG%#fvG%#fuG%"uG%"vG%&deltaG%'delta1G%%averG%'deltaMG%'deltaNGFP%#IRG%#ORGF%
%#v1G%(deltaMbG%(deltaNbG%&fnetiG%%netiG%$UMbG%$UNbG%$IRbG%%netjG%&fnetjG%#TOG%
&inputG%+iterationsG%&alphaGFgdmF/Chn-F^em6#%\o~~The~purpose~of~this~procedure~
is~to~simulate~a~neural~network.|+G-F^em6#%E~~More~specifically,~this~procedure
|+G-F^em6#%ho~~~1.~~simulates~and~train~a~neural~net~using~the~back~propagation
~algorithm|+G-F^em6#%eo~~~2.~~checks~if~the~trained~net~recognizes~the~set~of~r
ules~by~producing|+G-F^em6#%Q~~~~~~~an~output~"close~to"~the~targeted~output|+G
-F^em6#%hn~~For~more~details,~please~see~Application~2.3~from~Unit~Two|+GFjdm@$
F]^nC'-F^em6#%enNeural~net~is~trained~using~an~iterative~process.~You~may|+G-F^
em6#%]ochoose~the~number~of~iterations.~How~many~iterations~do~you~want?|+G>8LF
^fm>FafmF`[p@$FcfmFcin@$Fc[n>F`[pFbjm-F^em6#%VPlease~enter~the~input~rules~in~t
he~form~of~a~matrix|+G-F^em6#%\oFor~example,~matrix([[1,0,1,0,0,1,0],[1,0,0,1,0
,0,1]])~represent|+G-F^em6#%intwo~input~rules.~(Input~rules~(a)~and~(b)~of~Appl
ication~2.3)|+G>8KF^fm>FafmF_\p@$FcfmFcin?(F/F[oF[oF/4-F:6$F_\pFhjmC&-F^em6#%1M
atrix~expected|+G>F_\pF^fm>FafmF_\p@$FcfmFcin>8G-Fa\o6#F_\p-F^em6#%enPlease~ent
er~the~targeted~output~of~each~rule~as~a~matrix|+G-F^em6#%MFor~example,~matrix(
[[1,0,0,0],[0,1,0,0]]);|+G>8JF^fm>FafmFh]p@$FcfmFcin?(F/F[oF[oF/4-F:6$Fh]pFhjmC
&Fg\p>Fh]pF^fm>FafmFh]p@$FcfmFcin>Fev-Fa\o6#Fh]p>8M.Fg^p-F^em6#%ZEnter~the~scal
ing~function~as~f:=t->1/(1+exp(-alpha*t));|+GF^fm>8<RF+F/6$%)operatorG%&arrowGF
/*$,&F[oF[o-%$expG6#,$*&T#F[oFKF[oF^vF[oF^vF/F/6$FcioFg^p-Fjfm6$%9You~entered~t
he~functionG-F]_pF+-F^em6#%hoChoose~alpha~appropriately~so~that~the~net~can~be~
trained~with~a~reasonable~|+G-F^em6#%[pnumber~of~iterations.~In~this~procedure~
alpha=50~is~a~good~choice.~However,~you|+G-F^em6#%@may~enter~your~choice~of~alp
ha|+G-F^em6#%@Please~enter~a~value~for~alpha|+G>Fg^pF^fm@%/Fg^p"#]C$>F]_pRF+F/F
__pF/*$,&F[oF[o-Fe_p6#,$FK!#]F[oF^vF/F/F/-Fjfm6$%8The~scaling~function~isGF^`pC
$>F]_pRF+F/F__pF/Fb_pF/F/Fj_pFhap>Fh`n-Fhjm6#7&7&$F^vF\o$F[oF\o$F]jmF\oFdbp7&Fc
bpFdbpFebp$Fi_nF\o7&FcbpFgbpFebpFdbp7&FcbpFdbpFgbpFdbp>Fcv-Fhjm6#7)7%Fcbp$F\oF\
oFcbp7%FdbpFebpFgbp7%FdbpFgbpFdbp7%FdbpFdbpFebp7%FgbpFebpFdbp7%FebpFdbpFebp7%Fc
bpFgbpFdbp-Fjfm6$%enThe~original~weights~of~the~input~layer~is~a~random~matrixG
/FcvF\eo-Fjfm6$%fnThe~original~weights~of~the~output~layer~is~a~random~matrixG/
Fh`nFgdoF_in@$FcfmFcin-F^em6#%WWe~start~training~the~net~and~updating~of~the~we
ights|+G-F^em6#%Fusing~the~back~propagation~algorithm|+G?(FfcnF[oF[oF`[pF_sC$?(
F`[lF[oF[o-F`_nF`]pF_sC8>8H-Fian6$-Fa\o6#-F\[o6$-Fcbn6$F^]pF`[lFhjmFcv>8I-%'vec
torG6#7&F[o-F]_p6#&F[ep6$F[oF[o-F]_p6#&F[ep6$F[oF]jm-F]_p6#&F[ep6$F[oFi_n>Fial-
Fian6$-Fa\o6#-F\[o6$FeepFhjmFh`n>Fet-Fgep6#7&-F]_p6#&FialF]fp-F]_p6#&FialFafp-F
]_p6#&FialFefp-F]_p6#&Fial6$F[o""%-F^em6$%FThis~is~net~output~for~input~rule~%d
|+GF`[l-Fjfm6#Fet>F[x-Fgep6#F^hp?(F_yF[oF[oF^hpF_s>&F[xFjy*(&FetFjyF[o,&F[oF[oF
[ipF^vF[o,&&-Fcbn6$FevF`[lFjyF[oF[ipF^vF[o>8A-Fhjm6$F^hpF^hp?(F_yF[oF[oF^hpF_s?
(FganF[oF[oF^hpF_s>&Fbip6$F_yFgan,$*&FihpF[o&FeepFg\oF[o$Fi_nF^v/F`gn-Fhjn6#-Fa
\o6#Fbip>8E-Fhjn6#,&FajpF[oFh`nF[o>Fh`n-Fhjn6#Fdjp>F_^l-Fgep6#Fi_n?(F_yF[oF[oFi
_nF_s>&F_^lFjy-%$sumG6$*&&F[xF\]lF[oF_anF[o/Fdz;F[oF^hp>Fd]lF\[q?(F_yF[oF[oFi_n
F_s>&Fd]lFjy*(&FeepFjyF[o,&F[oF[oF]\qF^vF[oF`[qF[o>8B-Fhjm6$""(Fi_n?(FganF[oF[o
Fc\qF_s?(F_yF[oF[oFi_nF_s>&F`\q6$FganF_y,$*&F[\qF[o&FbepFg\oF[oF]jp/F`\q-Fhjn6#
F`\q>8F-Fhjn6#,&F`\qF[oFcvF[o>Fcv-Fhjn6#F`]q@$2FfcnF`[p-Fjfm6$%8This~is~iterati
on~roundG,&FfcnF[oF[oF[oF_in@$FcfmFcin-Fjfm6#%?These~are~the~trained~matricesG-
Fjfm6#%QThe~adjusted~weight~matrix~of~the~input~layer~isG-Fjfm6#FicpFjdm-Fjfm6#
%RThe~adjusted~weight~matrix~of~the~output~layer~isG-Fjfm6#F]dp-Fjfm6#%foDo~you
~want~to~check~whether~the~above~matrices~actually~do~train~the~net?|+G-F^em6#%
4Answer~yes;~or~~no|+G>F[uF^fm>FafmF[u@$FcfmFcin@$/F[uFi^oC+Fjdm?(FganF[oF[oFhd
pF_sC=-F^em6$%HEnter~your~input~rule%d~as~a~1x7~matrixGFgan>FboF^fm>FafmFbo@$Fc
fmFcin?(F/F[oF[oF/F[jnC&Fg\p>FboF^fm>FafmFbo@$FcfmFcin>Ffcn-Fian6$FboFcv-Fjfm6$
%;The~intermediate~output~isG/Ffcn*&FgjnF[oF\eoF[oFjdm>8?-Fgep6#7&F[o&FfcnF]fp&
FfcnFafp&FfcnFefp>F]_pRF+F/F__pF/Fb_pF/F/Fj_p>F]aq-Fgep6#7&-Fban6#-F]_p6#&F]aqF
\em-F]_p6#&F]aqF^[n-F]_p6#&F]aqF][q-F]_p6#&F]aqFfhp-Fjfm6$%_oThe~firing~of~the~
intermediate~output~by~the~scaling~function~yieldsGF]aqFjdm>F]aq-F\[o6$F]aqFhjm
>Fial-Fian6$-Fa\o6#F]aqFh`n-Fjfm6#%hoMultiplying~the~fired~intermediate~output~
by~the~updated~output~matrix~yieldsG-Fjfm6#/Fial*&-Fhjn6#FacqF[oFgdoF[oFjdm>Fia
l-F\[o6$FialFgep>8@-Fgep6#7&-Fban6#-F]_p6#&FialF\em-F]_p6#&FialF^[n-F]_p6#&Fial
F][q-F]_p6#&FialFfhp-Fjfm6$%2The~net~output~isGF`dqFjdm-F^em6$%FCompare~with~th
e~targeted~output~%d~|+GFgan-Fjfm6#-Fcbn6$FevFganFjdmF_in@$FcfmFcin-F^em6#%goIs
~the~net~output~within~a~"reasonable~tolerance"~from~the~targeted~output?|+G-F^
em6#%Jfor~each~input~rule?~Answer:~yes;~or~no;|+GFjdm>F[uF^fm>FafmF[u@$FcfmFcin
@%/F[uF_^o-Fjfm6#%_oYou~may~need~to~choose~more~iteration~in~trainnet()~to~trai
n~the~netGC%-Fjfm6#%FThe~updated~input~and~output~matricesG-Fjfm6$FicpF]dp-Fjfm
6#%Uhave~the~proper~weights~and~the~net~has~been~trainedGFjdm-Fjfm6#%]oYou~may~
try~to~train~the~net~with~different~input~and~output~rulesGF/FagmF/%)LUdecompGR
F/6N%&AUGGGG%$n11GFghnF`hnF^hmF_hmFbgo%"nG%$numGF_hn%#A1G%#b1G%$AUGG%(colflagGF
ihm%'norowsG%#IdG%&lowerG%$pdtG%%tempG%%EpdtGFfdm%'secmulG%%rsetG%)firstmulG%)f
irstrowG%'secrowG%%pdt1G%+multiplierG%'rownumG%%row2G%%leftG%%row1G%)typeflagG%
'rowaddG%(tempaugG%#L1G%#u1G%*undocountG%(leftrowG%'assrowG%(rowsaveG%"mG%%Rset
GFgdmF/C2Fjdm-Fjfm6#%goThe~purpose~of~this~procedure~is~to~demonstrate~the~step
s~needed~to~find~theG-Fjfm6#%[pLU~decomposition~of~a~given~matrix~and~to~intera
ct~with~the~function~to~find~theG-Fjfm6#%]pLU~decomposition~yourself.~For~more~
information~type~?LUdecomp~at~the~Maple~promptGFjdm@%F]^nC$-F^em6#%hnEnter~a~ma
trix.~For~example~type~:~A:=matrix([[0,1],[3,3]]);|+G>FcdnF^fm>FcdnFbjm?(F/F[oF
[oF/4-F:6$FcdnFhjmC&-F^em6#%<Matrix~Expected.~Try~again|+G>FcdnF^fm>FafmFcdn@$F
cfm-F_o6#%$ByeG>FcvFbal?(FganF[oF[o-%%rankG6#FcdnF_s@$/-Fj[o6#-%*submatrixG6%Fc
dn;F[oFganFi\rFbal>FcvF[o@$/FcvF[oC&Fjdm-Fjfm6%%,The~matrix~GFcdn%I~does~not~ha
ve~a~unique~LU~factorizationG-Fjfm6#%DType~?LUdecomp~for~more~informationGFjdm-
Fjfm6#%CSelect~one~of~the~following~modes:G-F^em6#%:~~~1.~Demonstration~mode|+G
-F^em6#%8~~~2.~Interactive~mode|+G-F^em6#%V~~~3.~No~Intermediate~steps.~Just~gi
ve~me~the~answer|+G>Fh`nF^fm@(/Fh`nF[oCT>FhvF[o@$Fc[n>FgdnF[[o@$F]^n>Fgdn-F\[oF
b[r>Fbo-Fhjn6#Fgdn>F`[l-%(vectdimG6#-Fcbn6$FgdnF[o>FetF`[l>F^cn-Fhjm6$FetFet?(F
ganF[oF[oFetF_sC$?(F_yF[oF[oFetF_s>&F^cnFh\qFbal>&F^cn6$FganFganF[o>85-Fgep6#,$
*&F`[lF[o,&F`[lF[oF[oF[oF[o#F[oF]jm>F[uFd`r?(FganF[oF[oF[uF_s>&Fa`rFg\oFf_r>&Fa
`rF\emF^cn-Fjfm6#%RThe~LU-Decomposition~of~the~matrix~is~obtained~byG-Fjfm6$%Ep
erforming~the~gauss~elimination~on:GFgdnF_in@$55Fdfm/Fafm%%ExitGFffmFj[r>F`[l-F
b_nF]_r>8N-F`_nF]_r@$2F\brF`[l>F`[lF\br?(FganF[oF[oF`[lF_sC)>FdzFgan?(F/F[oF[oF
/3/&Fgdn6$FdzFganFbal2FdzF\br>Fdz,&FdzF[oF[oF[o>FjdnFbal?(F_y,&FganF[oF[oF[oF[o
F\brF_s@$0&FgdnFiipFbal>FjdnF[o@$0&FgdnF_`rFbalC$@$/FjdnF[oC$-Fjfm6$%A~Perform~
elimination~on~column~:GFgan-Fjfm6#%E************************************G@$0Fe
crF[oC.Fjdm>F[xFecr-Fjfm6(%+Divide~rowGFgan%4of~the~given~matrixGFgdn%#byGFecr>
Fgdn-%'mulrowG6%FgdnFgan*$FecrF^vFjdm-Fjfm6$%4to~get~the~matrix~:GFgdn>&Fa`rFdw
-F[er6%F^cnFgan*$F[xF^v-Fjfm6$%RPerform~the~same~operation~on~the~identity~matr
ixG-FhjnFdcn-Fjfm6$%Lto~get~the~corresponding~elementary~matrix:G/&%"EGFdw-Fhjn
6#Fber>Fhv,&FhvF[oF[oF[oF_in@$Fear-F_o6#%*Try~LaterGFjdm?(F_yF^crF[oF\brF_s@$F`
crC.>F[x,$FacrF^v-Fjfm6*%-Multiply~rowGFganFgdrFgdn%$~byGF\gr%4and~add~that~to~
rowGF_y>Fgdn-%'addrowG6&FgdnFganF_yF\grFjdm-Fjfm6$%3to~get~the~matrix:GFgdn>Fbe
r-Fdgr6&F^cnFganF_yF[xFjdm-Fjfm6$FherF^cnFjer>FhvFcfrF_in@$FearFj[r@$30&Fgdn6$F
`[lF`[lFbal0FchrF[oC*-Fjfm6(%(The~rowGF`[l%.of~the~matrixGFgdn%2~is~multiplied~
byG*$FchrF^v>Fber-F[er6%F^cnF`[lF\ir>F[xF\ir>Fgdn-F[er6%FgdnF`[lF\irFjdm-Fjfm6(
FihrF`[lFjhrF^cnF[irF[xFjer>FhvFcfr>FialF]ar>Fd]l&F_frF\em?(FganF]jmF[o,&FhvF[o
F^vF[oF_sC$>Fial-Fhjn6#-%#&*G6$F[arFial>Fd]l*&&F_frFg\oF[oFd]lF[o-Fjfm6#%[pAppl
ying~elementary~row~operations~on~the~given~matrix~to~obtain~its~row~echelonG-F
jfm6#%ioform~U,~is~equivalent~to~pre-multiplying~the~matrix~by~the~elementary~m
atricesG>%"UG.F][s-Fjfm6#/*&Fd]lF[o-Fhjn6#FacnF[oF][sFjdm-Fjfm6$%>The~upper~tri
angular~matrix~:G/F][s-Fhjn6#-&F_\o6#%*gausselimGF]_r-Fjfm6#%Ois~the~row~echelo
n~form~of~the~original~matrixGF_in@$FearFj[rFjdm>%"PG.Fd\s-Fjfm6$%SThe~product~
of~the~elementary~matrices~is~given~byG/Fd\sFd]l-Fjfm6$%4Which~is~equal~to~:G/F
d\s-FhjnF\cl>Fd\sFialF_in@$FearFj[r-Fjfm6#%hnThe~lower~triangular~matrix~L~is~t
he~inverse~of~this~matrix~PG-Fjfm6#/F'-Fd\oF\clFjdm>F'Fg]sF_in-Fjfm6#%WTherefor
e~the~LU~decomposition~of~the~given~matrix~is:G-Fjfm6#/Fgjn*&-Fhjn6#F'F[oF\_rF[
oFjdm-Fjfm6#%doThis~is~the~end~of~the~~LU~decomposition~demostration~version.~T
ry~other~G-Fjfm6#%Bexamples~to~learn~this~procedure.G/Fh`nF]jmCao>8O-Fgep6#71%#
R1G%#R2G%#R3G%#R4G%#R5G%#R6G%#R7G%#R8G%#R9G%$R10G%$R11G%$R12G%$R13G%$R14G%$R15G
>8;-Fgep6#71FbhnFabo%#r3G%#r4G%#r5G%#r6G%#r7G%#r8G%#r9G%$r10G%$r11G%$r12G%$r13G
%$r14G%$r15G>FhvF[o@$Fc[n>FgdnF[[o@$F]^n>FgdnFj^r>FboF\_r>F`[lF__r>FetF`[l@$Fh^
sC*>FetF__r>F^cnFf_r?(FganF[oF[oFetF_sC$?(F_yF[oF[oFetF_s>F\`rFbal>F^`rF[o>Fa`r
Fb`r>F[uFd`r?(FganF[oF[oF[uF_s>F[arFf_r>F]arF^cn-%*gaussmenuGF/>F_^l%&BEGING-Fj
fm6$%7The~orginal~matrix~is:GFgdnFjdm>FcvF[o>FhvF[o?(F/F[oF[oF/30FcvFbal0FcvF]j
mC*-F^em6#%hpPlease~enter~~at~each~step~a~row~operation~as~in~the~above~menu~to
~get~the~LU-~decomposition|+G-F^em6#%jn<enter~operations~as~Ri~<>~Rj~or~Ri~=~c*
Rj~+~Ri~or~Ri~=~c*Ri~>|+G>F_^lF^fm>FafmF_^l@$5FcfmFgar-F_o6#%&AdiosG@-5/F_^lFgf
m/F_^lFefmC$>FcvFbal>Fh]pFbal5/F_^l%&rowecG/F_^l%&ROWECGC'>FgdnF[\s?(FganF[oF[o
FetF_s@$Fdcr>FgdnFjdr-Fjfm6$%DThe~matrix~in~row~echelon~form~is~:GFgdn>FcvF]jm>
Fh]pFbal5/F_^l%%undoG/F_^l%%UNDOGC%@(/FdjpF[o>Fgdn-%(swaprowG6%Fgdn8DF`\q/FdjpF
]jm>Fgdn-F[er6%FgdnFbip*$F`dqF^v/FdjpFi_n?(FganF[oF[oFjarF_s>&Fgdn6$F`[pFgan&Fg
^pFg\o@%/Fh]pFbalC$-Fjfm6$%FBefore~your~selection~the~matrix~was~GFgdn>Fh]p,&Fh
]pF[oF[oF[o-Fjfm6#%Xonly~one~level~of~undo~is~permitted~in~the~demo~versionG>Fh
vF[jr-F:6$-%$rhsGFc_l%)monomialGC%?(FganF[oF[oFetF_s@$50-%&coeffG6$F]hs&F_`sFg\
oFbal0-Ffhs6$F]hs&F[_sFg\oFbalC%@$Fdhs>F`dqFehs@$Fihs>F`dqFjhs>FbipFgan@%/F`dqF
[oC.>F`\qFbip>8C-%$lhsGFc_l?(FganF[oF[oFetF_sC$@$/FhhsFhis>FcfsFgan@$/F\isFhis>
FcfsFgan>FgdnF`fs-Fjfm6&%0Interchange~rowGFcfs%$andGF`\q-Fjfm6$%8This~is~the~ne
w~matrix:GFgdnFjdm>Fber-Fafs6%F^cnFcfsF`\q-Fjfm6$%CApply~the~same~row~operation
s~to~:GF^cn-Fjfm6$%PWe~obtain~the~corresponding~elementary~matrix~:GF]fr>FdjpF[
o>FhvFcfrC$?(FganF[oF[oFetF_s@$50-Ffhs6$FiisFhhsFbal0-Ffhs6$FiisF\isFbal>F_\pFg
an@%/F_\pFbipC+>Fgdn-F[er6%FgdnFbipF`dq-Fjfm6&F_grFbipFhdrF`dq-Fjfm6$%7This~is~
the~new~matrixGFgdnFjdm>Fber-F[er6%F^cnFbipF`dq-Fjfm6$%IBy~applying~the~same~ro
w~operations~to~:GF^cn-Fjfm6$%Pwe~obtain~the~corresponding~elementary~matrix~:G
F]fr>FdjpF]jm>FhvFcfr-Fjfm6#%foinvalid~row~multiplication.~Multiplied~row~must~
be~assigned~to~the~same~rowG>Fh]pFbal-F:6$F]hs%(polynomGC0>Fi`lFbal>FevFbal?(Fg
anF[oF[oFetF_sC$@$Fi[t@&Fj[t>F`[pFganF]\t>F`[pFgan@$Fchs@%/Fi`lFbalC$>Fi`lFgan@
&Fdhs>F]_pFehsFihs>F]_pFjhsC$>FevFgan@&Fdhs>FauFehsFihs>FauFjhs>Fg^p-%$rowG6$Fg
dnF`[p?(FganF[oF[oFjarF_s>F\gs&-Fhjn6#,&*&F]_pF[o-Ff_t6$FgdnFi`lF[oF[o*&FauF[o-
Ff_t6$FgdnFevF[oF[oFg\o>Fber-Fdgr6&F^cnFi`lFevF]_p@'3/F]_pF[o/Fi`lF`[p-Fjfm6(%.
multiply~row~GFevFhdrFau%/and~add~to~rowGFi`l3/FauF[o/FevF`[p-Fjfm6(F]atFi`lFhd
rF]_pF^atFev-F^em6-%G~~~~~~~%s~%d~%s~%d~%s~%d~%s~%d~%s~%d~|+G%-multiply~rowGFi`
lFhdrF]_p%(and~rowGFevFhdrFau%6add~and~assign~to~rowGF`[p-Fjfm6$%4The~new~matri
x~is~:GFgdnFjdm-Fjfm6$%QApply~the~same~row~operations~to~identity~matrixGF^cn-F
jfm6$%Pto~obtain~the~corresponding~elementary~matrix~:GF]fr>FdjpFi_n>Fh]pFbal>F
hvFcfrC$-Fjfm6#%@incorrect~selection.~Try~again.G>Fh]pFbal>F^]pF[\s@$/-Fb^o6$-F
hjn6#,&F^]pF[oFgdnF^vF[oFbalC%>F_yF]br?(F/F[oF[oF//&Fgdn6$F_yF_yFbal>F_y,&F_yF[
oF^vF[o@$3/FgctF[oFbcsC%F_in-Fjfm6$%LYou~obtained~the~matrix~in~row~echelon~for
mGFgdn>FcvF]jm@$3/FcvFbalFbcs-Fjfm6#%Kwarning.~Matrix~is~not~in~row~echelon~for
mGFjdm-F^em6#%HWhat~is~the~upper~triangular~matrix~U?|+G-Fjfm6#%LEnter~your~ans
wer~as~a~matrix.~For~example,G-Fjfm6#%Atype:~>U:=matrix([[1,2],[0,2]]);GFjdm>F]
[sF^fm?(F/F[oF[oF/4-F:6$F][sFhjmC&>FafmF][s@$FcfmFcin-F^em6#%1Matrix~Expected|+
G>F][sF^fm@&FdetC$Fe]n>F][sF^fm50-F`_r6#-Ff_t6$F][sF[o-F`_r6#-FcbnFfft0Fcft-F`_
r6#-Ff_tFc_rC&-Fjfm6#%TThe~matrix~is~not~square~or~dimensions~incompatibleG-Fjf
m6#%0Enter~a~matrix|+G>F][sF^fm@$FcfmFcin@%0-Fb^o6$-Fhjn6#,&FgdnF[oF][sF^vF[oFb
alC%-Fjfm6#%PYour~answer~is~incorrect.~The~correct~matrix~isG-FjfmF]_r-Fjfm6#%N
which~is~the~matrix~A~in~its~row~echelon~formG-Fjfm6$%?Good!~The~correct~matrix
~is~:~GFgdnFjdm>FialF]ar>F]aqFf_r>F]aqFial?(FganF]jmF[oF[jrF_sC$>Fial-Fhjn6#-Fa
jr6$F[arF]aq>F]aqFialFjdm-F^em6#%P~~How~would~find~the~lower~triagular~matrix~L
?|+G-F^em6#%en~~~~~~1.~Take~the~inverse~of~the~upper~triangular~matrix.|+G-F^em
6#%`o~~~~~~2.~Take~the~inverse~of~the~product~of~the~elementary~matrices.|+G-F^
em6#%A~~~~~~3.~Take~the~inverse~of~A.|+G-Fjfm6#%=<choose~from~1;~or~2;~or~3;>GF
jdm>Fh`nF^fm@$FcfmFcin@%0Fh`nF]jmC$-Fjfm6#%XIncorrect.~The~lower~triangular~mat
rix~L~is~obtained~byG-Fjfm6#%gntaking~the~inverse~of~the~product~of~the~element
ary~matricesGC$-Fjfm6#%+Very~Good!GF_inFjdm-Fjfm6$%ZThe~product~of~the~elementa
ry~matrices~E1,~E2,~..~is~P~=~GFial>Fd\sFialFjdm-F^em6#%HWhat~is~the~lower~tria
ngular~matrix~L?|+G-F^em6#%8~~~~~~1.~inverse~of~P.|+G-F^em6#%R~~~~~~2.~transpos
e~of~the~row~echelon~form~of~P.|+G-F^em6#%N~~~~~~3.~product~of~the~elementary~m
atrices.|+GF`jtFjdm>F[epFg]s>Fh`nF^fm>FafmFh`n@$FcfmFcin@%0Fh`nF[oC$-Fjfm6#%gnI
ncorrect.~The~lower~triangular~matrix~L~is~the~inverse~of~PG-Fjfm6#/)Fd\s%#-1G-
Fhjn6#F[epC$F_[uF_in>F'F[ep>F][sFgdn-F^em6#%TTo~verify~the~answer,~multiply~the
~matrices~L~and~UG-Fjfm6$%$LU=G/*&F`^sF[o-Fhjn6#F][sF[o-Fhjn6#-Fajr6$F'F][sFjdm
-Fjfm6#/%OIs~this~product~the~same~as~the~given~matrix~AGF`^u-F^em6#%4Answer~ye
s;~or~no;|+G>Fh`nF^fm>FafmFh`n@$FearFcin@%/Fh`nF_^o-Fjfm6#%1Redo~the~problemGC$
-Fjfm6#%WYou~obtained~the~LU~decomposition~of~the~given~matrix:G-Fjfm6#/F_^sF`^
uFjdm-Fjfm6#%foThis~is~the~end~of~the~interactive~LUdecomp~procedure.~Please~en
ter~anotherG-Fjfm6#%hnmatrix~and~execute~the~procedure~to~learn~the~steps~invol
ved.G/Fh`nFi_nC6>FhvF[o@$Fc[n>FgdnF[[o@$F]^n>FgdnFj^r>FboF\_r>F`[lF__r>FetF`[l>
F^cnFf_r?(FganF[oF[oFetF_sC$?(F_yF[oF[oFetF_s>F\`rFbal>F^`rF[o>Fa`rFb`r>F[uFd`r
?(FganF[oF[oF[uF_s>F[arFf_r>F]arF^cn?(FganF[oF[o,&F`[lF[oF^vF[oF_sC%?(FdzFganF[
oF`[lF_s@$0FgbrFbalC%@$0FdzFganC(>Fber-Fafs6%F^cnFdzFgan>Fgdn-Fafs6%FgdnFdzFgan
Fjdm>FhvFcfrF_in@$FearFefrFjdm%&breakG@$FdcrC%>FjdnF[o?(F_yF^crF[oF`[lF_s@$F`cr
>FjdnFbal@$FadrC%>Fber-F[er6%F^cnFganF]er>FgdnFjdr>FhvFcfr?(F_yF^crF[oF`[lF_s@$
F`crC%>Fber-Fdgr6&F^cnFganF_yF\gr>FgdnFcgr>FhvFcfr@$FahrC%>FberF^ir>FgdnFbir>Fh
vFcfr>FialF]ar?(FganF]jmF[oF[jrF_s>FialF^jr>Fd\sFial>F'Fg]s-Fjfm6#%NThe~LU~deco
mposition~of~the~given~matrix~is~:GF\^sF/6&FafmFd\sF'F][sF/%*matrixmulGRF/6'F_h
n%"BGF^hmF_hmFehqFgdmF/C+-Fjfm6#%XThe~purpose~of~this~procedure~is~to~show~the~
process~ofG-Fjfm6#%enmultiplying~two~matrices~A~and~B~and~obtaining~the~entries
G-Fjfm6#%9of~the~product~matrix~ABGFjdm@)F[jmC$>FboFbjm>F[uF][nFc[nC(>FboFbjm-F
jfm6#/%<The~matrix~you~entered~is~AGFgjn-F^em6#%;Please~enter~the~matrix~B|+G>F
[uF^fm?(F/F[oF[oF/Fa]nC$F_jn>F[uF^fm-Fjfm6#/%<The~matrix~you~entered~is~BGFicoF
]^nC*-F^em6#%<Please~enter~the~matrix~~A|+G>FboF^fm?(F/F[oF[oF/F[jnC$F_jn>FboF^
fmFcjn-F^em6#%BPlease~enter~the~second~matrix~B|+G>F[uF^fm?(F/F[oF[oF/Fa]nC$F_j
n>F[uF^fmFfco-Fj]n6#%(Error!!G@%/Fd[oF__nC'>Fgan-Fhjm6$Fb[oFa_n-Fjfm6#/%DThe~pr
oduct~of~the~two~matrices~AB~G*&FgjnF[oFicoF[o-Fjfm6#%7is~obtained~as~followsG?
(Fh`nF[oF[o-F`_nFg\oF_s?(FcvF[oF[o-Fb_nFg\oF_s>&Fgan6$Fh`nFcv%"*G?(Fh`nF[oF[oFb
[oF_s?(FcvF[oF[oFa_nF_sC+F_in@$FbinFcin-F^em6%%inMultiply~row~%d~of~the~matrix~
A~and~column~%d~of~the~matrix~B|+GFh`nFcv-F^em6%%Pto~get~the~(%d,%d)-th~entry~o
f~the~product~AB~|+GFh`nFcv-Fjfm6#/*&-Fhjn6#-Ff_t6$FboFh`nF[o-Fhjn6#-F\[o6$-Fcb
n6$F[uFcvFhjmF[o-%*innerprodG6$FfjuF\[vFjdm>FciuF^[v-Fjfm6#%Land~write~the~entr
y~into~the~product~matrixG-Fjfm6#/%#ABG-FhjnFg\o-Fjfm6#%ZThe~matrix~dimensions~
are~incompatible~for~multiplicationGFjdm-Fjfm6#%HEnd~of~matrix~multiplication~p
rocedure.G-Fjfm6#%KTry~your~own~examples~to~learn~the~processGF/FagmF/%*trprodu
ctGRF/62F`hnF_hnF^euF^hq%#B1GFahnF`boFbhnFabo%(mult_ABG%)trans_ABGF^bo%(trans_B
G%&tA_tBGFghn%&tB_tAGFgdmF/C3>F[u.F[u>Fh`n.Fh`n-Fjfm6#%YThe~purpose~of~this~pro
gram~is~to~check~the~conjecture~:G-Fjfm6#%Yif~the~transpose~of~the~product~of~t
wo~matrices~is~equalG-Fjfm6#%Lto~the~product~of~their~transpose;~that~is,G-Fjfm
6#/)Fh[vFjgm*&FjboF[o)Fh`nFjgmF[oF_in@$FbinFcin@&F]^nC*-F^em6#%aoPlease~Enter~t
he~first~matrix.~For~example,~A:=matrix([[1,2],[2,3]]).|+G>FcvF^fm?(F/F[oF[oF/4
-F:6$FcvFhjmC$F_jn>FcvF^fm-Fjfm6$Fejn/F[uF\eo-F^em6#%BPlease~Enter~the~second~m
atrix~B|+G>FganF^fm?(F/F[oF[oF/4-F:6$FganFhjmC$F_jn>FganF^fm-Fjfm6$Fejn/Fh`nFi[
vF[jmC'>FcvF[[o>Fgan-F\[oFa[n-Fjfm6$%6The~first~matrix~is~:GF__vFjdm-Fjfm6$%=Th
e~second~given~matrix~is:~GF\`vF_in@$FbinFcin>F`[lFceo>F_yFeeo>F`gnF_iu>FdzFaiu
@%3/F`gnF_y/F`[lFdzC=>Facn-&F_\o6#Fian6$FcvFgan>Fcdn-F^\oFd[s>Ffcn-F^\oFix>Fgdn
-F^\oFg\o>Fjdn-Fcav6$FfcnFgdn>Fet-Fcav6$FgdnFfcn-Fjfm6$%OThe~transpose~of~the~p
roduct~transpose(AB)~is~G/F]^v-FhjnFa\rF_in@$FbinFcin-Fjfm6#%ZThe~product~of~th
e~transpose~transpose(A)transpose(B)~is~G-Fjfm6#-Fhjn6#FjdnF_in@$FbinFcin-Fjfm6
#%ZThe~product~of~the~transpose~transpose(B)transpose(A)~is~G-Fjfm6#-FhjnFchpF_
in@$FbinFcin-F^em6#%enWhich~one~of~the~following~results~will~then~hold~for~any
|+G-F^em6#%fnmatrices~A~and~B?~Enter~your~answer~by~selecting:~1;~or~2;|+G-F^em
6#%Q~~~1.~transpose(AB)~=~transpose(A)*transpose(B)|+G-F^em6#%Q~~~2.~transpose(
AB)~=~transpose(B)*transpose(A)|+G>FhvF^fm?(F/F[oF[oF//FhvF[oC$-Fjfm6#%JThis~is
~not~correct.Try~the~other~choice.G>FhvF^fm-Fjfm6#%LIndeed,~for~the~matrices~A~
&~B~you~entered,G-Fjfm6#%Jtranspose(AB)~=~transpose(B)*transpose(A)GFjdmFeao-Fj
fm6#%_oThe~two~matrices~are~not~compatible.~We~cannot~verify~this~statementGFjd
mF/FagmF/Ff\oRF/6PFghnF_hnF^hqF^hmF_hmFbgo%#i1G%#j1GF\jqF\hq%&cofacG%#IDGF`hqFi
hqF]jqFfhqFdgo%"CG%$AdjG%"dG%%AUG1GFfdmF_hqF`hnFhhqFdiqFaiqFciqFjhqF[iqF\iqF`iq
Fbiq%$eqsGF^iqFbhqF_iqFeiqF^eu%#i2G%#j2G%&temp1GFhiqFiiqFjiqF[jqFgdmF/C3Fjdm-Fj
fm6#%ipThe~purpose~of~this~function~is~to~demonstrate~the~steps~to~find~the~inv
erse~of~a~matrix~usingG-Fjfm6#%jpeither~the~Gauss-Jordan~algorithm~or~the~Adjoi
nt~method.~In~selecting~the~interactive~mode,~youG-Fjfm6#%ipcan~immediately~che
ck~if~you~have~learned~this~algorithm.~The~nostep~mode~dispalys~the~inverseG-Fj
fm6#%^pwith~no~intermediate~steps.~For~more~information~type~?inverse;~at~the~M
aple~promptGFjdm@%F]^nC%-Fjfm6#%gnEnter~a~matrix.~For~example~type~:~A:=matrix(
[[0,1],[3,3]]);G-F^em6#%7Please~enter~a~Matrix|+G>Fh`nF^fm>Fh`nFbjm?(F/F[oF[oF/
4-F:6$Fh`nFhjmC&-F^em6#%3A~matrix~Expected|+G>Fh`nF^fm>FafmFh`n@$FcfmFcin@&0F_e
oFaeoC%-Fjfm6$%8The~matrix~you~entered~G/F[uFgdo-Fjfm6#%8~is~not~a~square~matri
xGFcin/-Fj[oFbdoFbalC%Fghv-Fjfm6#%gnhas~a~zero~determinant.~Therefore~the~inver
se~does~not~existGFcin-Fjfm6$Fg_oFjhvFjdmFe]rFh]rF[^rF^^r>FboF^fm@(/FboF[oC(>Fa
cnF_eo>F`gnFaeo@%/F`gnFacnC*-F^em6#%=Which~method~do~you~prefer?|+G-F^em6#%>~~~
~~~1.~Gauss-Jordan~method|+G-F^em6#%9~~~~~~2.~Adjoint~method|+G-F^em6#%8<Select
~from~1;~or~2;>|+G>FboF^fm>FafmFbo@$FcfmC$FifmF]gm@&/FboF]jmC9Fjdm-Fjfm6#%ioIn~
this~method~we~find~the~cofactor~of~each~entry~and~form~the~cofactor~matrixG-Fj
fm6#%aoThe~cofactor~of~(i,j)th-entry~is~given~by~(-1)^(i+j)*det(minor(A,i,j))GF
jdm>Fcdn-Fhjm6$FacnFacn?(FdzF[oF[oFacnF_s?(F`[lF[oF[oFacnF_s>&Fcdn6$FdzF`[lFeiu
>F[uFh`n-Fjfm6$%9The~original~matrix~is:~GFhcoF_inFjdm?(FdzF[oF[oFacnF_sC$-F^em
6$%R~Now~find~the~cofactors~of~the~entries~in~row~%d|+GFdz?(F`[lF[oF[oFacnF_sC,
@$FcfmFcin>F^\w*&)F^v,&FdzF[oF`[lF[oF[o-Fj[o6#-%&minorG6%F[uFdzF`[lF[o-Fjfm6$%/
In~the~matrix~GFhco-F^em6&%KThe~minor~of~the~(%d,%d)-th~entry~%d~is~:|+GFdzF`[l
&F[uF_\w-Fjfm6#/&F^cnF_\w-&F_\o6#Fc]wFd]wFjdm-Fjfm6$%7and~it's~cofactor~is:~G/&
Fa`rF_\wF^\wFjdm-Fjfm6$%=The~updated~cofactor~matrix:G/Fa`rFfbvF_in@$FcfmFcin-F
jfm6#%`oThe~adjoint~matrix~is~the~transpose~of~the~cofactor~matrix.~ThereforeG-
Fjfm6#/%>The~adjoint~of~the~matrix~is:G-F^\oFa\rF_in@$FcfmFcin-Fjfm6#%_oThe~inv
erse~of~the~matrix~A~is~(1/det(A))*Adjoint(A)~and~therefore~:GFjdm>Fet-&F_\o6#F
j[oFav@$0FetF[oC%-Fjfm6$%GSince~the~determinant~of~the~matrix~isGFet-Fjfm6$%7Th
e~inverse~of~A~~is~:G/)F[uF`]u*&-Fhjn6#Fd_wF[oFetF^vFjdm-Fjfm6$%7which~is~equiv
alent~toG-Fhjn6#-Fd\oFavF_in-Fjfm6$%>We~can~verify~this~by~showingG/*&FicoF[oF^
awF[o-Fhjn6#-Fajr6$F[uF`awFhivC>-Fjfm6#%foWe~find~the~inverse~of~the~matrix~A~b
y~forming~the~augmented~matrix~[A~|gr~I]G-Fjfm6#%gowhere~I~is~the~identity~matr
ix.~Then~use~elementary~row~operations~to~reduceG-Fjfm6#%`oit~to~a~matrix~of~th
e~form~[I~|gr~B].~The~matrix~B~is~the~inverse~of~A.GFjdm>FfcnFi[w?(FcvF[oF[oFac
nF_s?(FganF[oF[oFacnF_s>&FfcnFeavFbal?(FcvF[oF[oFacnF_s>&Ffcn6$FcvFcvF[o>F[uFh`
n>Fgdn-%(augmentG6$F[uFfcn>FacnF__r>F`gnF[gt-Fjfm6$%CThe~original~augmented~mat
rix~is~:GFgdnFjdm-Fjfm6#%enApply~the~elementary~row~operations~to~[A~|gr~I]~and
~convertG-Fjfm6#%Sthe~matrix~to~its~reduced~row~echelon~form~[I~|gr~B]GF_in@$Fc
fmFcinFjdm-Fjfm6#%NWe~employ~the~Gauss~-~Jordan~elimination~on~:GFbhtF_in@$Fcfm
Fcin?(FcvF[oF[oFacnF_sC*@$/&FgdnF\cwFbalC(>F_yF]jm?(F/F[oF[oF//&Fgdn6$F_yFcvFba
l>F_y,&F_yF[oF[oF[o-Fjfm6&%5Interchange~the~row~GFcv%%and~GF_y>Fgdn-Fafs6%FgdnF
_yFcv-Fjfm6$%)To~get~:GFgdnF_in@$0FfdwF[oC&-Fjfm6&%,Divide~row~GFcv%$by~GFfdw>F
gdn-F[er6%FgdnFcv*$FfdwF^v-Fjfm6$%*To~get~:~GFgdnF_in>F_y,&FcvF[oF[oF[o@$1F_yFa
cn?(F/F[oF[oF/3Fjdw2F_yFacn>F_yF^ew@$3/F_yFacn0F[ewFbal-Fjfm6$%>Perform~elimina
tion~on~columnGFcvFjdm?(FganFhfwF[oFacnF_s@$0&Fgdn6$FganFcvFbalC'-Fjfm6(%.multi
pliy~rowGFcvFhdr,$FigwF^vFagrFgan>Fgdn-Fdgr6&FgdnFcvFganF_hwFfewF_in@$FcfmFcin?
(Fgan,&FcvF[oF^vF[oF^vF[oF_s@$FhgwC'F\hw>FgdnFahwFfewF_in@$FcfmFcin-Fjfm6#%goTh
e~augmented~matrix~[A~|gr~I]~is~now~in~the~form~[I~|gr~B],~with~B~=~inverse(A)G
-Fjfm6$%<Therefore~the~inverse~of~A=GF[u-Fjfm6$%$is~GF`awF_inFaaw-Fjfm6%%+The~m
atrixGF[u%Vis~not~square~.~Type~in~a~square~matrix~and~try~againGFjdm-Fjfm6#%fo
This~is~the~end~of~the~inverse~procedure.~Choose~other~matrices~and~executeG-Fj
fm6#%cothe~function~again~to~learn~these~two~algorithms~for~finding~the~inverse
GF`[wC>>Facn-F`_r6#-Fcbn6$Fh`nF[o>F`gn-F`_r6#-Ff_tFbjw>F[xF_iv>FfcnFi[w?(FcvF[o
F[oFacnF_s?(FganF[oF[oFacnF_s>FhbwFbal?(FcvF[oF[oFacnF_s>F[cwF[o>F[uFh`n>Fd]lF_
cw-Fjfm6$%BThe~determinant~of~this~matrix~isGF[xFjdm-F^em6#%I~~Does~the~inverse
~of~the~matrix~exist?|+G-F^em6#%8~~answer~~yes;~or~no;~|+G>FboF^fm?(F/F[oF[oF/4
5/FboFi^o/FboF_^oC&>FafmFbo@$FcfmFcinFh^u>FboF^fm@&3F^\x0F[xFbal-Fjfm6#%WThis~i
s~not~correct.~The~inverse~of~the~matrix~exists.G3F]\xFe\x-Fjfm6$%PGood!~Procee
d~to~find~the~inverse~of~the~matrixGF[uFjdm-F^em6#%Z~~Which~one~of~the~followin
g~methods~you~prefer~to~use~?|+G-F^em6#%jn~~~~1.~Show~that~the~matrix~is~row~eq
uivalent~to~the~identity.|+G-F^em6#%S~~~~2.~Use~the~Adjoint~method~to~find~the~
inverse|+G-F^em6#%8<select~from~1;~or~2;>|+G>FboF^fm>FafmFbo@$FcfmFcin@&FhivCI-
F^em6#%QAugment~the~matrix~A~with~the~identity~matrix.~|+G-F^em6#%Ieg:~AUG:=mat
rix([[1,2,1,0],[3,4,0,1]]);|+G>FgdnF^fm?(F/F[oF[oF/4-F:6$FgdnFhjmC&>FafmFgdn@$F
cfmFcinFjet>FgdnF^fm?(F/F[oF[oF/50F]br-F`_n6#Fd]l0Fjar-Fb_nFa_xC'-F^em6%%9%d~x~
%d~matrix~expected|+GF`_xFc_x-F^em6#%+Try~again|+G>FgdnF^fm>FafmFgdn@$FcfmFcin@
%/-Fb^o6$-Fhjn6#,&Fd]lF[oFgdnF^vF[oFbalC$Fjdm-Fjfm6$%@You~entered~the~correct~m
atrix.G/FgdnF\_r-Fjfm6$%JThis~is~not~correct.The~correct~matrix~isG/Fgdn-FhjnFa
_xFjdm-F^em6#%hnWhat~is~the~best~route~you~will~select~to~find~the~inverse?~|+G
-F^em6#%C~~~1.~Find~the~determinant~of~AUG|+G-F^em6#%O~~~2.~Use~the~Gauss-Jorda
n~elimination~on~AUG|+G-F^em6#%Q~~~3.~Use~the~backsubstitution~algorithm~on~AUG
|+G-F^em6#%;<select~from~1;~2;~or~3;>|+G>FboF^fm>FafmFbo@$FcfmFcin?(F/F[oF[oF/5
Fhiv/FboFi_nC&-Fjfm6#%@You~can~have~a~better~selectionG>FboF^fm>FafmFbo@$FcfmFc
in-Fjfm6#%KGood~choice.~Proceed~to~find~the~inverse!!G>FgdnFd]l>FhvF\_s>FjdnF``
s>F^]pF]brFfbs-F^em6#%boPlease~enter~a~row~operation~to~reduce~the~matrix~as~in
~the~menu~above|+G>F_^lFibs-Fjfm6$%AThe~orginal~augmented~matrix~is:GFgdnFjdm>F
_`sF[o>F\brFbal?(F/F[oF[oF/30F_`sFbal0F_`sF]jmC'Fgcs>F_^lF^fm@-FbdsC$>F_`sFbal>
F\brFbalFhdsC'>FgdnF[\s?(FcvF[oF[oF^]pF_s@$0FfdwFbal>FgdnFafwFbes>F_`sF]jm>F\br
FbalFgesC$@(/F]aqF[o>Fgdn-Fafs6%FgdnFcfsFhis/F]aqF]jm>Fgdn-F[er6%FgdnF[ep*$F`]q
F^v/F]aqFi_n?(FcvF[oF[oFjarF_s>&Fgdn6$8PFcv&8QFix@%/F\brFbalC$Fbgs>F\br,&F\brF[
oF[oF[oC%-Fjfm6#%Tonly~one~level~of~undo~is~permitted~in~this~versionG-Fjfm6#%Z
If~you~wish~to~exit~the~program~type~exit;~or~to~continueGFjdmF[hsC%?(FcvF[oF[o
F^]pF_s@$50-Ffhs6$F]hs&FjdnFixFbal0-Ffhs6$F]hs&FhvFixFbalC%@$F`gx>F`]qFagx@$Fdg
x>F`]qFegx>F[epFcv@%/F`]qF[oC*>FhisF[ep>FevFiis?(FcvF[oF[oF^]pF_sC$@$/FcgxFev>F
cfsFcv@$/FggxFev>FcfsFcv>FgdnFaex-Fjfm6&FfjsFcfsFgjsFhisFhjsFjdm>F]aqF[oC$?(Fcv
F[oF[oF^]pF_s@$50-Ffhs6$FiisFcgxFbal0-Ffhs6$FiisFggxFbal>F[_sFcv@%/F[_sF[epC'>F
gdn-F[er6%FgdnF[epF`]q-Fjfm6&F_grF[epFhdrF`]qFi\tFjdm>F]aqF]jmFg]t>F\brFbalF[^t
C,>FbipFbal>F`\qFbal?(FcvF[oF[oF^]pF_sC$@$Fbix@&Fcix>F]fxFcvFfix>F]fxFcv@$F_gx@
%/FbipFbalC$>FbipFcv@&F`gx>F`dqFagxFdgx>F`dqFegxC$>F`\qFcv@&F`gx>F]_pFagxFdgx>F
]_pFegx>F_fx-Ff_t6$FgdnF]fx?(FcvF[oF[oFjarF_s>F[fx&-Fhjn6#,&*&F`dqF[o-Ff_t6$Fgd
nFbipF[oF[o*&F]_pF[o-Ff_t6$FgdnF`\qF[oF[oFix@'3Fdis/FbipF]fx-Fjfm6(F]atF`\qFhdr
F]_pF^atFbip3Fi`t/F`\qF]fx-Fjfm6(F]atFbipFhdrF`dqF^atF`\q-F^em6-FfatFgatFbipFhd
rF`dqFhatF`\qFhdrF]_pFiatF]fxFjatFjdm>F]aqFi_n>F\brFbalC$Fgbt>F\brFbal>Feep-&F_
\o6#%%rrefGF]_r@$/-Fb^o6$-Fhjn6#,&FeepF[oFgdnF^vF[oFbalC%>FganF]br?(F/F[oF[oF//
FecrFbal>Fgan,&FganF[oF^vF[o@$3/FecrF[oF^dxC$-Fjfm6$%TYou~obtained~the~matrix~i
n~reduced~row~echelon~formGFgdn>F_`sF]jm@$/F_`sFbalC&-Fjfm6$%boWarning.~You~are
~existing~too~soon.~Here~is~the~reduced~echelon~form~ofGFgdnFjdm>FgdnFi]y-Fjfm6
$%@The~reduced~row~echelon~form~isGFgdnFjdm-F^em6#%]oFrom~this~reduced~form~ext
ract~the~inverse,B,~of~the~given~matrix|+G-F^em6#%@eg~:~B:=matrix([[1,2],[3,2]]
);|+G>Fh]pF^fm?(F/F[oF[oF/F\^pC&Fjet>Fh]pF^fm>FafmFh]p@$Fcfm-F_o6#%,Bye~for~now
G>F[u-Fg\r6%Fgdn;F[oFacn;,&FacnF[oF[oF[o,$FacnF]jm@$50-F`_nFe^pFacn0-Fb_nFe^pFa
cnC$-Fjfm6$%SIncorrect~matrix~dimensions.~The~correct~matrix~isGF[u>Fh]pF[u@%/-
Fb^o6$-Fhjn6#,&Fh]pF[oF[uF^vF[oFbal-Fjfm6$%ICorrect,~the~inverse~of~the~matrix~
is~:~G/Fg`w-FhjnFe^p-Fjfm6$%DIncorrect.~The~correct~inverse~is~:GFgbyFjdmF`[wCJ
>Fh]pFi[w-Fjfm6$%7The~original~matrix~isGFh`nF_in-Fjfm6#%hnTo~find~the~cofactor
~of~the~(i,j)~entry~use~the~Maple~commandG-Fjfm6#%=>(-1)^(i+j)*det(minor(A,i,j)
GFjdm>FcdnFi[w?(FcvF[oF[oFacnF_s?(FganF[oF[oFacnF_s>&FcdnFeavFeiu>F[uFh`n?(FcvF
[oF[oFacnF_s?(FganF[oF[oFacnF_s>&Fh]pFeav*&)F^v,&FcvF[oFganF[oF[o-Fj[o6#-Fc]w6%
F[uFcvFganF[o?(FcvF[oF[oFacnF_s?(FganF[oF[oFacnF_sC*Fjdm-F^em6%%JEnter~the~cofa
ctor~of~the~entry~(%d,%d):|+GFcvFgan-Fjfm6$%/of~the~matrix~GF[u>FboF^fm>FafmFbo
@$Fcfm-F_o6#%1Too~soon~to~quitG>F[dyF`dy@%0-%$absG6#,&FboF[oF`dyF^vFbalC%-Fjfm6
$%EIncorrect.~The~correct~cofactor~is~:GF`dyFjdm-Fjfm6$%=The~new~cofactor~matri
x~is~:GFcdnC$-Fjfm6#%hnYou~entered~the~correct~cofactor.~The~new~cofactor~matri
x~is:G-FjfmFa\rFjdm-F^em6#%`oHaving~the~cofactor~matrix,~what~is~the~next~step?
~Enter~your~choice|+G-F^em6#%U~~~~~1.~Find~the~determinant~of~the~cofactor~matr
ix|+G-F^em6#%K~~~~~2.~Construct~the~adjoint~matrix~of~A|+G-F^em6#%H~~~~~3.~Find
~the~LU-decomposition~of~A|+G>FboF^fm>FafmFbo@$FcfmFcin@$0FboF]jm-Fjfm6#%`oThis
~is~not~a~good~choice!!.~You~need~to~construct~the~adjoint~matrixGFjdm>Fial-F^\
oFe^p-F^em6#%enGood.~Construct~the~adjoint~matrix~of~A.~Enter~the~matrix|+G>Fbo
F^fm?(F/F[oF[oF/F[jnC&-F^em6%%DA~%dx%d~matrix~expected.~Try~again|+G-F`_nF\cl-F
b_nF\cl>FboF^fm>FafmFbo@$Fcfm-F_o6#%.See~you~laterGFjdm@%3/Fb[oFihy/Fd[oFjhy@%0
-Fb^o6$-Fhjn6#,&FboF[oFialF^vF[oFbalC%-Fjfm6#%EYou~did~not~enter~the~right~matr
ix!!G-Fjfm6$%enThe~correct~matrix~is~the~transpose~of~the~cofactor~matrixGFcdn-
Fjfm6$%*That~is~:G/)-%'cofactGFavFjgmF^]s-Fjfm6$%<Good.The~adjoint~matrix~is:GF
_hyC%-F^em6%%NIn~this~case~Adjoint~of~A~is~a~%dx%d~matrix.|+GFihyFjhyF`jyFcjyFj
dm-F^em6#%]oWhich~of~the~following~will~give~the~inverse~of~the~given~matrix?|+
G-F^em6#%J~~~~1.~The~inverse~of~the~adjoint~matrix|+G-F^em6#%ao~~~~2.~The~cofac
tor~matrix~times~1/det(A)~where~A~is~the~given~matrix|+G-F^em6#%`o~~~~3.~The~ad
joint~matrix~times~1/det(A)~where~A~is~the~given~matrix|+GFjdm>FboF^fm>FafmFbo@
$Fcfm-F_o6#%&AdoisG@$0FboFi_n-Fjfm6#%WThis~is~not~the~correct~route.~The~correc
t~choice~is~3GFjdm-F^em6#%QThe~inverse~of~the~matrix~based~on~choice~3~is~:G-Fj
fm6#/Fg`w*&-Fj[oFavF^v-Fhjn6#-&F_\o6#%(adjointGFavF[oF_inFaaw-Fjfm6#%`oThis~is~
the~end~of~the~Interactive~inverse~procedure.~You~may~choose~G-Fjfm6#%[oother~e
xamples~to~enforce~your~understanding~of~these~algorithmsGFcbx@'Fehv-Fjfm6#%CTh
is~matrix~is~not~a~square~matrixGF^iv-Fjfm6#%IThe~inverse~of~the~matrix~does~no
t~existGC$-Fjfm6#%>The~inverse~of~the~matrix~is~G-Fjfm6#/Fg`w-Fhjn6#-Fd\oFbdoF/
FagmF/%&trsumGRF/6/F`hnF_hnF^euFahnF`boFbhnFabo%&addABG%,trans_addABGF^boFi\v%)
add_tAtBGFghnFgdmF/C:Fjdm-Fjfm6#%hnThe~purpose~of~this~program~is~to~check~the~
conjecture~if~theG-Fjfm6#%entranspose~of~the~sum~is~the~sum~of~the~transpose;~t
hat~is,G-Fjfm6#/),&F[uF[oFh`nF[oFjgm,&FjboF[oF_^vF[oF_in@$FbinFcin@&F]^nC*-F^em
6#%jnEnter~the~first~matrix~|frfor~example,~A:=matrix([[1,2],[2,3]])|hr|+G>F[uF
^fm?(F/F[oF[oF/Fa]nC$-F^em6#%>Matrix~is~expected.Try~again|+G>F[uF^fmFfco-F^em6
#%[oEnter~the~second~matrix~|frfor~example,B:=matrix([[1,-1],[4,3]])|hr|+G>Fh`n
F^fm?(F/F[oF[oF/FjgvC$Fi`z>Fh`nF^fm-Fjfm6$FejnF]dpF[jmC$>F[uF[[o>Fh`nF``v-Fjfm6
$Fc`vFhco-Fjfm6$%9~The~second~matrix~is~:~GF]dp>F_yF__n>FcvFa_n>FdzF_eo>FganFae
o@$3/F_yFdz/FcvFganC.>F`[l-Fhjn6#F^`z>F`gn-F^\o6#F`[l>FacnF_do>FcdnFado>Ffcn-Fh
jn6#,&FacnF[oFcdnF[oF_in@$FbinFcinFjdm-Fjfm6$%?The~transpose~of~their~sum~~=~G/
F]`z-Fhjn6#F`gnF_in@$FbinFcin-Fjfm6$%=The~sum~of~their~transpose=~G/F_`z-Fhjn6#
FfcnF_in@$FbinFcinFjdm-F^em6#%jnIs~the~sum~of~the~transpose~equal~to~the~transp
ose~of~the~sum?|+GFg]o>FgdnF^fm@&/FgdnFi^o-Fjfm6%%9Yes~indeed,~The~propertyGF\`
z%>~holds~for~these~two~matricesG/FgdnF_^oC$-Fjfm6%%7Incorrect,The~propertyGF\`
zF[ez>FboF[oFjdm-Fjfm6#%`oThe~transpose~of~the~sum~is~always~equal~to~the~sum~o
f~the~transposesG-Fjfm6#%GThis~is~the~end~of~the~trsum~procedureGF/FagmF/%(comm
uteGRF/61F`hnF_hnF^euF^hqFf\vFahnF`bo%#n1G%#p1GFbhnFaboF\hq%"pGFfhqFghnFgdmF/C8
-Fjfm6#%QThe~purpose~of~this~procedure~is~to~check~if~theG-Fjfm6#%Qcommutative~
property~of~multiplication~holds~forG-Fjfm6#%1any~two~matricesGF_in@$FbinFcin@&
F]^nC*-F^em6#%inEnter~the~first~matrix~|frfor~example,A:=matrix([[1,2],[2,3]])|
hr|+GF^fm>F[uFb_o-Fjfm6#/%8The~first~matrix~is~:~AGFicoF]azF^fm>Fh`nFb_o-Fjfm6#
/%:~The~second~matrix~is~:~BGFgdoF[jmC'>F[uF[[o>Fh`nF``vF_gzFjdm-Fjfm6#/%8The~s
econd~matrix~is:~BGFgdo>FacnF__n>F_yFa_n>FcdnF_eo>FdzFaeo@$3/F_yFcdn/FdzFacnC,>
Ffcn-Fhjm6$FacnFdz>Fgdn-Fhjm6$FcdnF_y>Ffcn-FcavFjanF_in@$FbinFcin-Fjfm6$%IThe~p
roduct~AB~of~the~two~matrices~is~:~G/Fh[vF_dzF_in@$FbinFcin>Fgdn-Fcav6$Fh`nF[u-
Fjfm6$%IThe~product~BA~of~the~two~matrices~is~:~G/%#BAGF\_rF_in@$FbinFcinFjdm-F
^em6#%YDoes~the~commutative~property~hold~for~these~matrices~?|+GFe`o>FhvF^fm@*
3/FhvFi^o/-Fb^o6$-Fhjn6#,&FfcnF[oFgdnF^vF[oFbal-Fjfm6#%\oYes~indeed,~The~commut
ative~property~holds~for~these~two~matricesG3Fejz0FgjzFbalC$-Fjfm6#%coIncorrect
,~The~commutative~property~does~not~hold~for~these~two~matricesG>FboF[o3/FhvF_^
oFfjz-Fjfm6#%gnWrong,~the~commutative~property~holds~for~these~two~matricesG3Fg
[[lF`[[lC$-Fjfm6#%]oYes,~the~commutative~property~does~not~hold~for~these~two~m
atricesG>FboF[o@$FhivC*Fjdm-F^em6#%@Choose~from~the~following~menu|+G-F^em6#%G~
~1.~Show~me~two~commutative~matrices|+G-F^em6#%]o~~2.~Give~me~another~chance.~I
~will~find~two~commutative~matrices|+G-F^em6#%E~~3.~I~am~tired.~Get~me~out~of~h
ere|+G-Fjfm6#%;<choose~from~1;~2;~or~3;~>G>FhvF^fm@(Fedv-%'ABeqBAGF//FhvF]jm-Fh
ezF//FhvFi_n-Fjfm6#%+Bye!~Bye!!GFjdm-Fjfm6#%UThis~is~the~end~of~the~commute~pro
cedure.~Try~other~G-Fjfm6#%Xexamples~to~see~whether~any~two~matrices~always~com
muteGF/FagmF/F/,
I/SMARTPLOTColorR6#'F'<$Fg_nFdjm6&Fjev%(LocListGF^hmF\hqF-F/C*>F[u7#F\\l@$/<#-F
exFav<#-Fex6#-F`cmF/>F[uFg`l>Fcv-F`vFb_[l@$4-F56#%5SMARTPLOTColor_IndexG>F[`[lF
bal>Fbo&Fc_[l6#,&F[`[lF[oF[oF[o?(Fh`nF[oF[oF/3-%'memberG6$FboF[u1Fh`nFcvC$>F[`[
l-%$modG6$F``[lFcv>FboF^`[l>F[`[lFi`[lFboF/6$F`cmF[`[lF/6",
I.CM_InertFuncs<*%$IntG%$SumG%&DESolG%&RESolG%%DiffG%&LimitG%'NormalG%(ProductG
6",
IBcontext/cfparse/splitconditionalsR6#'%%CondGFcy6*F,%'resultGFhrFaqFdq%,LocAnc
estorG%2ConditionalsTableG%2ConditionalsGroupG6#FeclF\vC)>6$8&8(-Fjs6$7$Q%TestF
^wQ)Ancestor6"9$>Fcv-%'removeG6%F:Fac[l%"=G>8*Fht?&8$FcvF_sC&-F26#-F:6$Fjc[lFcw
>8)-Fjs6$F_c[l&Fgt6#Fjc[l@$/Fad[l%%FAILG>Fad[lFjb[l-F"6$Fjc[l&Fhc[l6#F_y>8%F\p?
&Fjc[l7#-%(indicesGF\]lF_sC%>Fjb[l-FexFed[l>8+-F\[o6$-F]bl6#&Fhc[l6#Ffe[l%4cont
ext/StackToListG@%/Fjb[lFhd[l>F_e[l6$F_e[l-Fcy6$/F^c[lFib[l-Fex6#Fhe[l>F_e[l6$F
_e[l-Fcy6%Fff[l/F_c[lFjb[lFgf[l/Fac[lF_e[lF\vF\vF\vF\v,
I-ToMatrixformRF/6+F_hnFggo%%varsG%$equGF\jqF\hqF^hmF_hm%%eqnsGFgdmF/C(>F`gnFbj
m>Fh`nF][n>Fbo-Fhjm6$-F`vFicz,&-F`_rFbdoF[oF[oF[o>F[u-Fhjm6$F[h[lF[o?(FdzF[oF[o
F[h[lF_sC$?(F`[lF[oF[oF]h[lF_s>&FboF_\w-Ffhs6$-Fjis6#&F`gnF\]l&Fh`nF\cz>&Fbo6$F
dzF\h[l-F^hsFih[lFboF/F/F/F/,
I?context/cfparse/act_LINEARFUNCR6#'%%CodeGF\`lF/6#FeclF/C$@$F]sF`s-FjelF]\lF/F
/F/6",
I0type/CF_SUBMENURFOF/F-F/3-F:6$FK-%)specfuncG6$%)anythingGFg]l/-Fdo6$FKF_uFgoF
/F/F/6",
I)rrefautoRF/6?FghnF_hnFggoF^hmF_hmFbgoF`hnF`hqF\hqFciqF\jqF^fvFfdmF^hqF_hqFhhq
FdiqFaiqFjhqF[iqF\iqFihqF`iqFbiqF^iqFbhqF_iqFeiq%%prevGFgdmF/C<>F`[lFbjm>F`[l-F
\[o6$F`[lFhjm-Fjfm6$%9The~original~matrix~is~:GF`[l>F`gn-F`_nF\cz>Fcdn-Fb_nF\cz
>Fdz-%)chkrowecGF\cz@$/FdzF[oC)-Fjfm6%%;Note~that~the~given~matrixGF`[l%<~is~no
t~in~row~echelon~formG-F^em6#%C~~~~~Choose~one~of~the~following:|+G-F^em6#%[q~~
~~~1.~if~you~want~to~observe~the~steps~to~display~the~row~echelon~form;~~type:~
>rowec;~~or~1;|+G-F^em6#%[q~~~~~2.~If~you~do~not~want~to~observe~the~steps~of~t
he~row~echelon~form;~~~type:~>bypass;~or~2;|+G>FboF^fm?(F/F[oF[oF/4555/FboFjds/
Fbo%'bypassGFhivF`[wC&>FafmFbo@$F]dsFcin-F^em6#%8Type~bypass;~or~rowec;|+G>FboF
^fm@&55F`]\l/FboF\esFhivC'>%(rreflagGF[o-%*gaussautoGF\cz>F`^\lFbal>F`[l-F\\sF\
cz?(FcvF[oF[oFd[\lF_sC%>FganF[o?(F/F[oF[oF/3/&F`[lFeavFbal2FganFf[\l>FganF^cr@$
0F\_\lFbal>F`[l-F[er6%F`[lFcv*$F\_\lF^v55/Fbo%'BYPASSGFa]\lF`[wC%>F`[lFe^\l>F`g
n-F`\rF\cz?(FcvF[oF[oFd[\lF_sC%>FganF[o?(F/F[oF[oF/Fj^\l>FganF^cr@$F`_\l>F`[lFb
_\l>FcvFd[\l?(F/F[oF[oF/3-%'iszeroG6#-Ff_t6$F`[lFcv2F[oFcv>FcvFehw>FganF[o?(F/F
[oF[oF/Fj^\l>FganF^cr@$3F`_\l0F\_\lF[o>F`[lFb_\lFjdm-Fjfm6#%DThe~matrix~in~ROW-
ECHELON~form~is~:G-FjfmF\cz-Fjfm6#%YNow~let~us~put~this~matrix~into~Reduced~Row
~echelon~FormGF_in@$F]dsFefr>F`dqF[o>F`gnFf[\l>FcdnFd[\l?(FganF[oF[oF`gnF_sC%>F
_yFd[\l?(F/F[oF[oF/3/&F`[lFiipFbal2F[oF_y>F_yFjct@%/F_yF[oF/@$2F`dqF_yC$>F`dqF_
y?(FcvFjctF^vF[oF_sC%-Fjfm6$FegwFgan-Fjfm6#%:*************************G@$0Ffb\l
FbalC(-Fjfm6*F^hwF_yFjhrF`[lFhdr,$F\_\lF^vFagrFcv>F`[l-Fdgr6&F`[lF_yFcvF[d\lFjd
m-Fjfm6$F\btF`[lF_in@$F]dsF^ds-Fjfm6$%LThe~matrix~in~reduced~row~echelon~form~i
s~:GF`[lFjdm-Fjfm6#%jnThis~is~the~end~of~demonstration~version~of~the~reduced~e
chelonG-Fjfm6#%[oform~procedure.~You~may~try~other~examples~to~learn~the~proces
s.GF/6$FafmF`^\lF/F/,
I<convert/context/ListToStackR6#'F'Fg_n6#%#STGFei[lF/C&F1>Fbo-&F*6#FDF/-Fio6%&F
*6#FIFKFbo-F]blF`gmF/F/F/6",
I:context/cfparse/joinblockR6#'%'blocksG-Fg_n6#%+CM_IFBLOCKG6'F^r%'KeySetGFjr%(
NewCodeG%-continuationG6#F.6"C)-FS6#.Fcil>8(Fdgm>8'Q!6">8%<"?&8&9$F_sC&>8$-Fjs6
$Q$KeyF^wFig\l@%-Fd`[lFe`[l-Fj]n6#-Fiu6$Q1Repeated~Key:~%sFjf\lF]h\l>Ffg\l-F_bn
6$Ffg\l<#F]h\l>Fbg\l-Fcil6%Q"~Fdg\lFbg\l-%<context/cfparse/ifblockcodeG6$Fig\lF
`g\l>F`g\lF_s@$/F`g\lF_s>Fbg\l-Fcil6%F_i\lFbg\lQ$fi;6"-F\`l6#FcvFjf\lFjf\lFjf\l
Fjf\l,
I0context/fullkeyR6$'%%RootG<$F)%'stringG'%(CurrentGFcj\lF/Fei[lF/C$@$5-F:6$FKF
)/FKFcg\l-F_o6#F<-%$catG6%FKQ"_6"F<F/F/F/Fb[]l,
I(trsumegRF/6&F_hnF^euF^hmF_hmFgdmF/C2>Fbo-Fhjm6$Fi_nFi_n>F[uFh[]l-Fjfm6$%%Let~
GFfjnFjdm-Fjfm6$FbewFhcoF_in@$FbinFcin-Fjfm6$F]\]l/Fjev,&FboF[oF[uF[o-Fjfm6$%%T
henG/&FjevFdiu,&&FboFdiuF[o&F[uFdiuF[oF_in@$FbinFcin-Fjfm6$%;Now~the~(i,j)th~el
ement~ofG/)FjevFjgm%5(j,i)th~element~of~CG-Fjfm6$%#SoG/)Fd\]lFjgm,&&Fbo6$FcvFh`
nF[o&F[uF[^]lF[o-Fjfm6$%<and~also~(i,j)th~element~ofG/,&)FboFjgmF[oFjboF[oFi]]l
Fjdm-Fjfm6$%*ThereforeG/Fh]]lFa^]lF/FagmF/F/,
I/type/CM_MatrixRFOF/Fei[lF/-%&evalbG6#0-F_bn6$-%'selectG6%F:-Fdo6$FK%&arrayG<$
FhjmFgep-Fdo6$FK-F`j[l6$Fbj[l<&Fhjm%'MATRIXGFgep%'VECTORGFgoF/F/F/6",
I<context/cfparse/conditionalR6#'F^rF^[l6%Fdi[l%,ConditionalGFhrF-F/C--FS6#.Fjs
-FS6#.F\`lF\g\l-%(raadlibGF_il-F26#-F56#&FgtF]\l>F[uFaa]l>Fh`n-Fe`]l6$F^c[lF[u-
F26#0Fh`nFcg\l>Fbo-Fiu6%Qenif~not~(~%s~)~then~stack[push](~%a~=~NULL,~Exclusion
s)~fi;6"-Fcil6$F_i\lFh`nFK@$4-FjhlF`gm-Fj]n6$%>unable~to~parse~condition~forGF[
u-F\`lF`gmF/F/F/F]b]l,
I4help/text/gausselim-F`p6D%+HELP~FOR:~G%#~~G%`o-~linsys[gausselim]~:~the~purpo
se~of~this~function~is~to~To~reduce~a~G%cq~a~matrix~OR~an~augmented~matrix~repr
esenting~a~system~of~linear~equations~Ax=b~to~its~row~echelon~form.G%fn~Gauss~e
limination~is~performed~on~the~rows~of~the~matrix~AG%N~or~on~the~rows~of~the~au
gmented~matrix~[A,b]G%en~There~are~THREE~versions~of~the~gauss~elimination~prog
ramGF[c]l%]p~VERSION~1~:~~GAUSSAUTO~:~demonstrates~a~step~by~step~process~of~ga
uss~eliminationG%[o~~~~~~~~~~~~~~using~the~row~operations~selected~by~the~machi
ne.~GFbp%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%?~~~~~~~~~~~~~~~gauss
auto(A,b);GFbp%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]]);GFbp%\o~VERSION~2~:~GAUSSMAN~:~T
his~procedure~performs~gauss~eliminationG%V~~~~~~~~~~~~~using~the~row~operation
s~provided~by~youGFbp%J~~example~:~~A~:=~matrix([[1,2],[3,43]]);G%:~~~~~~~~~~~~
~gaussman(A);GFbp%co~VERSION~3~:~gausselim~:~This~is~the~general~procedure.~It~
allows~you~toG%_o~~~~~~~~~~~~~select~the~row~operations~first.~However~you~can~
switchG%]o~~~~~~~~~~~~~to~auto~mode~any~time~and~observe~the~machine~performG%=
~~~~~~~~~~~~~row~operations.GFbp%I~~example~:~~A~:=~matrix([[1,2],[3,3]]);G%;~~
~~~~~~~~~~~gausselim(A);GFbpF[c]lF/,
I;context/cfparse/recognizerR6$'F]qF_]l'%%DATAGFit6'FgqF^rF\rFdi[l%'IfCodeGFei[
lF/C0Fhs-FS6#Fbel-FS6#Fjhl-FS6#Fcil-FS6#F\`l>F[u-Fjs6$F`h\lFK>Fh`n-FbelF]\l>Fga
n-Fdc[l6%F:Fh`nFfc[l-F26#/-F`vFg\oF[o-F26#-Fjhl6#-F\`l6#-FexFg\o>Fcv-F\`l6AQ(pr
oc(f)6"QWlocal~RememberFlag,Exclusions,Vars,Nvars,Funcs,NFuncs;F[g]lQ`pglobal~`
context/CM_GLOBALG`,~ContextData,~`context/Initialized`,~`context/InitProcs`,F[
g]lQB~~~~~_EnvContextData,~ContextKey;F[g]lQ/option~system;F[g]lQ9"ContextMenu~
Procedure";F[g]lQ`oif~`context/Initialized`~<>~true~then~readlib('`context/init
`')()~fi:F[g]l-Fiu6$Q2ContextKey~:=~%a;F[g]lF[u-Fiu6$Q7_EnvContextData~:=~%a;F[
g]lQ%DATAF[g]lQFContextData~:=~eval(_EnvContextData);F[g]l-Fiu6$QHif~`context/I
nitProcs`[%a]~<>~true~thenF[g]lF[u-Fiu6$QJreadlib('`context/InitializeProcs`')(
%a);F[g]lF[u-Fiu6$QA`context/InitProcs`[%a]~:=~true;F[g]lF[uFii\lQ0readlib(stac
k);F[g]lQ>Exclusions~:=~stack['new']():F[g]lQ6RememberFlag~:=~true;F[g]lFff]lQT
Context~:=~_EnvContextData[~CM_BUILD(ContextKey)~];F[g]lQGif~stack['depth'](Exc
lusions)~>~0~thenF[g]lQgnExclusions~:=~convert(~Exclusions~,~`context/StackToLi
st`~);F[g]lQGContext~:=~subs(Exclusions~,~Context);F[g]lFii\lQ<if~RememberFlag~
=~true~thenF[g]lQhn~~~~Context~~:=~convert(Context,'CONTEXTMENU',~"Top",~args~)
;F[g]lQ?~~~~procname(args)~:=~Context;F[g]lQ%elseF[g]lQgn~~~~Context~:=~Convert
(Context,'CONTEXTMENU',~"Top",~args~);F[g]lFii\lQ)Context;F[g]lQ$end6"@$4-FjhlF
ixC&>F`blF\bl>%+GlobalCodeGFcv-%(fprintfG6#-Fcil6$Q"|+F[g]lFcv-Fj]n6#Q;Unable~t
o~parse~RecognizerF[g]l>Fcv-FgilFix>Fcv-F\y6$<#/Fhg]lF\bl-F]blFixF/6%F`blFhi]l%
3context/CM_GLOBALGGF/F[g]l,
I8context/cfparse/topmenuR6&%3SimplificationTypeGF`rF[r'%&AListGFg_n6'%(Content
G%&MenusG%+ActionListGFjr%(EntriesG6#Fecl6"C,-FS6#F*-FS6#F]t>8%-FitFh[^l>&F_\^l
6#Q%MainFh[^l-&F*6#FDFh[^l-Fio6%%9context/cfparse/pushmenuG9'F_\^l>8$-F\[o6$Fb\
^lF_f[l-F26#-F:6$F]]^lFg_n>F]]^l-Fio6$R6#FjrFh[^lFh[^lFh[^l@%-F:6$9$Ff^l-F]t6$F
\^^lT#F\^^lFh[^lFh[^l6$Fd[^lF_\^lF]]^l>F]]^l-%Acontext/cfparse/removeduplicates
G6#F]]^l-%)CM_BUILDG6&F\^^l9%9&F]]^lFh[^l6#FgtFh[^lFh[^l,
I>convert/CONTEXTMENU/recursiveR6#%'ActionG6,%+TypeStringGF`rF[r%+SelectionsG%+
ActionProcGFjrF^hoFbg[l%)submenusG%(varprocG6#Fecl6"C(-Feu6%""#<#%,ContextMenuG
-F>6$F`bl-%%copyG6#Fgt>8)&9"6#;Fj_^l!""-F26#3-F:6$9$Fg_n/-F`v6#F^a^l""%>6&8%8&8
'8(-FexFaa^l@-/Fga^l7#Q$VAR6"C&>8+%,_EnvCM_VarsG@$2"""-F`v6#Fab^l>Fab^l-%%sortG
6#7#-FexFgb^l-F26#-F:6$Fab^l<$-Fg_n6#F)-FdjmFdc^l>8,-F^x6$7%-Fiu6$%#%aG8*Ffa^l&
Fha^l6#F^d^l/F^d^lFab^l/Fga^l7#Q%FUNCF^b^lC%>Fab^l-Fijl6#Fc`^l-F26#-F:6$Fab^l<$
-Fdjm6#%)functionG-Fg_nF_e^l>Fgc^lFhc^l/Fga^l7#Q&PAIRSF^b^lC'>Fab^l-Fjb^l6#7#-F
ex6#-FajlFhd^l-F26#/Ffb^lFj_^l>Fab^l-&%)combinatG6#.%(permuteGFgb^l-F26#-F:6$Fa
b^l%)listlistG>Fgc^l-F^x6$7%-Fiu6$%&%a,%aG&F^d^lFf_^lFfa^l&Fha^l6#-FexF`d^lFad^
l-F:6$Fga^l7#-Fg_n6#<%F)%(numericGFcj\lC$>Fab^l-Fex6$Feb^lFga^l>Fgc^lFhc^l-F:6$
Fga^l-Fg_n6#Fcj\lC(>8--Fcil6%Q"/Ff_^lFa[m-Fex6#Fcv-F26#-F:6$Fjh^lF)@$4-F:6$-F]b
l6#Fjh^l%*procedureG-Fj]n6$%CSelection~type~not~implemented~yetGFga^l>Fab^l-Fjh
^lFhd^lFhf^l>Fgc^l-F^x6$7%-Fhn6$F[d^l;Fj_^l!"#Q.sub-selectionFf_^lFeg^lFad^lF[j
^l-F_o6#-%,CONTEXTMENUG6%Fea^lFfa^l7#Fgc^lFf_^l6%F`blFgtFbb^lFf_^lFf_^l,
I+DELETEPLOTR6$'%)positionG%'posintG'%%PlotGF`e^lF/F-F/-F_o6#-%'subsopG6$/FKF\p
F<F/F/F/6",
I<convert/CONTEXTMENU/TRIPLESRF/6#Fbg[lFei[lF/C%>Fbo-Fjb^l6#7#-Fex6#--FS6#.Fajl
6#Fcjm-F26#/F[anFi_n>Fbo-&Fdf^l6#.Fgf^lF`gmF/F/F/6",
I0help/text/topic-F`p6&Fbp%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`p67Fjb]lFbp%en~linsys[graph]~:~The~purpose~of~this~procedur
e~is~to~graphG%K~a~system~of~linear~equations~in~2D~or~3D~GFbp%H~An~example~on~
how~to~use~this~functionG%hnTo~graph~2~equations~of~3~variables~we~can~type~the
~followingGFbp%9~>~eq1~:=~x~+~y~+~z~=10;G%9~>~eq2~:=~2*x~-~3*y~=~z;G%4~>~graph(
eq1,eq2);~GFbp%fn~or~just~type~at~the~Maple~prompt~>graph();~and~the~programG%9
~will~prompt~for~input.~GFbp%]o~In~2D~case~the~equations~will~be~displayed~alon
g~with~the~graphs.GFbp%Z~The~graph~will~dispaly~consistency~or~inconsistency~in
foGFbp%Z~Plot~Range~:~The~program~will~select~the~best~plot~rangeG%I~necessary~
to~display~all~intersections.GF/,
I1help/text/lindep-F`p6BFjb]lFbp%8~~~~~~Linear~DependenceG%:~~~~~~-------------
------GFbp%_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:GFbp%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()GFbp%D~an
d~enter~the~vectors~upon~requestGFbp%9~Background~Information:GFbp%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:GFb
p%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~dependentGFbpF/,
I3help/text/trainnet-F`p6.Fbp%;HELP~SCREEN~for~:~trainnetGFbp%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();GFbpFbpF/,
I4type/CF_CONDITIONALRFOF/F-F/-F:6$FK-F`j[l6$Fbj[lFbyF/F/F/6",
I3help/text/solveqns-F`p64Fjb]lF[c]l%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[
c]l%\o~The~set~of~variables~you~are~solving~for~must~also~be~entered~asG%I~a~se
t~or~list.~Example~:~>~vars:=[x,y];GFbp%jn~If~no~arguments~are~entered~the~prog
ram~will~prompt~for~input.G%9~Then~follow~directions.GFbp%E~~example~1~:~~S:=[x
-y=3,2*x+3*y=9];G%<~~~~~~~~~~~~~~~vars:=[x,y];G%A~~~~~~~~~~~~~~~solveqns(S,vars
);GFbp%<~~example~2~:~~solveqns();~GF[c]lF/,
I2type/CF_SEPARATORRFOF/F-F/-F:6$FK-F`j[l6$Fbj[l.F^`lF/F/F/6",
I2help/text/backsub-F`p6-Fjb]lFbp%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~~~~~~~~~~~belowGFbp%K~~examp
le~1~:~~A~:=~matrix([[1,2],[0,1]]);G%;~~~~~~~~~~~~~~~backsub(A);GF/,
I9type/context/Unbounded@@RFOF/Fei[lF/3-F:6$FKF`e^l53-F:6$-Fex6$FbalFKF`e^l-F:6
$-Fex6$FbalFcd_l-.%#@@G6$Fbj[lF)-F:6$Fcd_lFid_lF/F/F/6",
I(rrefmanRF/6@FghnF_hnFggoF^hmF_hmFbgoF`hnF`hqF\hqFciqF\jqF^fvFfdmF^hqF_hqFhhqF
diqFaiqFjhqF[iqF\iqFihqF]jqF`iqFbiqF^iqFbhqF_iqFeiqFjj[lFgdmF/C2>FauF\_s>F_^lF`
`s>F`[lFbjm>FevFd[\l-%,rdgaussmenuGF/>FgdnFibs-Fjfm6$%BThe~original~augmented~m
atrix~is:GF`[lFjdm>FdzF[o?(F/F[oF[oF/30FdzFbal0FdzF]jmC)-F^em6#%^pPlease~enter~
an~appropriate~row~operation~from~the~above~menu~to~reduce~the~matrix|+G>FgdnF^
fm>FafmFgdn@$F]dsFj[r@/5/FgdnFgfm/FgdnFefm>FdzFbal5/FgdnFjds/FgdnF\esC%>F`[lFe^
\l?(FcvF[oF[oFd[\lF_sC%>FganF[o?(F/F[oF[oF/Fj^\l>FganF^cr@$F`_\l>F`[lFb_\l-Fjfm
6$FdesF`[l5/Fgdn%$recG/Fgdn%$RECGC$>F`[l-Fj]yF\cz>FdzF]jm5/FgdnFies/FgdnF[fsC$@
(/FacnF[o>F`[l-Fafs6%F`[lF]_pF_`s/FacnF]jm>F`[l-F[er6%F`[lF]aq*$Fi`lF^v/FacnFi_
n>F`[l-Fdgr6&F`[lF[xF^cn,$FialF^v-Fjfm6$FdgsF`[l-F:6$-F^hsF]_rF_hsC$?(FcvF[oF[o
FevF_s@$50-Ffhs6$F]j_l&F_^lFixFbal0-Ffhs6$F]j_l&FauFixFbalC%@$Fbj_l>Fi`lFcj_l@$
Ffj_l>Fi`lFgj_l>F]aqFcv@%/Fi`lF[oC*>F_`sF]aq>Fa`r-FjisF]_r?(FcvF[oF[oFevF_sC$@$
/Fej_lFa`r>F]_pFcv@$/Fij_lFa`r>F]_pFcv>F`[lF]i_l-Fjfm6&FfjsF]_pFgjsF_`s-Fjfm6$F
jjsF`[lFjdm>FacnF[oC'>F`[l-F[er6%F`[lF]aqFi`l-Fjfm6&F_grF]aqFhdrFi`l-Fjfm6$F[]t
F`[lFjdm>FacnF]jm-F:6$F]j_lF]^tC+>F[xFbal>Fd]lFbal?(FcvF[oF[oFevF_sC$@$50-Ffhs6
$Fe[`lFej_lFbal0-Ffhs6$Fe[`lFij_lFbal>F^cnFcv@$Faj_l@%/F[xFbalC$>F[xFcv@&Fbj_l>
FialFcj_lFfj_l>FialFgj_lC$>Fd]lFcv@&Fbj_l>FetFcj_lFfj_l>FetFgj_l@$0FialF[oC$>F`
[l-Fdgr6&F`[lF[xF^cnFial-Fjfm6(F_grF[xFhdrFialF^atF^cn@$F^`wC$>F`[l-Fdgr6&F`[lF
d]lF^cnFet-Fjfm6(F_grFd]lFhdrFetF^atF^cn@$3/FialF[o/FetF[oC$>F`[lF^_`lF`_`lF_d\
lFjdm>FacnFi_n-Fjfm6#%BChoose~a~selection~from~the~menu.G>F`dqFdh_l@$/-Fb^o6$-F
hjn6#,&F`dqF[oF`[lF^vF[oFbalC$-Fjfm6$%KThe~matrix~in~reduced~row~echelon~form~i
s:GF`[l>FdzF]jm@$3/FdzFbalFbf_l-Fjfm6$%Swarning.~Matrix~is~not~in~reduced~row~e
chelon~formGF`[lFjdm@$Fhiv-Fjfm6#%[oTo~solve~the~system~apply~backsubstitution~
to~the~reduced~matrixGFjdm-Fjfm6#%doThis~is~the~end~of~the~interactive~Gauss-Jo
rdan~procedure.~You~may~selectG-Fjfm6#%coother~examples~to~enforce~your~underst
anding~of~this~important~algorithmGF/FagmF/F/,
I*lasterror%OIEEE~Math:~SignalNaN,~nan,~0,~000000000000FC7FGF/,
I2type/CM_SEPARATORRFOF/F-F/-F:6$FK-F`j[l6$Fbj[l.F_`lF/F/F/6",
I+APPENDPLOTR6$%"gG'%*dimensionGFe[_l6+%,CoordinatesG%)NewPlotsGFP%&PlistGFd\s%
&PlotsG%/ExcludedColorsG%)newcolorG%(OptionsGF-F/C-@$F]s--FS6#FdsF/@'/F[pFham>F
cv7#FK/F[pF\bm>FcvFK>FcvF`d`l-F26#-F:6$Fcv-Fg_n6#<%Fg_n%*algebraicG/F[e`lF[e`l-
F26#5/F<F]jm/F<Fi_n>F_y-F`_]l6$R6#FjrF/F/F/-F:6$FKFg_nF/F/F/7#&Fcjm6#;Fi_nF\jm>
F`gn-Fdc[l6$RFfe`lF/F/F/Fge`lF/F/F/Fie`l>F[`[lFbal>Fdz--FS6#FdcmFjy>F[uF\p@&F`e
`lC%>Fbo--FS6#FhcmFjy>%6_EnvSMARTPLOTCoords2DGFbo?&Fh`nFcvF_sC$@'-F:6$Fh`n<$Ffc
[lF[e`lC'@$53-F:6$Fh`nF[e`l2F[o-F`v6#-FajlFbdo3-F:6$Fh`nFfc[l2F]jmF\h`l-Fj]n6#%
Eattempting~to~add~3D-Plot~to~2D-PlotG>F`[l-F\cmF\]l>Fdz7$-FexF\]lF`[l>Fgan-Fex
6#-%*smartplotGFbdo>Fgan-Fhbm6$Fgan/.FhenF`[l-F:6$Fh`nFg_nC%@$53-F:6$&Fh`nF\emF
[e`l2F[o-F`v6#-Fajl6#F]j`l3-F:6$F]j`lFfc[l2F]jmF_j`lFch`l>FganFh`n@%4-%$hasG6$F
gan<$Fdi`l.%'colourGC%>F`[lFgh`l>FdzFih`l>FganFai`l>Fdz7$Fjh`l-Fex6#-Fdcm6#7#Fg
an-Fj]n6#%2unknown~structureG>F[u6$F[uFganFae`lC%>Fbo--FS6#F\dmFjy>%6_EnvSMARTP
LOTCoords3DGFbo?&Fh`nFcvF_sC$@'Fbg`lC$@$53Fig`lFbh`l3F`h`l2Fi_nF\h`l-Fj]n6#%3to
o~many~variablesG>Fgan-Fex6#-%,smartplot3dGFbdoFei`lC$@$53F[j`lFfj`l3Fdj`l2Fi_n
F_j`lF`]al>FganFh`nF[\al>F[uF_\al6%-FexFjyF[u-FexFiczF/Fj_[lF/6",
I0CM_GetArrayVarsR6#'FPFd_]l6$%%valsGF,Fei[lF/C%>FboFgo?&F[u<#FcjmF_s@%-F:6$F[u
Fd_]l>Fbo-F_bn6$Fbo-Fio6$Fex<#-FhxFav>Fbo<$FboF[u-FajlF`gmF/F/F/6",
I)continueRF/6#%(defaultGFgdmF/C'@%Fc[n>FboFbjm>FboF[o-F[emF`gm-F^em6#%ao<~type
~;~and~press~Enter~to~continue~?topic~for~help~~exit;~to~quit~>|+G>FafmF^fmFd`a
lF/FagmF/F/,
I4help/text/trproduct-F`p6-Fbp%KHELP~SCREEN~for~:~Transpose~of~the~ProductGFbpF
dp%foCONTENTS~:~This~function~checks~if~the~product~of~the~transpose~is~equal~t
oG%Z~~~~~~~~~~~transpose~of~the~product~for~~matrices~A~and~BGFbp%U~~~~~~~~~~~~
To~Call~:~~>~A~:=~matrix([[1,2],[3,3]]);G%U~~~~~~~~~~~~~~~~~~~~~~~>~B~:=~matrix
([[1,0],[7,7]]);G%I~~~~~~~~~~~~~~~~~~~~~~~>~trproduct(A,B);GFbpF/,
I1context/cmglobalR6$'%#wwG<$%'symbolGFcj\l'FPFbj[l6'%"wG%,NotAssignedG%%exprG%
'newvarGF^hmF-F/C.>FboFK-F26#-F:6$FboFgaal>F[uRFfe`lF/Fei[lF/C$@$4Fij\l-F_o6#Fd
gm-Fj^]l6#/FK-F]blF]\lF/F/F/@$-F:6$F[p%(indexedGC$>&%7context/cmglobal/namesGF`
gm,&-Fex6$F[oF[pF[oF^vF[o-F26#3-F:6$Ffcal%(integerG1F^vFfcal@$2F[oF\jm>Fh`nF<@$
-F:6$FboFcj\lC&@$4-F56#Ffcal@%4-F[[al6$-Fdo6$Fh`nFcj\lFboC$>FfcalF^vF_gm>FfcalF
^v?(Fgan,&FfcalF[oF[oF[oF[oF/F_sC$>Fcv-F\[o6$-F_[]l6%%!GFboFganFcj\l@$4-F[[al6$
FaealFcvFgbu>FfcalFgan-F_oFix-F26#-F:6$FboFhaal>Fbo-F\[o6$FKFcj\l@&455-F5F]\lF[
eal-F[[al6$-Fdo6$Fh`nFhaalFbo>FfcalF^vFjdal>FfcalF^v@%-F:6$Fh`nFji^l?(FganFgeal
F[oF/F_sC$>FcvF\fal@$3-F[uFix4-F[[al6$-F]blFbdoFcvFgbu?(FganFgealF[oF/F_sC$>Fcv
F\fal@$3F_hal4-F[[al6$FcgalFcvFgbu>FfcalFganFdfalF/6#FgcalF/6",
I(cleanupRF/6(%#s2GF_hm%#chG%#s1GFjj[l%)prevprevGFgdmF/C(>FboFbp>FcvFbjm>F[uFi_
n?(F/F[oF[oF/1F[u-%'lengthGFixC&>F_y-Fhn6$Fcv;,&F[uF[oF\oF[oFcjal>Fgan-Fhn6$Fcv
;,&F[uF[oF^vF[oFhjal>Fh`n-Fhn6$Fcv;F[uF[u@%330F_y%")G0F_y%"(G0F_yFeiu@)33/F_y%"
+G/Fgan%"-G/Fh`n%"1GC$>Fbo-F_[]l6$Fbo%#~-G>F[u,&F[uF[oFi_nF[o3/F_yF[\bl/FganF]\
blC$>Fbo-F_[]l6$FboF[\bl>F[u,&F[uF[oF]jmF[oFg[blC$>FboFj\bl>F[uF]]blC$>Fbo-F_[]
l6$FboF_y>F[u,&F[uF[oF[oF[o>F[uFf]bl>Fbo-F_[]l6%FboFganFh`nF_gmF/F/F/F/,
I4help/text/trinverse-F`p6,Fbp%GHELP~SCREEN~for~:~Transpose~of~inverseGFbpFdp%f
oCONTENTS~:~This~function~checks~if~the~transpose~of~the~inverse~is~equal~toG%Q
~~~~~~~~~~~inverse~of~the~transpose~of~matrix~A~GFbp%U~~~~~~~~~~~~To~Call~:~>~~
A~:=~matrix([[1,2],[3,3]]);G%F~~~~~~~~~~~~~~~~~~~~~~>~trinverse(A);GFbpF/,
I.type/CM_CLASSRFOF/F-F/-F:6$FK-F`j[l6$Fbj[l.F_]lF/F/F/6",
I)chkvalidR6$'F%Fhjm'FggoFgep6#F^hmF/F/C$?(FboF[oF[o-F`_nF]\lF_s@$/-Fb^o6$-Fhjn
6#,&F<F[o-Ff_t6$FKFboF^vF]jmFbal>%'answerG%#okGF_oF/6#F^`blF/F/,
I*constructRF/6-%&eqsetG%#eqG%(varlistGF^hmF_hmFbgoF\jqF\hqF,FfhqF`hqFgdmF/C+>F
boFbjm>Fh`nF][n>FdzF[an?(FcvF[oF[oFdzF_s>&F[uFix&FboFix>F`[l-F`vFbdo>Fcdn-Fhjm6
%FdzFf`rFbal?(FcvF[oF[oFdzF_sC$>Facn<#-Fex6#-Fjis6#F]abl?(FganF[oF[o-F`vFd[sF_s
?(F_yF[oF[oF`[lF_s@$0-Ffhs6$&FacnFg\o&Fh`nFjyFbal>&Fcdn6$FcvF_yFabbl?(FcvF[oF[o
FdzF_s>&Fcdn6$FcvFf`r-F^hsF[bbl-F_oFa\rF/F/F/F/,
I&blankRF+F`_blF/F/?(FboF[oF[oFKF_s-FjfmF^gmF/F/F/F/,
I0type/CM_NOPRINTRFfe`lF/Fei[lF/-F:6$FK-F`j[l6$Fbj[l<$%&PADICG%&CFRACGF/F/F/6",
I0type/CM_SUBMENURFOF/F-F/-F:6$FK-F`j[l6$Fcj\l.Ff^lF/F/F/6",
I'bksub1RF/6?F^hmFfevF_hmFbgoF\jqF\hq%&soln1GF]hoF`hq%(charsetG%)solnvarsG%)sol
nvectG%(tempsumGF_hnFggoFjrFfhq%)solneqnsGFe`blFdg[l%&eqns1G%%solnG%&rightG%)ty
epflagGF^fvF`hn%&countGFbg[l%)templistGFgdmF/C5>Facn-Fgep6#7.&%#_tGF\em&FieblF^
[n&FieblF][q&FieblFfhp&Fiebl6#""&&Fiebl6#""'&Fiebl6#Fc\q&Fiebl6#"")&Fiebl6#""*&
Fiebl6#F^gn&Fiebl6#"#6&Fiebl6#"#7>F`gnFbjm>F]aqF][n>Fgan-F`_r6#-Fcbn6$F`gnF[o>F
_y,&-F`_r6#-Ff_tFigblF[oF^vF[o>Fet.Fet>Fet-FgepFjy?(FboF[oF[o-F`_rFbcqF_s>&FetF
`gm&F]aqF`gm>F[x-Fhjn6#-Fajr6$-Fg\r6%F`gnFi\r;F[oF_y-F\[o6$FetFhjm>Fd]lFihbl>Fa
u-Fg\r6%F`gnFi\r;F^ewF^ew>F`gn-F\\sFicz?(FboF[oF[oFganF_s@$0&F`gn6$FboFboFbal>F
`gn-F[er6%F`gnFbo*$F\jblF^v>F[xFihbl>FjdnF]ibl>FfcnFbhbl?(FboF[oF[oF_yF_s>&Ffcn
F`gmFfhbl>Fa`r-Fian6$FjdnFfcn@%/-F`\rF^cv-F`\rFiczC4>FboFgan?(F/F[oF[oF/-Fh`\l6
#-Ff_t6$F`gnFbo>Fbo,&FboF[oF^vF[o>FganFfgbl>F_yF[hbl@$2FboFgan>F[xFihbl@%/FganF
_yF/C$-Fjfm6#%JThis~system~has~infinitely~many~solutionsGFjdm>F_^lFbhbl?(FboF[o
F[oF_yF_s>&F_^lF`gmFfhbl>Fi`l-Fgep6$F_yFbal>FevF_y?(FcvF^[clF^vF[oF_sC,>FboF[o?
(F/F[oF[oF//&F`gn6$FcvFboFbal>Fbo,&FboF[oF[oF[o>&Fi`lF`gmF[o?(Fh`nFc]clF[oF_yF_
s@$0&Fi`lFbdoF[oC%>&F_^lFbdo&Facn6#,(F_yF[oFevF^vF[oF[o>Fi]clF[o>Fev,&FevF[oF^v
F[o>FgdnFbal?(Fh`nFc]clF[oF_yF_s>Fgdn,&FgdnF[o*&&F`gnF[^]lF[oF\^clF[oF[o>Ff\cl*
&,&&F`gn6$FcvF^ewF[oFgdnF^vF[oF`]clF^v@%0FboF_yC(@%/FcvF^[cl-Fjfm6#%KChoose~the
~free~variable(s)~and~substituteG-Fjfm6#%-substitute~:G?(Fh`nFc]clF[oF_yF_s-Fjf
m6#/&FetFbdoF\^clFjdm-Fjfm6$%/in~equation~:~G/&Fa`rFixF\_cl-Fjfm6$%5to~find~the
~variableG/FfhblFf\clFjdmC$-Fjfm6#%FBy~using~the~last~equation~we~can~getG-Fjfm
6#/F[ipF`[qF_in@$FcfmFcin-Fjfm6#%STherefore~the~values~of~the~variables~are~giv
en~byG>F`dqFbhbl?(FboF[oF[oF_yF_s>&F`dqF`gmFf`cl>F^cnFbhbl?(FboF[oF[oF_yF_s?(F[
uFboF[oF_yF_s@$/&F]aqFav-Fjis6#&F`dqFav>&F^cnFav/F[bcl-F^hsF]bcl-FjfmFdcnFjdmC%
Fjdm-Fjfm6#%coIn~the~above~system~we~observe~that~one~or~more~of~the~statement(
s)~leadG-Fjfm6#%foto~inconsistent~equation(s).~Therefore~the~original~system~is
~Inconsistent.GF/FagmF/F/,
I'bksub2RF/6?F^hmFfevF_hmFbgoF\jqF\hqFgdblF]hoF`hqFhdblFidblFjdblF[eblFfhqF_hnF
ggoFjrF\eblFe`blFdg[lF]eblF^eblF_eblFciqF^fvF`hnFaeblFbg[lFbeblFgdmF/C6>FacnFee
bl>F`gnFbjm>F]aqF][n>FganFfgbl>F_yF[hbl?(FboF[oF[o-F`_nFiczF_s?(Fh`nF[oF[o-Fb_n
FiczF_sF/>F^cn.F^cn>F^cnFbhbl?(FboF[oF[oFdhblF_s>&F^cnF`gmFghbl>F[x-Fhjn6#-Fajr
6$F]ibl-F\[o6$F^cnFhjm>Fd]lF_dcl>FauFdibl>F`gnFhibl?(FboF[oF[oFganF_s@$F[jbl>F`
gnF_jbl>F[xF_dcl>FhvF]ibl>FfcnFbhbl?(FboF[oF[oF_yF_s>FgjblF]dcl>Fa`r-Fian6$FhvF
fcn@%/-F`\rFdwF^[clC7>FboFgan?(F/F[oF[oF/Fb[cl>FboFg[cl>F`gn-Fg\r6%F`gn;F[oFbo;
F[oF^ew>FganFfgbl>F_yF[hbl@$F[\cl>F[xF_dcl@%F^\clF/F/Fjdm>F_^lFbhbl?(FboF[oF[oF
_yF_s>Ff\clF]dcl>Fi`lFh\cl>FevF_y?(FcvF^[clF^vF[oF_sC)>FboF[o?(F/F[oF[oF/F_]cl>
FboFc]cl>Fe]clF[o?(Fh`nFc]clF[oF_yF_s@$Fh]clC%>F\^clF]^cl>Fi]clF[o>FevFb^cl>Fgd
nFbal?(Fh`nFc]clF[oF_yF_s>FgdnFf^cl>Ff\clFj^cl>F`dqFbhbl?(FboF[oF[oF_yF_s>Feacl
/F]dclFf\cl-Fjfm6#%IThe~values~of~the~variables~are~given~byG>F`dqFbhbl?(FboF[o
F[oF_yF_s>FeaclF\hcl>FjdnFbhbl?(FboF[oF[oF_yF_s?(F[uFboF[oF_yF_s@$Fjacl>&FjdnFa
vFabcl-FjfmF^cvC%FjdmFebclFhbclF/FagmF/F/,
I3help/text/LUdecomp-F`p6UFbp%CHELP~SCREEN~for~:~LU-DecompositionGFbpF[a_l%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..jGFbp%
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.GFbp%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~inverseGFbp%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~=~LUGFbp%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[c]lF_a_lF`a
_l%G~~~~~~~~~~~>A:=~matrix([[1,2],[4,4]]);G%9~~~~~~~~~~~>LUdecomp(A);GFbp%hn~~~
~~~~~~~~The~procedure~can~also~be~called~without~argumentsG%8~~~~~~~~~~~>LUdeco
mp();GFbpFbpF/,
I8context/InitializeProcsR6#'F^rFcj\l6$Fjr%%KeysGFei[lF/C&>F[u-F`_]l6%F:7#-Fce[
lF^bl7#Fhaal>F[u-Fio6$FexF[u?&FboF[uF_sC$-F26#-F:6$&FgtF`gmFji^l-F>6$Fbo-F]bl6#
Fj\dl>&%2Context/InitProcsGF]\lF_sF/6$FgtFa]dlF/6",
I1help/text/linsys-F`p6<Fjb]l%'~~~~~~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~seetingGFbp%gnCALLING~SEQUENCE:~You~can~
call~any~function~BY~the~statementG%5<function>(args)~~ORG%9linsys[<function>](
args)GFbp%*SYNOPSIS:G%P-~to~use~a~linsys~function,~invoke~the~functionG%D~using
~the~form~linsys[<function>].G%$~~~G%@-~The~functions~available~are~:GFe^dl%C~~
~~gausselim~~gaussauto~gaussman~GFe^dl%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_completeRFbi[lF/Fei[lF/C%-FS6#.FgilF\g\l-Fgil6#-F\`l6ZQgn
proc(s1::uneval,s2::uneval,s3::uneval,s4::uneval,s5::uneval)6"Q_o~~~~local~Eval
,~EvalLabel,Parameters,AutoAssignFlag,ParseFlag,Nargs,Fg_dlQO~~~~~~~~F,Labels,P
arams,i,ProcBody,CodeString;Fg_dlQ;~~~~global~none,CM_Global;Fg_dlQ@~~~~"Comple
te~Action~Template";Fg_dlQD~~~~readlib(~'`context/cmglobal`');Fg_dlQB~~~~readli
b(~'`context/testeq`');Fg_dlF\`dlQQ~~~~Parameters~:=~select(~type~,~[args]~,~`=
`~);Fg_dlQjn~~~~AutoAssignFlag~:=~subs(~op(Parameters),~"autoassign"=FAIL~,Fg_d
lQ6~~~~~~"autoassign"~);Fg_dlQX~~~~ParseFlag~:=~~subs(~op(Parameters),~"parse"=
false~,Fg_dlQ1~~~~~~"parse"~);Fg_dlQG~~~~Nargs~:=~nargs~-~nops(Parameters);Fg_d
lQap~~~~if~not~type([args[1]]~,~list(list)~)~then~ERROR(`list~of~lists~expected
`,args)~fi;Fg_dlQD~~~~Eval~:=~()~->~op(~eval(~subs(~|frFg_dlQM~~~~~~~~~~~PIECEW
ISE=`context/fixpiecewise`,Fg_dlQ:~~~~~~~~~~~MATRIX=matrix,Fg_dlQI~~~~~~~~~~~VE
CTOR=vector|hr~,~[args]~)));~Fg_dlQ\p~~~~Labels~:=~vector(~map(~convert~,~map2(
~op~,~1~,~[args[1..Nargs]]~)~,~name~));Fg_dlQ6~~~~for~i~to~Nargs~doFg_dlQ<~~~~~
~~~if~Labels[i]~=~noneFg_dlQK~~~~~~~~or~(not~type(Labels[i],name))~thenFg_dlQS~
~~~~~~~~~~~~Labels[i]~:=~Eval(op(2..-1,args[i]));Fg_dlQ\p~~~~~~~~elif~not~`cont
ext/testeq`(eval(Labels[i]),~eval(op(2..-1,args[i]))~)~thenFg_dlF]adlQ,~~~~~~~~
fi;Fg_dlQ(~~~~od;Fg_dlQG~~~~if~type(~procname~,~indexed~)~thenFg_dlQdo~~~~~~~~P
arams~:=~seq(~CF_ARG.i~=~op(i,procname)~,~i=1..nops(procname)~);Fg_dlQ)~~~~else
Fg_dlQ8~~~~~~~~Params~:=~NULL;Fg_dlQ(~~~~fi;Fg_dlQ8~~~~ProcBody~:=~subs(~|frFg_
dlQ]p~~~~~~~'f'=~~~`if`(~type([Labels[1]],[indexed])~and~evalb(op(0,Labels[1])=
Labels),Fg_dlQN~~~~~~~~~~~~~~~~eval(Labels[1],1),Labels[1]),Fg_dlQar~~~~~~~'ARG
S'~=~(seq(`if`(~type([Labels[i]],[indexed])~and~evalb(op(0,Labels[i])=Labels),e
val(Labels[i],1),Labels[i]),Fg_dlQ:~~~~~~~~~~~i=1..Nargs~)),Fg_dlQ/~~~~~~~Param
s,Fg_dlQS~~~~~~~'`CM_Global`(a)'~=~`context/cmglobal`('a'),Fg_dlQS~~~~~~~'`CM_G
lobal`(b)'~=~`context/cmglobal`('b'),Fg_dlQS~~~~~~~'`CM_Global`(x)'~=~`context/
cmglobal`('x'),Fg_dlQL~~~~~~~'CM_GLOBALG'~=~`context/CM_GLOBALG`,Fg_dlQ1~~~~~~~
~NULL~|hr,~Fg_dl-Fcil6&F_i\lQ9proc()~global~CM_ASSIGN;Fg_dlF\\lF`i]lQ'~~~~);Fg_
dlQ]o~~~~CodeString~:=~readlib('`context/BuildInputString`')(ProcBody);Fg_dlQho
~~~~CodeString~:=~readlib('`context/autoassign`')(CodeString,AutoAssignFlag);Fg
_dlQ>~~~~if~ParseFlag~=~true~then~Fg_dlQP~~~~~~~~CodeString~:=~`context/jointex
t`(~"~"~,Fg_dlQR~~~~~~~~~~~~"if~true~then"~,~CodeString~,~"fi"~);Fg_dlQJ~~~~~~~
~~parse(~CodeString~,~statement~);Fg_dlFcadlQ4~~~~~~~~CodeString;Fg_dlFeadlF`i]
lF/F/F/Fg_dl,
I6SMARTPLOTFindCoords2DR6#'Fdc`lFg_n6)F^hmF]fz%+candidatesG%(nextposG%*newcoord
sG%"XG%"YGFei[lF/C)>F_y.%%_NoXG>Fdz.%%_NoYG-FSF6>FganF@>Fcv-&F*6#.%&depthGFg\o?
&F[uFK2FcvF]jmC(@$1Fi_nF`ddlFgbu>Fh`n-Fdc[l6%Fd`[l&F[u6#;F]jmFi_n<$F_yFdz@$/Fh`
nFg`l%%nextG-Fio6%FFFh`nFgan?(FboF[oF[o-%$minG6$F]jmF`ddlF_s@&Fhiv>F_y&FganF`gm
F`[w>FdzF\fdl>FcvF`ddl7$F_yFdzF/F/F/6",
I.EDITSMARTPLOTRF/6(%1characterizationGF^hm%*newoptionGFdc`lFgc`lFd\sF-F/C%@$F]
sFjc`l>F[`[lFbal@%-F:6$FbjmF`dalC)>F[uFbjm>Fh`nF][n>Fcv-F`_]l6$RFfe`lF/F/F/Fge`
lF/F/F/Fie`l>Fgan-Fdc[l6$RFfe`lF/F/F/Fge`lF/F/F/Fie`l>F_y-Fhbm6$&FcvFavFh`n>Fcv
-F[\_l6$/F[uF_yFcv-F_o6$-FexFixFff]lC(>Fh`nFbjm>Fcv-F`_]l6$RFfe`lF/F/F/Fge`lF/F
/F/7#&Fcjm6#;F]jmF\jm>Fgan-Fdc[l6$RFfe`lF/F/F/Fge`lF/F/F/Fhhdl@%-F:6$Fh`n/F)Fbj
[l>Fbo-FjisFbdo>FboFh`n>Fgan-Fdc[l6%R6$Fjr%"yGF/F/F/-F[[alFJF/F/F/FganFbo-F_o6%
FahdlFff]lFh`nF/Fj_[lF/6",
I4SMARTPLOTColor_ListRF/F/Fei[lF/7*.Fien.%&greenG.Fefn.%(magentaG.%'orangeG.%%p
lumG.%&coralG.%$tanGF/F/F/6",
I'ABeqBARF/6%FievF_hnF^euFgdmF/C9-F_`n6#F_\oFjdm>Fbo-Fhjm6%F]jmF]jmFbal>&FboF]f
pF[o>&Fbo6$F]jmF]jmF[o-F^em6#%enIn~general~matrices~do~not~commute.~However,~fo
llowing~is|+G-F^em6#%hna~brief~list~of~conditions~on~A~and~B~,~so~that~they~com
mute|+GFjdm-F^em6#%Z~~1.~A~and~B~commute~if~B~is~a~power~of~A~and~vice~versa|+G
-Fjfm6$%-~Proof~:~LetG/Fh`n)F[uF\hq-Fjfm6%%&Then~G/Fh[v)F[u,&F\hqF[oF[oF[o/F]jz
Fc]elFjdmF_in@$FbinFcin-F^em6#%jn~~2.~More~generally~A~and~B~commute,~if~B~is~a
~polynomial~of~A|+G>F[u-Fhjm6#7$7$F[oF]jm7$Fi_nF[o-Fjfm6$%$LetGFhco>Fh`n-Fhjn6#
,(-Fajr6$,$F[uF]jmF[uF[oF[u!"$FboF^hp-Fjfm6$F]\]l/Fh`n,(*$FicoF]jmF]jmFicoFj^el
FgjnF^hp-Fjfm6$%+That~is~:~GF]dpFjdm-Fjfm6$Fg\]l/Fh[v-Fhjn6#-FajrFjan-Fjfm6$Fbe
w/F]jz-Fhjn6#-FajrFhizF/FagmF/F/,
I0help/text/trsum-F`p6-Fbp%GHELP~SCREEN~for~:~Transpose~of~the~sumGFbpFdp%boCON
TENTS~:~This~function~checks~if~the~sum~of~the~transpose~is~equal~toG%T~~~~~~~~
~~~transpose~of~the~sum~of~matrices~A~and~BGFbpFgp%T~~~~~~~~~~~~~~~~~~~~~~>~B~:
=~matrix([[1,0],[7,7]]);G%D~~~~~~~~~~~~~~~~~~~~~~>~trsum(A,B);GFbpF/,
I)linspace=F/FdgmE\[l(%-graphvectaddGRF/65FcgoFghnF^ho%%sizeG%)arrowsetGF^hmF_h
mFbgo%)colorsetG%'txtsetG%$txtG%'sumsetG%%sumxG%%sumyG%$ln1G%$ln2G%*sumstringG%
&psumxG%&psumyGF/F/C8>F`gn-Fgep6#7;%+aquamarineG%&blackGFefn%%navyGF_[el%%cyanG
%&brownG%%goldGFfjdl%%grayG%%greyG%&khakiGFijdl%'maroonGF[[el%%pinkGF][elFien%'
siennaGFa[el%*turquoiseG%'violetG%&wheatG%&whiteG%'yellowG-F_`n6#%*plottoolsGF^
`n-Fjfm6#%VThe~purpose~of~this~procedure~is~to~help~us~visualizeG-Fjfm6#%Ythe~c
oncept~of~vector~addition.~The~program~will~displayG-Fjfm6#%Xthe~original~vecto
rs~and~their~sums.The~sum~vectors~areG-Fjfm6#%6drawn~with~red~arrowsGF_in>Fh`n-
Fgep6#"#?@%F]^nC'-F^em6#%LHow~many~vectors~would~you~like~to~enter?~|+G>F[uF^fm
>FcvF[u>F_yF[o?(F/F[oF[oF/1F_yFcvC%-F^em6#%[oPlease~enter~a~vector~with~two~com
ponents.~eg:~>~vector([1,2]);|+G>FdbblF^fm@%4-F:6$FdbblFgep-Fjfm6#%GERROR!~a~ve
ctor~is~expected.~Try~againG>F_yF^ewC$?(F_yF[oF[oF\jmF_s@%-F:6$&FcjmFjyFgep>Fdb
blF_fel-Fj]n6#%:Vector~arguments~expectedG>FcvF\jm?(F_yF[oF[oFcvF_s@$0-F`_r6#Fd
bblF]jm-Fj]n6#%Fvectors~must~have~only~two~componentsG>Fbo-FgepFix>FcdnF^gel?(F
_yF[oF[oFcvF_sC$>F[cn-Fa_p6(7$FbalFbal7$&FdbblF\em&FdbblF^[nFdbp$F]jmF^v$FjfblF
\o/Fhen&F`gnFjy>&FcdnFjy-F\en6%7%FggelFhgel-F\[o6$FdbblFcj\l/Fcen<$%&RIGHTGFeen
/%%fontG7%%&TIMESG%%BOLDGF^gn>FganFgo>FacnFgo?(F_yF[oF[oFcvF_s>Fgan-F_bn6$Fgan<
$F[cnF^hel>FfcnFgo>Fa`r&F]j`lF\em>Fial&F]j`lF^[n?(F_yF[oF[oFehwF_sC.>FgdnFbal>F
jdnFbal>F^cnF^fal?(F`[lF]jmF[oF_yF_s>F^cn-F_[]l6$F^cnFc[bl?(FdzF[oF[oF^ewF_sC%>
Fgdn,&FgdnF[o&&Fh`nF\]lF\emF[o>Fjdn,&FjdnF[o&FfjelF^[nF[o@%0FdzF^ew@%1F]jmFdz>F
^cn-F_[]l6%F^cn-F\[o6$FfjelFcj\l%#)+G>F^cn-F_[]l6%F^cnFa[flFi[bl>F^cn-F_[]l6$F^
cnFa[fl>F[cn-Fa_p6(Fegel7$FgdnFjdnFdbpFigel$F[oF^vFgen>F^hel-F\en6%7%FgdnFjdnF^
cnFben/Fhhel7%FjhelF[ielFgfbl>&FhvFjy-%%lineG6&7$Fa`rFialF]\flFdfn/%*linestyleG
F]jm>F[ip-Fh\fl6&7$&&Fh`n6#F^ewF\em&Fb]flF^[nF]\flFdfnF[]fl>Ffcn-F_bn6$Ffcn<&F[
cnF[ipF^helFf\fl>Fa`rFgdn>FialFjdn-&Fa`n6#Ffgn6$-F_bn6$FganFfcn/%%axesG%'normal
GF/F/F/%&basisGRF/6HF^hmF_hmFfevFgevF\jqF\hqF^hoF[balF`hq%'chkansGFghnFfdmF]fvF
afvFd\sF]fz%"sGFigmFcgo%)basissetG%)basisaugG%'bassetG%'orgsetGFbgo%%sum0G%%sum
1GF^eblFfhq%'vecsetGF_hn%$augGF]ho%%mat1G%%aug1GF`hn%(tempsetGFaebl%#m1GFgdmF/C
0Fjdm-Fjfm6#%]pThe~purpose~of~this~procedure~is~to~learn,~how~to~verify~if~a~se
t~of~vectors~formsG-Fjfm6#%\pa~basis.~Recall~that~a~basis~for~a~linear~space~is
~a~minimal~linearly~independentG-Fjfm6#%]psubset~of~the~given~vectors~that~span
s~the~space.~For~more~information~type~?basisGFjdm-F^em6#%DSelect~one~of~the~fo
llowing~modes:|+G-F^em6#%<~~~~~1.~Demonstration~mode|+G-F^em6#%:~~~~~2.~Interac
tive~mode|+G-F^em6#%6~~~~~3.~No-Step~mode|+G>FcdnF^fm>FafmFcdn@$FcfmFcinFjdm@(/
FcdnF[oCP@%0F\jmFbalC'?(FboF[oF[oF\jmF_sC$>&FdzF`gm&FcjmF`gm@$4-F:6$FgaflFgep-F
_o6#%EError!~All~arguments~must~be~vectorsG>F_yF\jm>Fgan-F`_r6#&FdzF\em>F`gn-Fh
jmFh\q?(FboF[oF[oFganF_s?(F[uF[oF[oF_yF_s>&F`gnFe`[l&&FdzFavF`gmC.-F^em6#%gnPle
ase~Enter~the~set~of~vectors.~All~vectors~must~be~of~the|+G-F^em6#%Asame~dimens
ion.~To~end~type~>0;|+G-F^em6#%WExample~:~>~v1:=vector([3,3,2]);~v2:=vector([0,
1,1]);|+G>FfcnF^fm>FafmFfcn@$FcfmFcin>FboF[o>FafmFfcn@$FcfmFcin?(F/F[oF[oF/0Ffc
nFbal@%-F:6$FfcnFgepC+>FafmFfcn@$FcfmFcin>FgaflFfcn@%Fhiv>F`gn-F\[o6$FdbflFhjm>
F`gn-F`cw6$F`gnFgafl>FboFc]cl-F^em6#%LEnter~the~next~vector;or~type~0~~when~don
e|+G>FfcnF^fm>FafmFfcn@$FcfmFcinC&-F^em6#%AA~vector~is~expected.~Try~again|+G>F
fcnF^fm>FafmFfcn@$FcfmFcin>F_yFg[cl@$1F[oF_y>FganFeccl?(FboF[oF[oF_yF_sC$>Fgafl
-FgepFg\o?(F[uF[oF[oFganF_s>&FgaflFav&F`gn6$F[uFbo>FauFgo?(FboF[oF[oF_yF_s>Fau-
F_bn6$Fau<#-F\[o6$-Fhjn6#FgaflFhjm-F^em6#%]qA~set~of~vectors~S~is~a~basis~for~a
~linear~space~V|+if~the~following~two~conditions~are~satisfied,|+G-F^em6#%E~~~~
~1.~~S~is~linearly~independent.|+G-F^em6#%F~~~~~2.~~S~spans~the~linear~space~V.
|+GFjdm-Fjfm6%%.The~given~setG/Fa`r-F]bl6#Fau%5may~form~a~basis~forG-Fjfm6%)%"R
GFgan%6or~for~a~subspace~of~GFdhflF_in@$FcfmFcin-Fjfm6#%_oFirst~we~check~to~see
~if~the~given~vectors~are~linearly~independent.G-Fjfm6#%fnConstruct~the~matrix~
AUG,~whose~rows~are~the~given~vectors.G-Fjfm6#/%/The~matrix~AUGG-Fhjn6#-Fa\oFic
zF_in@$FcfmFcin-Fjfm6#%MApply~Gauss-Jordan~Elimination~to~matrix~AUGG-Fjfm6#%Ht
o~obtain~its~reduced~row~echelon~form.G>Fgdn-Fj]yFcifl-Fjfm6#/%DThe~reduced~row
~echelon~form~of~AUGGF\_rFjdm-Fjfm6#%hnInspect~the~reduced~matrix~and~determine
~how~many~independentG-Fjfm6#%envectors~can~be~drawn~from~the~rows~of~the~reduc
ed~matrix~?GF_in@$FcfmFcin@*32-F`\rF]_rF_y/F\[glFganC'-Fjfm6#%epThe~reduced~row
~echelon~form~of~the~matrix~shows~that~at~least~one~of~the~original~vectorsG-Fj
fm6#%epis~a~linear~combination~of~the~others.~Therefore~the~set~of~vectors~is~l
inearly~dependent.G-Fjfm6&%DHowever,~we~can~extract~a~subset~ofGF\[gl%Jvectors~
that~forms~a~basis~for~the~space~GFdhflFjdm>F`]qF[o3F[[gl2F\[glFganC(F_[glFb[gl
-Fjfm6%Fg[glF\[gl%=linearly~independent~vectorsG-Fjfm6&%Fthat~forms~a~basis~for
~a~subspace~of~GFdhfl%-of~dimensionGF\[glFjdm>F`]qF]jm3/F\[glF_yF[\glC/-Fjfm6#%
aoThe~reduced~row~echelon~form~of~the~matrix~shows~that~no~vector~of~theG-Fjfm6
#%^ooriginal~set~of~vectors~can~be~written~as~linear~combination~of~theG-Fjfm6#
%aoothers.~Therefore~the~original~set~of~vectors~is~linearly~independent.GF_in@
$FcfmC$-Fjfm6#%(Bye~nowGF]gm-Fjfm6#/%=The~original~set~of~vectors~GF_hfl-Fjfm6%
%.does~not~haveGFgan%0vectors~to~spanG-Fjfm6%Fdhfl%Hbut~they~form~a~basis~for~a
~subspace~ofGFdhfl-Fjfm6$Fc\glF\[glFjdm>F`]qFi_n-Fjfm6#%;End~of~the~basis~proce
dureGFcin3Ff\glF][glC1Fh\gl-Fjfm6#%`ooriginal~set~of~vectors~can~be~written~as~
a~linear~combination~of~theGF^]glF_in@$FcfmC$Fc]glF]gmFf]gl-Fjfm6&%*does~haveGF
gan%>independent~vectors~that~spanGFdhflF_in-Fjfm6$%RTherefore~the~set~is~a~bas
is~for~the~linear~spaceGFdhflFa^glFjdm>F`]qF^hp-Fjfm6#%aoThis~is~the~end~of~the
~demonstration~version.~You~may~change~the~givenG-Fjfm6#%_ovectors~and~execute~
the~procedure~again~to~learn~the~steps~involved.GFcinF_in@$FcfmFcin@&F_hxC'-Fjf
m6#%hnSince~the~given~set~of~vectors~is~linearly~dependent,~we~mustG-Fjfm6%%4ex
tract~a~basis~forGFdhfl%>from~the~given~set~of~vectorsG-Fjfm6%%+Choose~anyGF\[g
l%5linearly~independentG-Fjfm6#%>vectors~from~the~original~setGFjdm/F`]qF]jmC(F
_`gl-Fjfm6&%Bextract~a~basis~for~a~subspace~ofGFdhflFc\glF\[gl-Fjfm6%Fh`glF\[gl
%Slinearly~independent~vectors~from~the~original~setGFjdmF_in@$FcfmFcin-Fjfm6#/
%FRecall~the~original~set~of~vectors~isGF_hflFjdm>F[x<#-FhjnFcbfl?(F_`sF]jmF[oF
]brF_sC&>Fd]l-F\[o6$-Fhjn6#&F[xF\emFhjm?(FboF]jmF[o-F`v6#F[xF_s>Fd]l-F`cw6$Fd]l
-F\[o6$&F[xF`gmFhjm>Fd]l-F`cw6$Fd]l-F\[o6$-Fhjn6#&Fdz6#F_`sFhjm@$/-F`\rFa_xFc_x
>F[x-F_bn6$F[x<#Fccgl>F_^lFgo?(FboF[oF[oFfbglF_s>F_^l-F_bn6$F_^l<#-F\[o6$-Fhjn6
#F]cglFhjm-F^em6$%YExtract~%d~linearly~independent~vectors~to~form~a~basis|+GFf
bgl-Fjfm6#/F^eu-F]blFc_lFjdmF_in@$FcfmFcin@$5F_hx/F`]qF^hpC6-Fjfm6$%inLet~us~ve
rify~that~the~above~set~is~basis~for~the~linear~spaceGFdhflFjdm-Fjfm6#%hn~~1.~C
learly~the~above~set~of~vectors~is~linearly~independentG-Fjfm6$%Z~~2.~Therefore
~it~is~sufficies~to~show~that~the~set~spansGFdhflF_in@$FcfmC$Fc]glF]gm>F`[l-Fg\
r6%-Fhjm6#7,7#F%7#Fggo7#%"cG7#F\fv7#%"eG7#FP7#F]c`l7#%"hG7#Fbo7#F[uFi\r;F[oF[o-
Fjfm6&%1Given~any~vectorGF`[l%+belongs~toGFdhfl-Fjfm6%%/We~shall~writeGF`[l%Xas
~a~linear~combination~of~the~basis~vectors~as~followsGFjdm>F]aq&F_^lF\em?(FboF]
jmF[o-F`vFc_lF_s>F]aq-F`cw6$F]aqFf\cl>F]aq-F`cw6$F]aqF`[l>Fev-&F_\o6#%(backsubG
6#-F\\sFbcq>Fi`l-Fgep6$FganFbal?(FboF[oF[oF^hglF_s>Fi`l,&Fi`lF[o*&&FevF`gmF[o-F
hjn6#Ff\clF[oF[o>F]_pFi`l>Fi`l.Fi`l>Fi`l,&F]_pF[oFi`lF^v-Fjfm6#/-FhjnF\czFi`l@$
5F]agl/F`]qFi_nC$-Fjfm6#%inThe~above~set~of~vectors~is~a~minimal~linearly~indep
endent~setG-Fjfm6&%9that~spans~a~subspace~ofGFdhflFc\glF\[glFjdmFf_glFi_gl/Fcdn
F]jmCU@%FbaflC'?(FboF[oF[oF\jmF_sC$>FgaflFhafl@$FjaflF]bfl>F_yF\jm>FganFbbfl>F`
gnFfbfl?(FboF[oF[oFganF_s?(F[uF[oF[oF_yF_s>FjbflF[cflC.F^cflFacflFdcfl>FfcnF^fm
>FafmFfcn@$FcfmFcin>FboF[o>FafmFfcn@$FcfmFcin?(F/F[oF[oF/F^dfl@%F`dflC+>FafmFfc
n@$FcfmFcin>FgaflFfcn@%Fhiv>F`gnFhdfl>F`gnF[efl>FboFc]cl-F^em6#%LEnter~the~next
~vector;~or~type~0~when~done|+G>FfcnF^fm>FafmFfcn@$FcfmFcinC&Feefl>FfcnF^fm>Faf
mFfcn@$FcfmFcin>F_yFg[cl@$F]ffl>FganFeccl?(FboF[oF[oF_yF_sC$>FgaflFbffl?(F[uF[o
F[oFganF_s>FefflFfffl-F^em6#%in~A~set~of~vectors~S~is~said~to~be~a~basis~for~a~
linear~space,|+G-F^em6#%M~if~the~following~conditions~are~satisfied:|+GFegfl-F^
em6#%=~~~~~2.~~S~spans~the~space.|+GF_in@$FcfmFcin>FauFbhbl>F^]pFgo?(FboF[oF[oF
_yF_sC$>&FauF`gmF^gfl>F^]p-F_bn6$F^]pF]gfl-F^em6%%ZEnter~the~%dx%d~matrix~whose
~rows~are~the~given~vectors.|+GF_yFgan-F^em6#%gnExample:~To~enter~a~matrix~type
:~matrix([[1,2,3],[1,1,1]]);|+G-Fjfm6$%AThe~original~set~of~vectors~is~:GFau>Fc
dnF^fm?(F/F[oF[oF/F`[rC&>FafmFcdn@$FcfmFcinFg\p>FcdnF^fm?(F/F[oF[oF/50-F`_nFa\r
F_y0-Fb_nFa\rFganC'-F^em6%%jnERROR!!~Incorrect~Dimensions.~%dx%d~matrix~expecte
d.~Try~again|+GF_yFgan-F^emFcgt>FcdnF^fm>FafmFcdn@$FcfmFcin>F[epF[o>F`gnFdifl?(
F/F[oF[oF/0-Fb^o6$-Fhjn6#,&FcdnF[oF`gnF^vF]jmFbalC(-Fjfm6#%DIncorrect~answer.~p
lease~try~again.G>FcdnF^fm>FafmFcdn@$FcfmFcin@$/F[epFi_nFgbu>F[ep,&F[epF[oF[oF[
o@%F[bhlC%-Fjfm6#%EYou~attempted~3~times~unsuccessfullyG-Fjfm6#%KThe~matrix~for
med~by~the~set~of~vectors~isG-FjfmFicz-Fjfm6$%>Good,~the~correct~matrix~is~:GFc
dn>FbipFcdnFjdm-F^em6#%gnSelect~the~most~appropriate~method~from~the~following~
menu:|+G-F^em6#%T~~1.~Find~the~LU-decomposition~of~the~above~matrix|+G-F^em6#%Y
~~2.~Find~reduced~row~echelon~form~of~the~above~matrix~|+G-F^em6#%N~~3.~Find~th
e~transpose~of~the~above~matrix.|+G-F^em6#%:~<Type~:~1;~or~2;~or~3;>|+GFjdm>Ffc
nF^fm>FafmFfcn@$FcfmFcin?(F/F[oF[oF/0FfcnF]jmC&-Fjfm6#%;Incorret~method.~Try~ag
ainG>FfcnF^fm>FafmFfcn@$FcfmC$FifmF]gm>F`\q-Fj]yFicz-Fjfm6#%OThe~reduced~row~ec
helon~form~of~the~matrix~is:G-FjfmF^]q>Feep-F`\rF^]qF_in@$FcfmC$Fc]glF]gm-F^em6
#%foHow~many~linearly~independent~rows~can~be~extracted~from~the~above~matrix?|
+G-F^em6#%IEnter~the~number~followed~by~semicolon;|+G>FcdnF^fm>FafmFcdn@$FcfmFc
in?(F/F[oF[oF/0FcdnFeepC&-F^em6#%6Incorrect.~Try~again|+G>FcdnF^fm>FafmFcdn@$Fc
fmFcinFjdm>Fgan-F`_nF^]q>F_y-Fb_nF^]q@%/F^ehlFganC.-Fjfm6%%-Correct,~allGFgan%L
vectors~of~the~set~are~linearly~independentGFjdm-F^em6#%SDoes~the~original~set~
of~vectors~form~a~basis~for|+G-F^em6$%;~~~1.~a~subspace~of~R%d~?|+GF_y-F^em6$%+
~~~2.~R%d|+GF_y-F^em6#%1Answer~1;~or~2;|+G>FcdnF^fm>FafmFcdn@$FcfmFcin@*3F_aflF
^\cl-Fjfm6$%NIncorrect.~The~original~set~forms~a~basis~forG)FehflF_y3F_afl0Fgan
F_y-Fjfm6$%enCorrect.~The~original~set~forms~a~basis~for~a~subspace~of~GFbhhl3F
ijglFdhhl-Fjfm6$%gnIncorrect.~The~original~set~forms~a~basis~for~a~subspace~of~
GFbhhl3FijglF^\cl-Fjfm6$%MCorrect.~The~original~set~forms~a~basis~for~GFbhhlFjd
m-Fjfm6$%-A~basis~is~:GF^]pC/-Fjfm6%%=Correct.~We~need~to~extract~GF^ehlFdaglFj
dm-F^em6$%inExtract~%d~Linearly~Independent~vectors~from~the~original~set|+GF^e
hl-F^em6#%JTo~enter~a~vector,~type~:~vector([1,2]);|+G>F_`sF[o>F[uF[o>F[xFgo?(F
/F[oF[oF/31F_`sF_y2FfbglF^ehlC/@$/F_`sF]jm>F`]qF[o-Fjfm6$%?The~original~set~of~
vectors~isGFauFjdm-F^em6#%SEnter~an~independent~vector~from~the~original~set|+G
>&FhisFfcglF^fm?(F/F[oF[oF/4-F:6$F_[ilFgepC&-F^em6#%6A~vector~is~expected|+G>F_
[ilF^fm>FafmF_[il@$Fcfm-F_o6#Fefm@$/-Fj`[l6$F_`sF]jmF[oFc_hl>F^`bl%&notokGFjdm?
(F/F[oF[oF//F^`blFb\ilC%-F[_bl6$F`gn-Fhjn6#F_[il@%Fd\ilC(-Fjfm6$%+The~vectorG-F
\[o6$Fh\ilFhjm-Fjfm6#%endoes~not~belong~to~the~original~set~of~vectors.~Try~aga
in.G-F^em6#%MPlease~enter~a~vector~from~the~original~set|+G>F_[ilF^fm>FafmF_[il
@$FcfmFcin-Fjfm6$%MYou~entered~a~vector~from~the~original~set~:GF_]ilFjdm@$/F[u
F[oC(-Fjfm6#%^oSince~a~set~consisting~of~a~non-zero~vector~is~linearly~independ
entG-Fjfm6%F^]ilF_]il%0is~in~the~basisG>Fcfs-F\[o6$-Fhjn6#&FhisF\emFhjm>F[x-F_b
n6$F[x<#Fg^il>F[uFf]bl>F`]qFbal@$32F[oF[u0F`]qFbalC9>F]aq-Fhjm6$-F`_n6#Fcfs,&-F
b_nF[`ilF[oF[oF[o?(Fh`nF[oF[oFj_ilF_s?(FcvF[oF[oF\`ilF_s@%0FcvF\`il>&F]aqFdiu&F
cfsFdiu>Fc`il&F_[ilFbdo>Fcfs.Fcfs>Fcfs-Fhjm6$-F`_nFbcq-Fb_nFbcq?(Fh`nF[oF[oF\ai
lF_s?(FcvF[oF[oF]ailF_s>Fd`ilFc`il>F]aq.F]aq>Fdjp-Fj]yF[`il@$0-F`\rFjjpF[uC(>Fe
v-Fghgl6#-F\\sF[`il>Fi`lF]igl?(FboF[oF[o-F`vFawF_s>Fi`l,&Fi`lF[o*&FciglF[o-Fhjn
6#-F\[o6$-Fcbn6$FcfsFboFhjmF[oF[o>F]_pFi`l>Fi`lFhigl>F]_pFjiglF_in@$FcfmFcin-Fj
fm6#%inLet~us~form~the~matrix~whose~rows~consist~of~the~current~basisG-Fjfm6$%6
and~the~new~vector~:~GF_]il-Fjfm6#-Fa\oF[`ilF_in@$FcfmC$Fc]glF]gm-Fjfm6$%EFind~
its~reduced~row~echelon~form~:~G-Fa\oFjjpFjdm-F^em6#%hoBased~on~the~reduced~row
~echelon~form,~are~the~vectors~linearly~independent?|+G-F^em6#%8<~answer~yes;~o
r~no;~>|+G>FcdnF^fm>FafmFcdn@$FcfmFcin@&/FcdnFi^o@%/FgailF[uC&-Fjfm6#%aoGood.~W
e~can~add~the~new~vector~to~form~a~new~linearly~independent~setG>F[x-F_bn6$F[x<
#F_]il-FjfmFgbgl>F[uFf]blC(-Fjfm6#%[oIncorrect.~The~new~vector~can~be~written~a
s~a~linear~combinationG-Fjfm6#%Jof~the~current~basis~vectors~as~follows~:G-Fjfm
6#/-Fhjn6#-F\[o6$F_[ilFhjmF]_pFjdm-Fjfm6#%VTherefore~the~new~vector~does~not~be
long~to~the~basisG>Fcfs-Fg\r6%FcfsF_ibl;F[oFhjal/FcdnF_^o@%FhdilC&-Fjfm6#%foinc
orrect.~We~can~add~the~new~vector~to~form~a~new~linearly~independent~setG>F[xF^
eilFaeil>F[uFf]bl@$FfailC(-FjfmFjjp-Fjfm6#%incorrect.~The~new~vector~can~be~wri
tten~as~a~linear~combinationGFgeilFjdmFafil>FcfsFefil>F_`s,&F_`sF[oF[oF[oF_in-F
jfm6%%2We~have~extractedGF^ehlFdaglFjdm-Fjfm6$%WTherefore,~the~basis~we~extract
ed~from~the~given~set~:GFau-Fjfm6$%%~is~GF[xFjdm-Fjfm6#%`pThis~is~the~end~of~th
e~interactive~version~of~the~basis~procedure.~Try~other~examplesG/FcdnFi_nC5@%F
baflC'?(FboF[oF[oF\jmF_sC$>FgaflFhafl@$FjaflF]bfl>F_yF\jm>FganFbbfl>F`gnFfbfl?(
FboF[oF[oFganF_s?(F[uF[oF[oF_yF_s>FjbflF[cflC.F^cflFacflFdcfl>FfcnF^fm>FafmFfcn
@$FcfmFcin>FboF[o>FafmFfcn@$FcfmFcin?(F/F[oF[oF/F^dfl@%F`dflC+>FafmFfcn@$FcfmFc
in>FgaflFfcn@%Fhiv>F`gnFhdfl>F`gnF[efl>FboFc]clF^efl>FfcnF^fm>FafmFfcn@$FcfmFci
nC&Feefl>FfcnF^fm>FafmFfcn@$FcfmFcin>F_yFg[cl@$F]ffl>FganFeccl?(FboF[oF[oF_yF_s
C$>FgaflFbffl?(F[uF[oF[oFganF_s>FefflFfffl>FauFgo?(FboF[oF[oF_yF_s>FauF[gfl>Fgd
nFhibl@*Fjjfl>F`]qF[oFj[gl>F`]qF]jmFe\gl>F`]qFi_nFg^gl>F`]qF^hp-Fjfm6#/%@The~or
iginal~set~of~vectors~is~GF_hflFjdm>FgdnF]jfl@*Fjjfl>F`]qF[oFj[gl>F`]qF]jmFe\gl
C(Ff]glFj]glF^^glFa^gl>F`]qFi_nFcinFg^glC(F^_glFb_glFa^glFjdm>F`]qF^hpFcinFjdm>
F[xF[bgl?(F_`sF]jmF[oF]brF_sC&>Fd]lF`bgl?(FboF]jmF[oFfbglF_s>Fd]lFibgl>Fd]lF_cg
l@$Fhcgl>F[xF[dgl>F_^lFgo?(FboF[oF[oFfbglF_s>F_^lFadgl-F^em6$%UA~basis~consists
~of~%d~linearly~independent~vectors|+GFfbglF[egl@$Faegl-Fjfm6$%NThe~above~set~i
s~a~basis~for~the~linear~spaceGFdhfl@$F`jgl-Fjfm6&%KThe~above~set~is~a~basis~fo
r~a~subspace~ofGFdhflFc\glF\[glF/6$FafmF^`blF/%-graphlincombGRF/6=FcgoFghnF^hoF
]aelF^aelF^hmF_hmF_aelF`aelFaaelFbaelFcaelFdael%'framesG%&frameGF\jq%#clG%%ctxt
G%#slG%%stxtGFgaelFbgoFhfglFeaelFfaelFhaelFiaelF/F/C6>F`[lF\belF`celF^`n-Fjfm6#
%ZThe~purpose~of~this~procedure~is~to~help~us~visualize~theG-Fjfm6#%Yconcept~of
~linear~combination.~The~program~will~take~twoG-Fjfm6#%envectors~and~create~lin
ear~combinations~of~the~two~vectors.G-Fjfm6#%gnThe~optional~third~argument~of~t
he~function~is~the~number~ofG-Fjfm6#%inlinear~combinations~you~want~to~view.~Cl
ick~on~the~play~buttonG-Fjfm6#%\oto~see~the~animations.~You~can~control~the~spe
ed~of~the~animationG-Fjfm6#%Iand~you~may~continuously~play~the~movie.GF_in@$Fcf
mC$Fc]glF]gm@%/F\jmFi_n>Fjdn&FcjmF][q>FjdnFi_n@'2Fi_nF\jm-Fj]n6#%gnThis~program
~display~linear~combinations~of~only~two~vectorsGF]^nC%>F_yF[o?(F/F[oF[oF/1F_yF
]jmC%F^eel>FdbblF^fm@%4-F:6$F]j`lFgep-F^em6#%IERROR!!~A~vector~is~expected.~Try
~again|+G>F_yF^ew>FcvF]jmC$?(F_yF[oF[oF]jmF_s@%F]fel>FdbblF_felFafel>FcvF]jm?(F
_yF[oF[oFcvF_s@$FgfelFjfel>FboF^gel>FacnF^gel?(F_yF[oF[oFcvF_sC$>F[cn-Fa_p6(Feg
elFfgelFdbpFigelFjgel/Fhen&F`[lFjy>&FacnFjyF_hel?(F_^lF[oF[oFjdnF_sC3@'/-Fj`[l6
$F_^lFi_nFbalC$>&FauF\em,&-Fj`[l6$-%%randGF/F^hpF[oF[oF[o>&FauF^[nFfcjl/FacjlF[
oC$>FecjlFfcjl>F\djl,&FgcjlF^vF^vF[oC$>FecjlFfcjl>F\djlFfcjl?(F_yF[oF[oFcvF_sC$
>&F^cnFjy-Fa_p6(Fegel7$*&&FauFjyF[oFggelF[o*&F]ejlF[oFhgelF[oFdbpFigelFjgel/Fhe
n&F`[l6#,&F_yF[oF]jmF[o>&Fa`rFjy-F\en6%7%F\ejlF^ejl-F_[]l6%-F\[o6$F]ejlFcj\lFei
uFbhelFdhelFghel>FfcnFbal>FgdnFbal>Fd]lF^fal?(FdzF[oF[oF]jmF_sC%>Ffcn,&FfcnF[o*
&&FauF\]lF[oFejelF[oF[o>Fgdn,&FgdnF[o*&FdfjlF[oFijelF[oF[o@%F[\\l>Fd]l-F_[]l6'F
d]l-F\[o6$FdfjlFcj\lFeiuFa[flFi[bl>Fd]l-F_[]l6&Fd]lF\gjlFeiuFa[fl>Fial-Fa_p6(Fe
gel7$FfcnFgdnFdbpFigelF^\flFgen>F[x-F\en6%7%FfcnFgdnFd]lFdhelFc\fl>F_`s-Fh\fl6&
7$*&FecjlF[oFeielF[o*&FecjlF[oFgielF[oFdgjlFdfnF[]fl>F]_p-Fh\fl6&7$*&F\djlF[o&&
Fh`nF^[nF\emF[o*&F\djlF[o&FehjlF^[nF[oFdgjlFdfnF[]fl>&Fhv6#,&F_^lFc\q!"'F[o-F\^
fl6$<$Ff_n&FacnF\emFa^fl>&Fhv6#,&F_^lFc\q!"&F[o-F\^fl6$<&Ff_nF`ijl&FacnF^[n&Fbo
F^[nFa^fl>&Fhv6#,&F_^lFc\q!"%F[o-F\^fl6$<(Ff_nF]arF`ijlFiijlFjijl&F^cnF\emFa^fl
>&Fhv6#,&F_^lFc\qFj^elF[o-F\^fl6$<*Ff_nF]arF`ijlFiijlFjijlFcjjl&F^cnF^[n&Fa`rF^
[nFa^fl>&Fhv6#,&F_^lFc\qF\oF[o-F\^fl6$<+Ff_nF]arF_`sF`ijlFiijlFjijlFcjjlF[[[mF\
[[mFa^fl>&Fhv6#,&F_^lFc\qF^vF[o-F\^fl6$<,Ff_nF]arF_`sF]_pF`ijlFiijlFjijlFcjjlF[
[[mF\[[mFa^fl>&Fhv6#,$F_^lFc\q-F\^fl6$<.Ff_nF[xFialF]arF_`sF]_pF`ijlFiijlFjijlF
cjjlF[[[mF\[[mFa^fl-Fjfm6#-Ffgn6%7#-F^x6$&FhvFc_l/F_^l;F[o,$FjdnFc\q/%+insequen
ceGF_s/%(scalingG%,constrainedGF/F/F/%'lindepGRF/6JF^hmF_hmFbgoF\jqF\hqF^hoF`hq
FghnFfdmF^_flF]_fl%(tempAUGGF]fv%%AUG3G%*setofvarsG%+solnvectorG%*eqnvectorGF[b
alF^ebl%%AUG0G%&soln0G%(homosumGFdg[l%+consisflagGF`hn%(zerovecGFfhq%%AUG2GFb[q
%(chkans1G%(chkans2G%(chkans3GFaebl%%res1G%&res11G%%AUGGG%)solutionG%(youransGF
[ebl%)smallsumGFgdmF/C.Fjdm-Fjfm6#%hoThe~purpose~of~this~procedure~is~to~determ
ine~if~a~set~of~vectors~is~linearlyG-Fjfm6#%hodependent~or~not.~It~includes~thr
ee~versions:~demonstration,~interactive,~andG-Fjfm6#%jnno-step.~For~more~inform
ation~type:~?lindep~at~the~Maple~promptGFjdmFjdmF_`flFb`flFe`flFh`fl>F`[lF^fm@(
/F`[lF[oCin@%F]^nC+-F^em6#%VPlease~Enter~the~set~of~vectors.~All~vectors~must~b
e|+G-F^em6#%Lof~the~same~dimension~as~w.~To~end~type~0;|+G>F`gnF[o>FboF[o-F^em6
#%@Please~enter~the~first~vector.|+G>&F_yF`gmF^fm?(F/F[oF[oF/0F`gnFbalC'>FafmFe
`[m@$FcfmFcin?(F/F[oF[oF/30Fe`[mFbal4-F:6$Fe`[mFgepC&>FafmFe`[m@$FcfmFcin-F^em6
#%4A~vector~expected.|+G>Fe`[mF^fm@$F]a[mC%>FboFc]cl-F^em6#%OPlease~enter~the~n
ext~vector.~To~end~type:~0;|+G>Fe`[mF^fm>F`gnFe`[m>FganFg[cl>Fcv-F`_r6#&F_yF\em
C%?(FboF[oF[oF\jmF_sC$>Fe`[mFhafl@$F^a[m-Fj]n6#%>All~arguments~must~be~vectorsG
>FganF\jm>FcvFbb[m>Fdz-FhjmFeav?(FboF[oF[oFcvF_s?(F[uF[oF[oFganF_s>&FdzFe`[l&&F
_yFavF`gm?(FboF[oF[o-Fb_nF\]lF_s?(F[uFc]clF[oFhc[mF_s@$/-Fb^o6$,&-Fcbn6$FdzFboF
[o-Fcbn6$FdzF[uF^vF]jmFbalC%-Fjfm6#%cpSince~among~the~vectors~entered,~two~vect
ors~are~the~same,~the~set~is~linearly~dependentG-Fjfm6#%hnEnter~distinct~vector
s~and~run~the~procedure~again.~Good~LuckG-F_o6#F^fal>Fhv-Fgep6#7,FahnF`bo%#c3G%
#c4G%#c5G%#c6G%#c7G%#c8G%#c9G%$c10G>Fi`l-F\[o6$-Fgep6$FcvFbalFhjm>Fhv-Fg\r6%-F\
[o6$FhvFhjmFi\rFbggl>F[xFdz?(FboF[oF[oFganF_s?(F[uF[oF[oFcvF_s>&Fe`[mFav&FdzFgf
fl>F_^lFbal?(FboF[oF[oFganF_s>F_^l,&F_^lF[o*&&Fhv6$FboF[oF[o-Fhjn6#-F\[o6$F_d[m
FhjmF[oF[o-Fjfm6#%hnIn~this~particular~example,~we~begin~with~the~vector~equati
onG-Fjfm6#/F_^l-Fhjn6#Fi`lFjdm-Fjfm6#%_oThe~problem~reduces~to~solving~a~homoge
neous~system~of~equations.~IfG-Fjfm6#%`othe~homogeneous~system~has~a~unique~sol
ution~(trivial~solution),~thenG-Fjfm6#%fothe~vectors~are~linearly~independent;~
otherwise,~the~vectors~are~dependent.GF_in@$FcfmC$FifmF]gm>FdzF`c[m?(FboF[oF[oF
cvF_s?(F[uF[oF[oFganF_s>Fdc[mFec[m>FetFbffl?(FboF[oF[oFganF_s>FfhblF]g[m>F^cn-F
ian6$FdzFet-Fjfm6#%]oThe~above~vector~equation~yields~the~following~homogeneous
~system:G?(FboF[oF[oFcvF_s-Fjfm6#/F]dclFbalF_in@$FcfmC$FifmF]gm-Fjfm6#%^oNow~co
nstruct~the~augmented~matrix~of~the~above~homogeneous~system.G>Fa`rF[f[m>Fdz-F`
cw6$FdzFa`r-Fjfm6$%0The~matrix~is~:G/Fdz-FhjnF\]lF_in@$FcfmC$FifmF]gm>Fgdn-F\\s
F\]l-Fjfm6#%coApply~the~Gauss~Elimination~to~obtain~the~row~echelon~form~of~the
~matrixG-Fjfm6#/%<The~row~echelon~form~of~AUGGF\_r>Fjdn-Fhjm6$F\[glFjar>FboF]br
?(F/F[oF[oF//-Fb^o6$-Ff_t6$FgdnFboF]jmFbal>FboFg[cl?(F[uF[oF[oFboF_s?(Fh`nF[oF[
oFjarF_s>&FjdnFjan&FgdnFjan@%3-F:6$&Fjdn6$FboFgan%(complexG0Fd\\mFbalC(>Fbo-F`_
nF^cv?(F/F[oF[oF/3-Fh`\l6#-Ff_t6$FjdnFbo2F[oFbo>FboFg[cl>F[uF[o?(F/F[oF[oF/3/&F
jdnFe`[lFbal2F[u-Fb_nF^cv>F[uFf]bl@$30Fg]\mFbal0Fg]\mF[o>Fjdn-F[er6%FjdnFbo*$Fg
]\mF^v>FgdnF]cvC%>F_y-Fgep6#Fi^y?(Fh`nF[oF[oFi^yF_s>&F_yFbdo&FjdnFgju@$34Fb\\m/
-Fb^o6$F_yF]jmFbalC*-Fjfm6%%.Assuming~thatGFg\\m%Lthe~echelon~form~of~the~augme
nted~matrix~isG>FboFj\\m?(F/F[oF[oF/F\]\m>FboFg[cl>F[uF[o?(F/F[oF[oF/Fe]\m>F[uF
f]bl@$F\^\m>FjdnF`^\m>FgdnF]cvFbhtF_in@$FcfmC$FifmF]gm-Fjfm6#%jnThis~row~echelo
n~form~of~the~augmented~matrix~yields~the~systemGFjdm>Fau-Fhjn6#-Fajr6$-Fg\r6%F
gdn;F[oF\[glFi\rFhv?(FboF[oF[oF\[glF_s-Fjfm6#/&FauF^g[mFbal-Fjfm6#%\oCan~you~te
ll~from~the~reduced~system~or~from~the~row~echelon~formG-Fjfm6#%gnof~the~augmen
ted~matrix,~if~it~has~the~unique~zero~solution?G-Fjfm6#%fnOr~if~the~system~has~
infinitely~many~non-trivial~solutions?GF_in@$FcfmC$FifmF]gm>Fial-FghglFj[s?(Fbo
F[oF[o,&FjarF[oF^vF[oF_s>&FialF`gm-F\y6(/FheblF^v/FjeblF^v/F[fblF^v/F\fblF^v/F]
fblF^vFbb\m>FacnFbal?(FboF[oF[oFganF_s>Facn,&FacnF[o-F[fy6#Fbb\mF[o@%/FacnFbalC
$-Fjfm6#%boThe~trivial~solution~is~the~only~solution~to~the~homogeneous~system~
andG-Fjfm6#%aotherefore~we~conclude~that~the~set~of~vectors~is~linearly~indepen
dent.GC4-Fjfm6#%VThe~homogeneous~system~has~a~non-trivial~solution~andG-Fjfm6#%
Wa~non-trivial~solution~to~the~above~system~is~given~byG-F^em6#%7~~~~~~~~~~~~~~
~~~~~~~~GFjdm?(FboF[oF[oFganF_sC$-F^em6%%0~~~~~~c%d~=~%a,GFboFbb\m@$/-Fj`[l6$Fb
oF^hpFbal-F^em6#%"|+GF_in-Fjfm6#%[oThis~implies~that~some~ci's~are~non-zero~in~
the~zero~combinationG-Fjfm6#%hnc1*v1+c2*v2+...+cn*vn~=~0~and~therefore~the~set~
of~vectors~isG-Fjfm6#%4linearly~dependent.GFjdmF_in-Fjfm6#%goSome~vectors~can~b
e~written~as~a~linear~combination~of~the~others~as~followsG>Ffcn-Fg\r6%F[x;F[o-
F`_nFgbgl;F[o,&-Fb_nFgbglF[oF^vF[o>Fgan-Fb_nF`dz?(F[uF[oF[oFi^yF_sC'>F[xFfcn>F[
x-%(swapcolG6%F[xF[uFgan>F_`s-%+consistentGFgbgl@%/F_`sF[oC*>Fd]l-Fghgl6#-F\\sF
gbgl?(FboF[oF[oFi^yF_s>&Fd]lF`gm-F\y6(Feb\mFfb\mFgb\mFhb\mFib\mFhg\m>FcdnF[f[m>
FacnF[f[m?(FboF[oF[oFi^yF_sC$>Facn,&FcdnF[o*&Fhg\mF[o-Fhjn6#-F\[o6$-Fcbn6$F[xFb
oFhjmF[oF[o>FcdnFacn>Fcdn.Fcdn>Facn,&FacnF[oFcdnF^v@%Fac\mC)>FevFacn>FcdnF[f[m>
FacnF[f[m?(FboF[oF[oFi^yF_s@$4-Fh`\l6#Ffh\mC$>Facn,&FcdnF[o*&%"0GF[oFbh\mF[oF[o
>FcdnFacn>FcdnFjh\m>FacnF\i\m-Fjfm6#/-Fhjn6#Fie[mFacn-Fjfm6#/-Fhjn6#-F\[o6$-Fcb
n6$F[xFganFhjmFacn-Fjfm6$Fgj\m%incannot~be~written~as~a~linear~combination~of~t
he~other~vectorsG@$0FevFbalC$F_in@$FcfmFcin>F[xFfcn>F_`sF]g\m@%F`g\mC*>Fd]lFcg\
m?(FboF[oF[oFi^yF_s>Fhg\mFig\m>FcdnF[f[m>FacnF[f[m?(FboF[oF[oFi^yF_sC$>FacnF`h\
m>FcdnFacn>FcdnFjh\m>FacnF\i\m@%Fac\mC(>FcdnF[f[m>FacnF[f[m?(FboF[oF[oFi^yF_s@$
Fdi\mC$>FacnFii\m>FcdnFacn>FcdnFjh\m>FacnF\i\mF_j\mFdj\mF][]mFjdm-Fjfm6#%foThis
~is~the~end~of~the~linear~dependence~procedure.~Please~change~the~givenG-Fjfm6#
%invectors~and~execute~the~procedure~again~to~learn~this~process~G/F`[lF]jmC^o@
%F]^nC+Fi_[mF\`[m>F`gnF[o>FboF[oFa`[m>Fe`[mF^fm?(F/F[oF[oF/Fg`[mC'>FafmFe`[m@$F
cfmFcin?(F/F[oF[oF/F\a[mC&>FafmFe`[m@$FcfmFcinFda[m>Fe`[mF^fm@$F]a[mC%>FboFc]cl
F[b[m>Fe`[mF^fm>F`gnFe`[m>FganFg[cl>FcvFbb[mC%?(FboF[oF[oF\jmF_sC$>Fe`[mFhafl@$
F^a[mFjb[m>FganF\jm>FcvFbb[m>FdzF`c[m?(FboF[oF[oFcvF_s?(F[uF[oF[oFganF_s>Fdc[mF
ec[m>FhvF]e[m>F[xFdz?(FboF[oF[oFganF_s?(F[uF[oF[oFcvF_s>Fff[mFgf[m>Fa`rF[f[m>Fg
dnF^j[m>Fh]p*&&FhvF\emF[o-F\[o6$-Fhjn6#-Fcbn6$FdzF[oFhjmF[o?(FboF]jmF[oFganF_s>
Fh]p,&Fh]pF[o*&&FhvF`gmF[o-F\[o6$-Fhjn6#F_d[mFhjmF[oF[o-F^em6#%^pHow~do~you~che
ck~if~the~given~set~of~vectors~is~linearly~independent~or~dependent?|+G-F^em6#%
X~~1.~Form~the~zero~combination~c1*v1+c2*v2+...+cn*vn=0|+G-F^em6#%fn~~2.~Visual
ly~inspect~vectors~to~find~a~dependency,~if~any|+G-F^em6#%9<~choose~from~1;~or~
2;>|+G>F`[lF^fm>FafmF`[l@$Fcfm-F_o6#Fhar@'Fe_[m-Fjfm6#%XVery~good.~This~is~the~
best~way~to~check~the~dependencyGFd]]mC$-Fjfm6#%`oThis~is~not~the~best~choice.~
In~most~cases~it~would~be~very~difficultG-Fjfm6#%\oto~detect~a~linear~dependenc
y~of~vectors~by~inspection.~Choose~1;G-Fjfm6#%EInvalid~choice.~The~best~choice~
is~1G-Fjfm6#%doThe~zero~combination~of~the~vectors~yields~the~following~vector~
equation:G-Fjfm6#/Fh]p-F\[o6$-Fhjn6#-Fcbn6$FgdnF^crFhjmFjdmFjdm-F^em6#%]oThe~pr
oblem~reduces~to~solving~a~homogeneous~system~of~equations:|+G-F^em6#%hnIf~the~
system~has~only~the~trivial~solution,~the~vectors~are|+G-F^em6#%;~~~1.~linearly
~dependent~|+G-F^em6#%=~~~2.~linearly~independent.|+GFha]m>F`[lF^fm>FafmF`[l@$F
cfmF^b]m@'Fe_[m-Fjfm6#%\pNo.~A~trivial~solution~to~the~system~implies~the~vecto
rs~are~linearly~independentGFd]]m-Fjfm6#%^pGood.~A~trivial~solution~to~the~syst
em~implies~the~vectors~are~linearly~independentG-Fjfm6#%hpInvalid~choice.~A~tri
vial~solution~to~the~system~implies~the~vectors~are~linearly~independentGFjdm-F
^em6#%WNow~from~the~zero~combination~of~the~vectors,~extract|+G-F^em6#%Mthe~hom
ogeneous~system~of~linear~equations.|+G?(FboF[oF[oFcvF_sC)-F^em6$%CPlease~enter
~equation~number~(%d)|+GFbo>&FeepF`gmF^fm?(F/F[oF[oF/4-F:6$F_f]m%)equationGC&>F
afmF_f]m@$FcfmF^b]m-F^em6#%DAn~equation~is~expected.~Try~again|+G>F_f]mF^fm>F_\
pFbal?(F[uF[oF[oFganF_s>F_\p,&F_\pF[o*&&FhvFavF[oFdc[mF[oF[o>&F`[lF`gm/F_\p&Fgd
n6$FboF^cr@$0Fcg]mF_f]m-Fjfm6$%DIncorrect.~The~correct~equation~is:GFcg]mF_in@$
FcfmF^b]m-F^em6#%TThe~system~of~linear~equations~resulting~from~the~|+G-F^em6#%
Mzero~combination~of~the~vector~equation~is:|+G?(FboF[oF[oFcvF_s-Fjfm6#Fcg]mFjd
m-F^em6%%hnEnter~the~%dx%d~associated~augmented~matrix~in~Maple~format;|+GFcvF^
cr-Fjfm6#%VFor~example~type:~AUG:=matrix([[1,2,2,0],[9,3,0,0]]);G>F`[lF^fm?(F/F
[oF[oF/4-F:F_[\lC&>FafmF`[l@$FcfmF^b]m-F^em6#%6A~matrix~is~expected|+G>F`[lF^fm
?(F/F[oF[oF/50Fd[\lFcv0Ff[\lF^crC'-Fjfm6#%=Incorrect~matrix~dimensions.G-F^em6%
%=Please~enter~a~%dx%d~matrix|+GFcvF^cr>F`[lF^fm>FafmF`[l@$FcfmFcin>FcfsF[o?(F/
F[oF[oF/30-Fb^o6$-Fhjn6#,&F`[lF[oFgdnF^vF]jmFbal1FcfsFi_nC'-F^em6#%KThis~is~not
~the~correct~matrix.~Try~again|+G>F`[lF^fm>Fcfs,&FcfsF[oF[oF[o>FafmF`[l@$FcfmC$
FifmF]gm@%F^[^mC%-Fjfm6#%AVery~good.~The~correct~matrix~isGFbhtFjdmC$-Fjfm6#%BI
ncorrect.~The~correct~answer~is:GFbht-F^em6#%9What~would~you~do~next?|+G>%(resf
lagGFbal?(F/F[oF[oF//Ff\^mFbalC%>F`gn-FcdmF/@*30-F`_nF\]lFhc[m5/F`gnF[o/F`gnF]j
m-Fjfm6#%>Please~make~another~selectionGFa]^m-%&menu1G6%FdzFcvFganFb]^m-%&menu2
GFh]^m/F`gnFi_n-%&menu3GFh]^m@$FcfmC$FifmF]gm-F^em6#%8Is~the~set~of~vectors:|+G
-F^em6#%=~~~1.~Linearly~independent?|+G-F^em6#%;~~~2.~Linearly~dependent?|+G-F^
em6#%3<~Type~1;~or~2;~>|+G>F`[lF^fm>FafmF`[l@$FcfmC$FifmF]gm>FialF^b\m@&Fe_[m@%
-Fh`\lF\cl-Fjfm6#%TCorrect.~The~set~of~vectors~is~linearly~independentGC$-Fjfm6
#%^oIncorrect.~The~set~of~vectors~is~linearly~dependent~as~shown~below.G>F]_pF[
oFd]]m@%Fc_^m-Fjfm6#%VIncorrect.~The~set~of~vectors~is~linearly~independentGC$-
Fjfm6#%\oCorrect.~The~set~of~vectors~is~linearly~dependent~as~shown~below.G>F]_
pF[oFjdm?(FboF[oF[oFhc[mF_s?(F[uFc]clF[oFhc[mF_s@$F[d[mC$-Fjfm6#%_oNote~two~vec
tors~are~the~same~and~thus~the~set~is~linearly~dependentGFjd[m@$Fi`tC)Fhe\m>Ffc
n-Fg\r6%F[xF^f\m;F[oFbf\m>FganFdf\m?(F[uF[oF[oFi^yF_sC'>F[xFfcn>F[xFif\m>F_`sF]
g\m@%F`g\mC*>Fd]lFcg\m?(FboF[oF[oFi^yF_s>Fhg\mFig\m>FcdnF[f[m>FacnF[f[m?(FboF[o
F[oFi^yF_sC$>FacnF`h\m>FcdnFacn>FcdnFjh\m>FacnF\i\m@%Fac\mC)>FevFacn>FcdnF[f[m>
FacnF[f[m?(FboF[oF[oFi^yF_s@$Fdi\mC$>FacnFii\m>FcdnFacn>FcdnFjh\m>FacnF\i\mF_j\
mFdj\mF][]m@$Fa[]mC$F_in@$FcfmFcin>F[xFfcn>F_`sF]g\m@%F`g\mC*>Fd]lFcg\m?(FboF[o
F[oFi^yF_s>Fhg\mFig\m>FcdnF[f[m>FacnF[f[m?(FboF[oF[oFi^yF_sC$>FacnF`h\m>FcdnFac
n>FcdnFjh\m>FacnF\i\m@%Fac\mC(>FcdnF[f[m>FacnF[f[m?(FboF[oF[oFi^yF_s@$Fdi\mC$>F
acnFii\m>FcdnFacn>FcdnFjh\m>FacnF\i\m-Fjfm6#/Fbj\m-F]blFd[s-Fjfm6#/Fgj\mFae^mF]
[]m-Fjfm6#%boThis~is~the~end~of~the~interactive~linear~dependence~procedure.~Yo
u~mayG-Fjfm6#%hochange~the~given~vectors~and~execute~the~procedure~again~to~lea
rn~the~processG/F`[lFi_nCE@%F]^nC+Fi_[mF\`[m>F`gnF[o>FboF[oFa`[m>Fe`[mF^fm?(F/F
[oF[oF/Fg`[mC'>FafmFe`[m@$FcfmFcin?(F/F[oF[oF/F\a[mC&>FafmFe`[m@$FcfmFcinFda[m>
Fe`[mF^fm@$F]a[mC%>FboFc]clF[b[m>Fe`[mF^fm>F`gnFe`[m>FganFg[cl>FcvFbb[mC%?(FboF
[oF[oF\jmF_sC$>Fe`[mFhafl@$F^a[mFjb[m>FganF\jm>FcvFbb[m>FdzF`c[m?(FboF[oF[oFcvF
_s?(F[uF[oF[oFganF_s>Fdc[mFec[m?(FboF[oF[oFhc[mF_s?(F[uFc]clF[oFhc[mF_s@$F[d[mC
%Fdd[mFgd[mFjd[m>FhvF]e[m>Fi`lFie[m>FhvF^f[m>F[xFdz?(FboF[oF[oFganF_s?(F[uF[oF[
oFcvF_s>Fff[mFgf[m>F_^lFbal?(FboF[oF[oFganF_s>F_^lF[g[m>FdzF`c[m?(FboF[oF[oFcvF
_s?(F[uF[oF[oFganF_s>Fdc[mFec[m>FetFbffl?(FboF[oF[oFganF_s>FfhblF]g[m>F^cnF^i[m
>Fa`rF[f[m>FdzF^j[m>FgdnFhj[m-Fjfm6#%\oSetting~the~linear~dependence~problem~yi
elds~the~augmented~matrixG-Fjfm6#Fdj[mF_in-Fjfm6#%:whose~row~echelon~form~isGFb
htF_in>FjdnFa[\m>FboF\[gl?(F/F[oF[oF/Fe[\m>FboFg[cl?(F[uF[oF[oFboF_s?(Fh`nF[oF[
oFjarF_s>F^\\mF_\\m@%Fa\\mC(>FboF][cl?(F/F[oF[oF/F\]\m>FboFg[cl>F[uF[o?(F/F[oF[
oF/Fe]\m>F[uFf]bl@$F\^\m>FjdnF`^\m>FgdnF]cvC%>F_yFf^\m?(Fh`nF[oF[oFi^yF_s>Fj^\m
F[_\m@$F]_\mC*-Fjfm6%Fe_\mFg\\m%Ithe~above~echelon~form~yields~the~matrixG>FboF
j\\m?(F/F[oF[oF/F\]\m>FboFg[cl>F[uF[o?(F/F[oF[oF/Fe]\m>F[uFf]bl@$F\^\m>FjdnF`^\
m>FgdnF]cvFbht>FauFf`\m>FialF^b\m>FacnFbal?(FboF[oF[oFganF_s>FacnF]c\m@%Fac\m-F
jfm6#%_oFrom~this~we~deduce~that~the~set~of~vectors~is~linearly~independent.GC*
-Fjfm6#%YThe~set~of~vectors~is~linearly~dependent~and~some~can~beG-Fjfm6#%Ywrit
ten~as~a~linear~combination~of~the~others~as~followsG>FfcnF\f\m>FganFdf\m?(F[uF
[oF[oFi^yF_sC'>F[xFfcn>F[xFif\m>F_`sF]g\m@%F`g\mC*>Fd]lFcg\m?(FboF[oF[oFi^yF_s>
Fhg\mFig\m>FcdnF[f[m>FacnF[f[m?(FboF[oF[oFi^yF_sC$>FacnF`h\m>FcdnFacn>FcdnFjh\m
>FacnF\i\m@%Fac\mC)>FevFacn>FcdnF[f[m>FacnF[f[m?(FboF[oF[oFi^yF_s@$Fdi\mC$>Facn
Fii\m>FcdnFacn>FcdnFjh\m>FacnF\i\mF^e^mFbe^mF][]m@$Fa[]m@$FcfmFcin>F[xFfcn>F_`s
F]g\m@%F`g\mC*>Fd]lFcg\m?(FboF[oF[oFi^yF_s>Fhg\mFig\m>FcdnF[f[m>FacnF[f[m?(FboF
[oF[oFi^yF_sC$>FacnF`h\m>FcdnFacn>FcdnFjh\m>FacnF\i\m@%Fac\mC(>FcdnF[f[m>FacnF[
f[m?(FboF[oF[oFi^yF_s@$Fdi\mC$>FacnFii\m>FcdnFacn>FcdnFjh\m>FacnF\i\mF_j\mFdj\m
F][]mF/6$FafmFf\^mF/%)subspaceGRF/6HFghnFigmF`hnF^hmF_hmFbgo%#k1G%)alphasetG%%x
setG%,relationsetG%*vectorsetGFjrF\jq%)subsflagGFfhqF]hoF^hoFc_fl%%tsetG%*relat
ion1G%*relation2G%'sumvecG%*relation3G%*subspflagG%#nuG%*relation4GF`hqFbg[lFdi
alFbialFaebl%+inrelationG%#kuG%$ku1G%%ans2G%#s3GF^ebl%+scalarflagGFgdmF/C/-Fjfm
6#%joThe~purpose~of~this~procedure~is~to~demonstrate~the~steps~needed~to~verify
~if~aG-Fjfm6#%`pgiven~set~is~a~subspace~or~not.~It~also~includes~an~interactive
~version~that~requiresG-Fjfm6#%\presponses~to~questions~posed~and~a~no-step~ver
sion~that~yields~the~result~withoutG-Fjfm6#%ioany~intermediate~steps.For~more~i
nformation~type~?subspace~at~the~Maple~promptGFjdmF_`fl-F^em6#%;~~~~1.~Demonstr
ation~mode|+G-F^em6#%9~~~~2.~Interactive~mode|+G-F^em6#%5~~~~3.~No-Step~mode|+G
>FboF^fm>FafmFbo@$FcfmFcin@(FhivCN@'F]^nC(-F^em6#%8Please~enter~the~set~S|+G-F^
em6#%EFor~example:~S:=|fr[x,y,z],~x+y-z=0|hr;|+G>F[uF^fm>FafmF[u@$FcfmFcin@$4-F
:6$F[uFdjm-F_o6#%:ERROR!!~A~set~is~expectedGFc[nC$>F[uFbjm@$F]e_mF`e_m-F_o6#%DE
RROR~!!~Only~one~argument~expectedG>F`[l-Fgep6#7;F%FggoFhfglF\fvF[gglFPF]c`lF_g
glFganF_yFcgoFgdnF\hq%"oGF]fz%"qG%"rGFh^flF,FetF^cnF[balFfcnF\jdl%"zG>F`gn-Fgep
6#7,FbhmFchm%#x3G%#x4G%#x5G%#x6G%#x7G%#x8G%#x9G%$x10GFjdm>Fial-Fgep6#7,FfhmFghm
%#t3G%#t4G%#t5G%#t6G%#t7G%#t8G%#t9G%$t10G>FacnFgo>FcdnFgo>F_yF[o>Fh`nF[o?(FcvF[
oF[oF_vF_s@(-F:6$F]abl%)relationG>Facn-F_bn6$Facn<#F]abl-F:6$F]ablFg_nC$>F]aqF]
abl>Fh`nFbal0Fh`nFbalC$>&F]aqFjyF]abl>F_yF^ew>F]aq-F\[o6$F]aqFg_n>Fh`nFbal?(Fcv
F[oF[oF]bblF_s@$4-F:6$&FacnFix%'linearG>Fh`nF[o>F`dq-F_[]l6%%(~S~=~|fr~G-F\[o6$
F]aqFcj\l%$~|gr~G?(FcvF[oF[oF]bblF_s>F`dq-F_[]l6%F`dq-F\[o6$Fii_mFcj\l%#~,G>F`d
q-F_[]l6$F`dq%$~|hr~G>F`dq-F_ial6#F`dq>FbipF`dq-Fjfm6$%8The~set~you~entered~is~
GF`dq>Fev-Fa`bl6$FacnF]aq>Fhv-F^g\mFaw@&/FhvFbalC$-Fjfm6$%CThe~set~is~the~trivi
al~subspace~ofG)Fehfl-F`vFbcqF]gmFedv>F[ep-Fghgl6#-F\\sFaw>Fa`rFgo?(FcvF[oF[oFa
\`mF_s>Fa`r-F_bn6$Fa`r<#/&FialFix&F]aqFix?(FcvF[oF[oFa\`mF_s>&F[epFix-F\y6$Fa`r
Fa]`m>F[ep-F\[o6$F[epFg_n>F`dq-F_[]l6%%'S~=~|fr~G-F\[o6$F[epFcj\l%>~|gr~where~x
~,~y,~z..etc~in~R~|hrG>F`dqF_[`mFjdm@$/Fh`nFbalC$-Fjfm6#%@This~set~can~also~be~
written~asG-FjfmF`[`m>Ffcn-F\[o6$F]aqFgep@$1-F`_rF`dzFi_nC+>FgdnF[_`mFjdm-Fjfm6
$%?Since~the~set~S~is~a~subset~ofGF`\`m-Fjfm6#%goFirst~determine~graphically~if
~the~necessary~condition~for~the~given~set~to~G-Fjfm6#%bobe~a~subspace~is~met;~
i.e~the~graph~of~the~set~must~contain~the~origin.GFjdm-F^em6#%[oWould~you~like~
to~see~the~graph~of~the~set?.~Answer~yes;~or~no;|+G>FboF^fm@%F]\x-%&graphG6$Fac
nF[_`m>FjdnF[o>FhvFbal?(FcvF[oF[o-F`vFb]uF_s@$0Fa]`mFbal>FhvF[o@$F[\`mC%-Fjfm6#
%]oSince~the~graph~does~not~pass~through~the~origin,~we~conclude~thatG-Fjfm6#%4
S~is~not~a~subspaceG>FjdnFbal>F_yF]bbl>FdzF[o@$0FjdnFbalCK-FjfmFbjp-Fjfm6%%0is~
a~subset~of~G)FehflFgdn%B~and~we~need~to~check~if,~for~anyG-Fjfm6#%Yu,v~in~S~an
d~n~a~constant,~if~the~two~properties~are~metGFjdm-F^em6#%:~~~~~~~~~~1.~u+v~is~
in~S|+G-F^em6#%:~~~~~~~~~~2.~n.u~is~in~S|+GF_in@$Fcfm-F_o6#Fgfm-Fjfm6#%8Show~th
e~first~propertyG>Fet-FgepF]_r>F^cnFjb`m?(FcvF[oF[oFgdnF_s>&FetFix&F`[lFix?(Fcv
F[oF[oFgdnF_s>&F^cnFix&F`[l6#,&FgdnF[oFcvF[o-Fjfm6(%'Let~u=GFet%'and~v=GF^cn%3b
e~two~vectors~in~GFha`mF_in@$FcfmFcin>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`r-F_bn6$Fa`r
<#/&FfcnFixF^c`m-Fjfm6'%&SinceGFet%&is~inGFbip%Git~satisfies~the~following~rela
tion(s)G?(FcvF[oF[oF_yF_sC&>&F[xFix-F\y6$Fa`rFii_m>F`dq-F_[]l6%%+~~~~~~~~(RG-F\
[o6$FdzFcj\lFa[bl-Fjfm6$F\e`mF`dq>FdzF[crFjdm@$FcfmFdb`m>Fa`rFgo?(FcvF[oF[oFgdn
F_s>Fa`r-F_bn6$Fa`r<#/Fcd`mFbc`m-Fjfm6%%*SimilarlyGF^cn%Dsatisfies~the~followin
g~relation(s)G?(FcvF[oF[oF_yF_sC&>&Fd]lFixF]e`m>F`dqF`e`m-Fjfm6$Fgf`mF`dq>FdzF[
crF_in@$FcfmFdb`m>F_^lFjb`m>F_^l-Fhjn6#,&FetF[oF^cnF[o-Fjfm6%%?For~the~sum~of~t
he~two~vectorsG-FhjnFc_l%,~to~be~in~SG-Fjfm6#%Kit~must~satisfy~the~following~re
lation(s)~G>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`r-F_bn6$Fa`r<#/Fcd`mFej_l?(FcvF[oF[oF_
yF_sC&>Fij_lF]e`m>F`dq-F_[]l6%%*~~~~~~~(RGFce`mFa[bl-Fjfm6$Fij_lF`dq>FdzF[crF_i
n@$FcfmFdb`m-Fjfm6#Ffd`m?(FcvF[oF[oF_yF_sC->F`dq-F_[]l6%Fbe`m-F\[o6$FcvFcj\lFa[
blFee`m-Fjfm6#%%plusG>F`dq-F_[]l6%Fbe`m-F\[o6$,&F_yF[oFcvF[oFcj\lFa[blFif`m-Fjf
m6#%,is~equal~toG-Fjfm6#,&F\e`mF[oFgf`mF[oFjdm@%/Fcj`mFij_lC%-Fjfm6#%2which~is~
equal~toG>F`dq-F_[]l6%Fbe`m-F\[o6$,&F_yF]jmFcvF[oFcj\lFa[blFhh`mC'-Fjfm6#%6whic
h~is~not~equal~toG>F`dqF[[amFhh`m>FjdnFbalFgbuF_in@$FcfmFdb`mFjdm@%Fba`mC$-Fjfm
6#%foThe~condition(s)~for~the~sum~of~the~vectors~u~+~v~to~be~in~S~can~be~deduce
dG-Fjfm6#%\ofrom~the~above~relation(s).~Therefore~S~is~CLOSED~UNDER~ADDITION.GC
$-Fjfm6#%joThe~condition(s)~for~the~sum~of~the~vectors,~u~+~v~to~be~in~S~cannot
~be~deducedG-Fjfm6#%_ofrom~the~above~relation(s).Therefore~S~is~NOT~CLOSED~UNDE
R~ADDITION.GFjdm@%/FjdnFbal-Fjfm6#%?Therefore~S~is~not~a~subspace.GC3F_in@$Fcfm
Fdb`m-Fjfm6#%_oNow~check~the~2nd~condition.~i.e.~closed~under~scalar~multiplica
tionG-Fjfm6#%^oIs~the~vector~n.u~is~in~S,~if~n~is~a~scalar~and~u~is~a~vector~in
~S?G>F]_p-Fhjn6#*&F\hqF[oFetF[o-Fjfm6$%;The~vector~n.u~is~given~byGF]_p>Fa`rFgo
?(FcvF[oF[oFgdnF_s>Fa`r-F_bn6$Fa`r<#/Fcd`m&F]_pFixFjdm-Fjfm6%%9For~this~vector~
to~be~inGFbip%Jit~must~satisfy~the~following~relation(s)G?(FcvF[oF[oF_yF_sC&>&F
i`lFixF]e`m>F`dq-F_[]l6%%(~~~~~(RGFce`mFa[bl-Fjfm6$Fi^amF`dq>FdzF[crFjdm?(FcvF[
oF[oF_yF_sC%F_in@$FcfmFdb`m@%/Fi^am-%'expandG6#*&F\hqF[oF\e`mF[oC'-Fjfm6&%8Mult
iply~the~relation~:GF\e`m%,by~n~we~getGFi_am-Fjfm6#%KThis~is~the~same~as~the~fo
llowing~relationG>F`dq-F_[]l6%Fgh`m-F\[o6$,&FcvF[oF_yFi_nFcj\lFa[blF^_am>F_`sF[
oC*Fjdm-Fjfm6&%3Multiply~relation~GF\e`mF^`amFi_am-Fjfm6#%SBut~this~is~NOT~the~
same~as~the~following~relationG>F`dqFc`amF^_am-Fjfm6#%VTherefore~S~is~not~close
d~under~scalar~multiplicationG>F_`sFbalFgbu@$Fba`m>FjdnF]jmFjdm@$/%)susbflagGFb
al-Fjfm6$%ATherefore~S~is~not~a~subspace~ofGFha`m@$/FjdnF]jm-Fjfm6#%RTherefore~
S~is~CLOSED~under~scalar~multiplicationG@$F^bamC(F_in@$FcfmFdb`m-Fjfm6#%GThe~se
t~S~satisfied~the~two~propertiesG-F^em6#%W~~~~~~~~~1.~The~set~S~is~closed~under
~vector~addition|+G-F^em6#%gn~~~~~~~~~2.~The~set~S~is~closed~under~scalar~multi
plication|+G-Fjfm6#%:Therefore~S~is~a~subspaceGFjdm-Fjfm6#%aoThis~is~the~end~of
~the~subspace~procedure.~Please~change~the~given~setG-Fjfm6#%Wand~execute~the~p
rocedure~again~to~learn~this~process.GF`[wC;>F`[lFje_m>F`gnFbf_m@%F]^nC&-F^em6#
%YPlease~enter~the~set~S;~for~example~S:=|fr[x,y],3*x+y=0|hr;|+G>F[uF^fm>FafmFd
_o@$FcfmFcin>F[uFbjm>FacnFgo>FcdnFgo>F_yF[o>Fh`nF[o?(FcvF[oF[oF_vF_s@(F_h_m>Fac
nFch_mFfh_mC$>F]aqF]abl>Fh`nFbalF[i_mC$>F^i_mF]abl>F_yF^ew>F]aqFai_m>F`dqF]j_m?
(FcvF[oF[oF]bblF_s>F`dqFej_m>F`dqF[[`m>F`dqF_[`m>FbipF`dqFb[`m>FevFf[`m>FhvFi[`
m@&F[\`mC$-Fjfm6$%?The~set~is~trivial~subspace~ofGF`\`mF]gmFedv>F[epFc\`m>FgdnF
dhbl@$1FgdnFi_nC)-Fjfm6#%coFirst~we~determine~graphically~if~the~given~set~is~a
~subspace~or~not.~A~G-Fjfm6#%eoA~necessary~condition~is~that~the~~graph~of~S~mu
st~passes~thru~the~origin.G-F^em6#%VWould~you~like~to~see~the~graph~?~answer~ye
s:~or~no;|+G>FboF^fm>FafmFd_o@$FcfmFcin@$5F]\x/Fbo%$YESGC0-F]``m6$FacnFgdnF_in@
$FcfmFcin>FjdnF[o-F^em6#%VBased~on~the~graph~can~you~determine~whether~the~set|
+G-F^em6#%jnis~a~possible~candidate~to~be~a~subspace~?.~Answer~YES;~or~NO;|+G>F
boF^fm>FafmFd_o@$FcfmFcin>FhvFbal>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`r-F_bn6$Fa`r<#/F
^]`mFbal?(FcvF[oF[oF]bblF_s>Fhv,&FhvF[o-F\y6$Fa`r,&-Fjis6#Fii_mF[o-F^hsF`iamF^v
F[o@%5FagamF]\x@%F[\`mC$-Fjfm6#%RYes,~the~graph~of~the~set~passes~thru~the~orig
in.G-Fjfm6#%WNow~we~will~try~to~prove~that~the~set~S~is~a~subspace.GC$>FjdnFbal
-Fjfm6#%inSet~is~not~a~subspace,~since~it~does~not~pass~thru~the~origin.G@%F[\`
mC$-Fjfm6#%ZIncorrect.~It~is~possible~that~S~is~a~subspace~since~the~G-Fjfm6#%Z
graph~passes~thru~the~origin.~~~~~~~~~~~~~~~~~~~~~~~~~~~~GC$-Fjfm6#%Fcorrect.~t
he~set~cannot~be~a~subspaceG>FjdnFbalFjdm@$Fba`mC]p-Fjfm6$%4The~set~you~entered
GFbip-Fjfm6$Fga`mFha`mFjdm-Fjfm6#%WCheck~the~two~subspace~properties.~If~u~and~
v~are~in~SG-Fjfm6#%Wand~k~is~a~scalar,~you~need~to~show~that~:~~~~~~~~~~~~GFjdm
-Fjfm6#%01.~u+v~is~in~S~G-Fjfm6#%02.~k*u~is~in~S.GFjdm-Fjfm6$%DChoose~any~two~g
eneral~vectors~fromGFha`m-F^em6#%jnFor~example~in~R^2~:~>~u~:=~vector([a,b]);~v
~:=~vector([c,d]);|+G>FetF^fm>FafmFd_o@$FcfmFcin?(F/F[oF[oF/4-F:6$FetFgep>Fet-F
_fm6#%=A~vector~expected.~Try~againG-F^em6#%>Please~enter~the~next~vector|+G>F^
cnF^fm>FafmFd_o@$FcfmFcin?(F/F[oF[oF/4-F:6$F^cnFgep>F^cnF_]bm>Fh`nFbal?(FcvF[oF
[o-F`_rFchpF_s@$-F:6$F^c`mF^h^l>Fh`nF[o@$Fc^rC%-Fjfm6#%KA~vector~with~symbolic~
entries~is~expectedG?(FcvF[oF[oFgdnF_s>F^c`mF_c`m-Fjfm6$%#u=GFecv>Fh`nFbal?(Fcv
F[oF[o-F`_rFdcnF_s@$-F:6$Fbc`mF^h^l>Fh`nF[o@$Fc^rC%Ff^bm?(FcvF[oF[oFgdnF_s>Fbc`
mFcc`m-Fjfm6$%#v=GFier@&33/F_^bmFgdn/F`_bmFgdnF`^`mC$-Fjfm6#%8you~entered~the~v
ectorsG-Fjfm6&F]_bmFecv%%~~v=GFierFg\uC%-Fjfm6#%UVector~dimensions~incorrect.~T
he~correct~vectors~areG?(FcvF[oF[oFgdnF_s>F^c`mF_c`m?(FcvF[oF[oFgdnF_s>Fbc`mFcc
`mFjdm>F_yF]bblF_in@$FcfmFcin>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`r-F_bn6$Fa`r<#/F^]`m
F^c`m?(FcvF[oF[oF_yF_s>F\e`mF]e`m-F^em6#%XSince~u~is~in~S,~what~is/are~the~rela
tion(s)~satisfied|+G-F^em6#%Oby~its~components~?~For~example~:~>~a+b-c=0;~|+G-F
jfm6$%-The~set~S~isGFbip?(FcvF[oF[oF_yF_sC&-F^em6#%9Please~enter~a~relation|+G>
&FhisFixF^fm>FafmFd_o@$FcfmFcin@%0-F\[o6$FhisFdjm-F\[o6$F[xFdjmC$-Fjfm6#%YYour~
answer~is~incorrect.~The~correct~relation(s)~is/areG-Fjfm6#FacbmC$-Fjfm6#%_oCor
rect.~The~set~of~relation(s)~satisfied~by~the~components~of~u~areGFgcbmF_in@$Fc
fmC$Fc]glF]gm>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`r-F_bn6$Fa`r<#/F^]`mFbc`m?(FcvF[oF[o
F_yF_s>Fgf`mF]e`m-F^em6#%XSince~v~is~in~S,~what~is/are~the~relation(s)~satisfie
d|+GF^bbmFjdmFabbm?(FcvF[oF[oF_yF_sC&Ffbbm>FjbbmF^fm>FafmFd_o@$FcfmFcin>F`\qFba
l@%0F_cbm-F\[o6$Fd]lFdjmC$Fdcbm-Fjfm6#FcebmC$FjcbmFfebmF_in@$FcfmFcin>F_^lFjb`m
>F_^lF_g`m-Fjfm6$%;The~sum~of~the~vectors~~isGF_^l>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa
`r-F_bn6$Fa`r<#/F^]`mFej_l?(FcvF[oF[oF_yF_s>Fij_lF]e`m-F^em6#%]oFor~the~sum~u+v
~to~be~in~S,~what~is/are~the~relation(s)~satisfied|+GF^bbmFjdm-Fjfm6$%9Recall~t
hat~the~set~S~isGFbip?(FcvF[oF[oF_yF_sC&Ffbbm>FjbbmF^fm>FafmFd_o@$FcfmFcin>F`\q
Fbal@%0F_cbm-F\[o6$FauFdjmC$Fdcbm-Fjfm6#FfgbmC$FjcbmFigbmF_in@$FcfmC$Fc]glF]gm-
Fjfm6$%AIs~it~possible~to~obtain~the~setGFfgbm-Fjfm6$%4From~relation~sets~GFacb
m-Fjfm6$FgjsFcebmFh^u>FboF^fm>FafmFd_o@$FcfmFcin>Fa`rFgo?(FcvF[oF[oF]bblF_s>Fa`
r-F_bn6$Fa`r<#Fcj`m>FjdnFbal?(FcvF[oF[oF]bblF_s@$0Fcj`mFij_l>FjdnF[o@%5F`gam/Fb
o%$YesG@%Fg\am-Fjfm6#%LGood.~Therefore~S~is~Closed~Under~Addition.GC&-Fjfm6$%<I
ncorrect.~The~relation~setGFfgbm-Fjfm6%%8Cannot~be~obtained~fromGFacbmFcebm-Fjf
m6#%ITherefore~S~is~NOT~CLOSED~UNDER~ADDITIONG>FjdnF[o@%Fg\am-Fjfm6#%GIncorrect
.~S~is~Closed~Under~Addition.GC$-Fjfm6#%OGood.~Therefore~S~is~NOT~CLOSED~UNDER~
ADDITIONG>FjdnF[oFjdm@%FhcrFh\amC9-Fjfm6#%NNow~let~us~check~the~second~property
.~That~isG-Fjfm6#%enif~k~is~any~scalar~and~u~is~a~vector~in~S,~.~Is~k*u~in~S~?G
-F^em6#%<please~enter~the~vector~ku|+G-Fjfm6#%fnFor~example,~if~u=vector([a,b])
~then~ku:=vector([k*a,k*b]);G>FcfsF^fm>FafmFd_o@$FcfmFcin>Fdjp-Fhjn6#*&F_yF[oFe
tF[o-Fjfm6$%<The~vector~you~entered~is:~GFcfs>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`r-F_
bn6$Fa`r<#/F^]`m&FcfsFix?(FcvF[oF[oF]bblF_s>Fi^amF]e`m-F^em6#%goIf~the~above~ve
ctor~ku~is~to~be~in~S,~what~is/are~the~relation(s)~satisfied|+G-F^em6#%Uby~its~
components~?~For~example~:~>~k*a+k*b-k*c=0;~|+GFjdmFabbm?(FcvF[oF[oF]bblF_sC&Ff
bbm>FjbbmF^fm>FafmFd_o@$FcfmFcin>FeepFbal?(FcvF[oF[oF]bblF_s@$0-Fg_am6#FjbbmFi^
am>FeepF[o@%/FeepF[oC$Fdcbm-Fjfm6#-F\[o6$Fi`lFdjmC$FjcbmFg^cmF_in@$FcfmC$Fc]glF
]gm@%Fg\amC$-Fjfm6#%TSince~the~set~S~is~closed~under~addition~and~scalarG-Fjfm6
#%B~multiplication,~~S~is~a~subspaceGC$-Fjfm6#%ROne~of~the~conditions~for~a~sub
space~is~violated.G-Fjfm6#%>Therefore~S~is~not~a~subspaceGFjdm-Fjfm6#%_pThis~is
~the~end~of~the~interactive~version~of~the~subspace~procedure.~Try~other~setsGF
cbxCH@'F]^nC(Fcd_mFfd_m>F[uF^fm>FafmF[u@$FcfmFcin@$F]e_mF`e_mFc[nC$>F[uFbjm@$F]
e_mF`e_mFfe_m>F`[lFje_m>F`gnFbf_mFjdm>FialF^g_mFjdm@%F]^nC&F\dam>F[uF^fm>FafmFd
_o@$FcfmFcin>F[uFbjm>FacnFgo>FcdnFgo>F_yF[o>Fh`nF[o?(FcvF[oF[oF_vF_s@(F_h_m>Fac
nFch_mFfh_mC$>F]aqF]abl>Fh`nFbalF[i_mC$>F^i_mF]abl>F_yF^ew>F]aqFai_m>FgdnFa\`m>
F`dqF]j_m?(FcvF[oF[oF]bblF_s>F`dqFej_m>F`dqF[[`m>F`dqF_[`m>FbipF`dqFb[`m>FevFf[
`m>FhvFi[`m@&F[\`mC$-Fjfm6$%PThe~set~is~Empty~and~is~the~trivial~subspace~ofGF`
\`mF]gmFedv>F[epFc\`m>Fa`rFgo?(FcvF[oF[oFa\`mF_s>Fa`rFi\`m?(FcvF[oF[oFa\`mF_s>F
a]`mFb]`m>F[epFe]`m>F`dqFh]`m>F`dqF_[`mFjdm>FfcnFg^`m>FhvFbal?(FcvF[oF[oFb``mF_
s@$Fd``m>FhvF[o>F_yF]bbl>FdzF[o@$Fba`mC2>FetFjb`m>F^cnFjb`m?(FcvF[oF[oFgdnF_s>F
^c`mF_c`m?(FcvF[oF[oFgdnF_s>Fbc`mFcc`m>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`rF_d`m?(Fcv
F[oF[oF_yF_s>F\e`mF]e`m>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`rF\f`m?(FcvF[oF[oF_yF_s>Fg
f`mF]e`m>F_^lFjb`m>F_^lF_g`m>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`rF]h`m?(FcvF[oF[oF_yF
_s>Fij_lF]e`m?(FcvF[oF[oF_yF_s@%Fej`m>FjdnF[oC$>FjdnFbalFgbu@%Fg\am-Fjfm6#%5S~i
s~not~a~subspace.GC*>F]_pFd]am>Fa`rFgo?(FcvF[oF[oFgdnF_s>Fa`rF]^am?(FcvF[oF[oF_
yF_s>Fi^amF]e`m?(FcvF[oF[oF_yF_s@%Fe_am>F_`sF]jmC$>FjdnFbalFgbu@$Fba`m>FjdnF]jm
Fjdm@$Fhaam-Fjfm6$%8~S~is~not~a~subspace~ofGFha`m@$F^bamC&FebamFhbamF[camF^camF
/FagmF/%1graphscalarmultiGRF/6(%#l1G%#l2GFghn%%ansvGFbgoF^hoF/F/C'@%F]^nC'-F^em
6#%:Please~enter~the~scalar~kG>FganF^fm-F^em6#%:Please~enter~the~vector~vG>F_yF
^fm>F_y-F\[o6$F_yFgepC$>FganFbjm>F_yF][nF`cel>Fbo-Fa_p6(Fegel7$Fdb[m&F_yF^[nFdb
pFigelFjgel/FhenFfjdl>F[u-Fa_p6(Fegel7$*&FganF[oFdb[mF[o*&FganF[oF`icmF[oFdbpFi
gelFjgelFaicm-F\^fl6$Fi_alFa^flF/F/F/%(lincombGRF/6DF^hmF_hmFbgoF\jqF\hqF^hoF[b
alF`hqFg^flFghnFfdm%'rhssumGF]fvF_^[mFf][m%&AUG33G%$eqnGFb[qFh][mFg][mF^ebl%%eq
n1GF[eblFf^[m%*setofeqnsG%(vectsetGF^^[mFg^[mF`hnF`^[mFh^[mFaebl%&done1G%+setof
vectsGFgdmF/C.Fjdm-Fjfm6#%^oThe~purpose~of~this~procedure~is~to~demonstrate~the
~steps~needed~toG-Fjfm6#%coverify~that~a~vector~w~is~a~linear~combination~of~a~
given~set~of~vectorsG-Fjfm6#%^o|fr~v1,v2,..|hr~or~not.~Also~the~function~can~be
~used~in~interactive~orG-Fjfm6#%gnor~no-step~mode.~For~more~information~type~?l
incomb~for~helpG@$FcfmC$Fc]glF]gmF_`flFb`flFe`flFh`fl>FacnF^fm@(F[i_lCH@$F]^nC0
-F^em6#%jnEnter~vector~w~that~is~to~be~expressed~as~a~linear~combination|+G-Fjf
m6#%FFor~example~type:~w:=vector([1,2,3]);G>FdzF^fm?(F/F[oF[oF/4-F:6$FdzFgepC&F
eefl>FdzF^fm>FafmFdz@$FcfmFcinFjdm-Fjfm6#%UPlease~Enter~the~set~of~vectors.~All
~vectors~must~beG-Fjfm6#%Kof~the~same~dimension~as~w.~To~end~type~0;G>FcdnF[o>F
boF[o-F^em6#%?Please~enter~the~first~vector|+G>Fe`[mF^fm?(F/F[oF[oF/0FcdnFbalC(
>FafmFe`[m@$FcfmFcin?(F/F[oF[oF/F\a[mC&>FafmFe`[m@$FcfmFcinFe[il>Fe`[mF^fm?(F/F
[oF[oF/3F]a[m0-F`_r6#Fe`[m-F`_rF\]lC&-F^em6#%RVector~dimensions~incompatible~wi
th~w.~try~again|+G>Fe`[mF^fm>FafmFe`[m@$FcfmFcin@$F]a[mC%>FboFc]clF[b[m>Fe`[mF^
fm>FcdnFe`[m>FganFbo@$1F[oFgan>FcvFbb[m@&FcdalC%?(FboF[oF[o,&F\jmF[oF^vF[oF_s@%
-F:6$FhaflFgep>Fe`[mFhafl-Fj]n6#%KAll~arguments~must~be~vectors.~Start~againG>F
dz&Fcjm6#F\jm>FganF\jmFc[nC(>FganF\jm@%-F:6$FbjmFgep>Fdb[mFbjm-Fj]n6#%?Vectors~
expected.~Start~~againG-F^em6#%^oPlease~enter~the~vector~w~to~be~written~as~a~l
inear~combination~of|+G-FjfmFcb[m>FdzF^fm?(F/F[oF[oF/Fa\dmC$-F^em6#%6a~vector~i
s~expected|+G>FdzF^fm>FcvFbb[m>F`[l-Fhjm6$FcvFi^y?(FboF[oF[oFcvF_s?(F[uF[oF[oFi
^yF_s>&F`[lFe`[lFec[m>F`gn-F`cw6$F`[lFdz>F[xF]e[m?(FboF[oF[oFf[\lF_s>Fe`[m-Fcbn
6$F`[lFbo>FgdnF`[l>Fjdn-F`cw6$FgdnFdz>FialFf^\m?(FboF[oF[oFi^yF_s>Fbb\mF]cgl>F]
_p-Fian6$FgdnFial>FauFbal?(FboF[oF[oFi^yF_s>Fau,&FauF[o*&F]cglF[o-F\[o6$-FhjnFb
^dmFhjmF[oF[o>F^cn/-F\[o6$Fdj[mFhjmFau-Fjfm6#%YExpress~w~as~a~linear~combinatio
n~of~the~other~vectors~:GFcbclF_in@$FcfmFcin-Fjfm6#%]oThe~system~of~equations~r
esulting~from~this~linear~combination~is:G>Fa`rF^gel?(FboF[oF[oFcvF_sC$>&Fa`rF`
gm/&F]_pF`gmFgafl-Fjfm6#FhddmF_in@$FcfmFcin-Fjfm6$%gnForm~the~augmented~matrix~
associated~with~the~above~system~:GFjdn-Fjfm6#%goApply~Gauss~Elimination~to~thi
s~matrix~to~check~if~the~system~is~consistent~G>Fhv-F\\sF^cv-Fjfm6#%XGauss~elim
ination~yields~the~following~row~echelon~formG-Fjfm6$%:of~the~augmented~matrix~
:GFhvFjdm-Fjfm6#%_oFrom~the~row~echelon~form,~can~you~tell~if~the~system~is~con
sistent?GF_in@$FcfmC$Fc]glF]gm@%/-F^g\mFdwFbalC'>Fet-Fhjm6$Feecl-Fb_nFdw>Fbo-F`
_nFdw?(F/F[oF[oF//-Fb^o6$-Ff_t6$FhvFboF]jmFbal>FboFg[cl?(F[uF[oF[oFboF_s?(Fh`nF
[oF[oFhfdmF_s>&FetFjan&FhvFjan@%35-F:6$&FetFe\\m%)rationalG-F:6$F\hdmFf\\m0F\hd
mFbalC'-Fjfm6#%IThe~system~is~inconsistent~and~thereforeG-Fjfm6%F^]il-F\[o6$Fdz
Fhjm%Qis~not~a~linear~combination~of~the~given~vectorsG>Fi`lFgo?(FboF[oF[oFi^yF
_s>Fi`l-F_bn6$Fi`l<#Ffcdm-FjfmFjg[m@$34FjgdmF`hdmC'-Fjfm6%%?The~system~is~incon
sistent~if~G&FhvFe\\m%B~is~~not~equal~to~0.~In~this~caseGFehdm>Fi`lFgo?(FboF[oF
[oFi^yF_s>Fi`lF]idmF`idmC.-Fjfm6#%eoFrom~this~augmented~matrix,~we~deduce~that~
the~system~is~consistent~and~weG-Fjfm6#%`oapply~the~back~substitution~algorithm
~to~solve~the~consistent~system.GFjdm>F_`s-Fghgl6#-F\\sFdw-Fjfm6$%AThe~solution
~to~the~system~is~:~GF_`sF_in@$FcfmC$-FjfmF_dsF]gm-Fjfm6#%inThus~the~vector~w~i
s~a~linear~combination~of~the~given~vectorsG-Fjfm6#%Dand~it~can~be~written~as~f
ollows~:~G?(FboF[oF[oFi^yF_s@'/&F_`sF`gmFbal>Fau-F\y6$<#/F]cglF[j\mFau/Fg[emF[o
>Fau-F\y6$<#/F]cglF]\blFau>Fau-F\y6$<#/F]cglFg[emFau-Fjfm6#/-Fhjn6#-F\[o6$-Fcbn
6$FjdnFganFhjmFau@$/%(varflagGF[o-Fjfm6$%'Where~G%$eq1GFjdm-Fjfm6#%gnThis~is~th
e~end~of~the~linear~combination~procedure.~You~mayG-Fjfm6#%Zchange~the~given~ve
ctors~and~execute~the~procedure~again.GF_i_lCin@$F]^nC0Fi[dmF\\dm>FdzF^fm?(F/F[
oF[oF/Fa\dmC&Feefl>FdzF^fm>FafmFdz@$FcfmFcinFjdmFh\dmF[]dm>FcdnF[o>FboF[oFa`[m>
Fe`[mF^fm?(F/F[oF[oF/Fe]dmC(>FafmFe`[m@$FcfmFcin?(F/F[oF[oF/F\a[mC&>FafmFe`[m@$
FcfmFcinFda[m>Fe`[mF^fm?(F/F[oF[oF/F_^dmC&-F^em6#%Svector~dimensions~incompatib
le~with~w.~Try~again.|+G>Fe`[mF^fm>FafmFe`[m@$FcfmFcin@$F]a[mC%>FboFc]clF[b[m>F
e`[mF^fm>FcdnFe`[m>FganFbo@$Fb_dm>FcvFbb[m@&FcdalC%?(FboF[oF[oFg_dmF_s@%Fi_dm>F
e`[mFhaflF\`dm>FdzF``dm>FganF\jmFc[nC(>FganF\jm@%Ff`dm>Fdb[mFbjmFi`dmF\admF_adm
>FdzF^fm?(F/F[oF[oF/Fa\dmC$Fe[il>FdzF^fm>FcvFbb[m>F`[lFiadm?(FboF[oF[oFcvF_s?(F
[uF[oF[oFi^yF_s>F^bdmFec[m>F`gnF`bdm>F[xF]e[mFjdm?(FboF[oF[oFi^yF_s>Fe`[mFebdm>
F`gnF`bdm-Fjfm6#%hnFor~the~given~vectors~can~you~find~scalars~c1,c2,..~such~tha
tG>Fau*&FdbglF[o-Fhjn6#-F\[o6$FhgblFhjmF[o?(FboF]jmF[oFi^yF_s>Fau,&FauF[o*&F]cg
lF[o-Fhjn6#-F\[o6$-FcbnFe[clFhjmF[oF[o-Fjfm6#/Fau-Fhjn6#-F\[o6$-Fcbn6$F`gnFganF
hjmFjdm-F^em6#%doEnter~the~system~of~linear~equations~resulting~from~this~vecto
r~equation|+G?(FboF[oF[oFcvF_sC+-F^em6$%Aplease~enter~equation~number~%d|+GFbo>
FghblF^fm?(F/F[oF[oF/4-F:6$FghblFdf]mC$-F^em6#%DThis~is~not~an~equation.~Try~ag
ain|+G>FghblF^fm>FafmFghbl@$FcfmC$-FjfmF[\rF]gm>F`\qFbal?(F[uF[oF[oFi^yF_s>F`\q
,&F`\qF[o*&&F[xFavF[oFjbflF[oF[o>&FacnF`gm/F`\q&F`gnFe\\m@$0FdeemFghblC$-Fjfm6$
F[h]mFeeemFjdmF_in@$FcfmFcin-Fjfm6#%aoThe~system~of~linear~equations~resulting~
from~the~vector~equation~is~:G?(FboF[oF[oFcvF_s-Fjfm6#FdeemFjdm-F^em6&%ioEnter~
the~%dx%d~augmented~matrix~whose~first~%d~columns~are~the~given~vectors|+GFcvFg
anFi^y-F^em6#%_pand~whose~last~column~is~the~vector~that~is~to~be~expressed~as~
a~linear~combination|+G-F^em6#%Lexample:~>~AUG:=matrix([[1,1,0],[3,2,1]]);|+G>F
acnF^fm?(F/F[oF[oF/4-F:6$FacnFhjmC&>FafmFacn@$FcfmFcinFci]m>FacnF^fm>FhisF[o>Fc
fsFdgm?(F/F[oF[oF//FcfsFdgmC$@%3/-F`_r6#-Fcbn6$FacnF[oFcv/-F`_r6#-Ff_tF`hemFgan
@%0-Fb^o6$,&FacnF[oF`gnF^vF]jmFbalC'-Fjfm6#%PThis~is~not~the~correct~matrix.Ple
ase~try~againG>FacnF^fm>FafmFacn@$FcfmC$F\eemF]gm>Fhis,&FhisF[oF[oF[oC%-Fjfm6#%
;This~is~the~correct~matrixG-FjfmFd[s>FcfsF_sC'-F^em6#%PERROR!!~incorrect~matri
x~dimensions.~Try~again|+G>FacnF^fm>FafmFacn@$FcfmC$F][emF]gm>FhisFciem@$/FhisF
i_nC$>FcfsF_sFgfm@%FdjemC$-Fjfm6$%OYou~attempted~3~times.~The~correct~answer~is
~:GF`gn>FjdnF`gnC%>FjdnF`gnFjdm-F^em6#%[oVery~Good.~You~entered~the~right~matri
x.~What~is~the~NEXT~STEP?|+GFjdm-F^em6#%[oChoose~the~most~appropriate~route~to~
proceed~from~the~following|+G-F^em6#%[oset.~At~the~prompt~sign~type~the~number~
followed~by~a~semicolon|+GFjdm-F^em6#%U~~~~~~~1.~Find~the~inverse~of~the~augmen
ted~matrix.|+G-F^em6#%hn~~~~~~~2.~Find~the~LU-decomposition~of~the~augmented~ma
trix.|+G-F^em6#%jo~~~~~~~3.~Find~the~row~echelon~form~of~the~matrix~to~solve~th
e~related~system.|+G>FcdnF^fm>FafmFcdn@$FcfmC$FifmF]gm?(F/F[oF[oF/0FcdnFi_nC&-F
jfm6#%JYou~can~do~better~with~a~different~choiceG>FcdnF^fm>FafmFcdn@$FcfmC$F\ee
mF]gmFjdm-Fjfm6#%]oTo~find~the~echelon~form~of~the~matrix,~apply~linalg[gaussel
im](*)G-Fjfm6#%Ito~the~augmented~matrix.~So~we~obtain~:~G>FhvFeedm-FjfmFdwFjdm-
F^em6#%jnDoes~the~echelon~form~of~the~matrix~imply~that~the~system~is:~|+G-F^em
6#%<~~~~1.~always~consistent~?|+G-F^em6#%7~~~~2.~inconsistent~?|+G-F^em6#%1Choo
se~1;~or~2;|+G>FacnF^fm>FafmFacn@$FcfmC$F\eemF]gm@&F[i_l@%/FcfdmF[o>F`dqF[o>F`d
qF]jmF_i_l@%Fbfdm>F`dqFi_n>F`dqF^hp@&Fdis-Fjfm6#%:Correct~answer.~Proceed..G/F`
dqF^hp-Fjfm6#%QIncorrect.~This~system~is~consistent.~Proceed...G@%5FdisFc_fmC=F
_in-F^em6#%hoSolve~the~system~by~selecting~the~appropriate~choice~from~the~foll
owing~menu|+GFjdm-F^em6#%X~~~~1.~Find~the~determinant~of~the~coefficient~matrix
~|+G-Fjfm6#-Fg\r6%Fhv;F[oFcv;F[oFi^y-F^em6#%X~~~~2.~Apply~backsubstitution~to~t
he~augmented~matrix|+~GFf]fm-F^em6#%Q~~~~3.~Find~the~inverse~of~the~augmented~m
atrix|+GFf]fm>FcdnF^fm>FafmFcdn@$FcfmFcin?(F/F[oF[oF/0FcdnF]jmC&-Fjfm6#%TAnothe
r~choice~would~be~more~appropriate.Try~Again!G>FcdnF^fm>FafmFcdn@$FcfmC$FifmF]g
mFjdm>F_`s-FghglFdw-Fjfm6$%enGood~Choice.~The~solution~using~the~backsub~algori
thm~is~:GF_`sF_in@$FcfmC$Fc]glF]gm-Fjfm6$%?Now~express~the~given~vector~wG-Fhjn
6#FghdmFjdm-Fjfm6#%;as~a~linear~combination~ofG>Fi`lFgo?(FboF[oF[oFi^yF_s>Fi`l-
F_bn6$Fi`l<#-F\[o6$-Fhjn6#FebdmFhjmF`idmFjdm?(FboF[oF[oFi^yF_s@'Ff[em>FauFi[emF
]\em>FauF_\em>FauFd\em-Fjfm6#/FacemFau@&/F`dqF]jm@%34-F:6$&FhvFeavFf\\m0FadfmFb
alC'-Fjfm6%%QIncorrect~answer.~This~system~is~inconsistent~ifGFbdfm%.in~which~c
aseGFehdm>Fi`lFgo?(FboF[oF[oFi^yF_s>Fi`lF]idmF`idmC'-Fjfm6#%OIncorrect~answer.~
This~system~is~inconsistent.GFehdm>Fi`lFgo?(FboF[oF[oFi^yF_s>Fi`lF]idmF`idm/F`d
qFi_nC'>FetFffdm>FboFjfdm?(F/F[oF[oF/F\gdm>FboFg[cl?(F[uF[oF[oFboF_s?(Fh`nF[oF[
oFhfdmF_s>FegdmFfgdm@%FhgdmC'-Fjfm6#%RCorrect.~The~system~is~inconsistent~and~t
hereforeGFehdm>Fi`lFgo?(FboF[oF[oFi^yF_s>Fi`lF]idmF`idm@$34F^hdmF`hdmC'-Fjfm6%%
HCorrect.~The~system~is~inconsistent~if~GF\hdmFiidmFehdm>Fi`lFgo?(FboF[oF[oFi^y
F_s>Fi`lF]idmF`idmFjdm-Fjfm6#%^pThis~is~the~end~of~the~interactive~mode~of~the~
linear~combination~procedure.~Try~toG-Fjfm6#%^pwork~more~examples~by~changing~t
he~given~vectors~and~executing~the~function~again.~GFdi_lC>@$F]^nC0Fi[dmF\\dm>F
dzF^fm?(F/F[oF[oF/Fa\dmC&Feefl>FdzF^fm>FafmFdz@$FcfmFcinFjdmFh\dmF[]dm>FcdnF[o>
FboF[oFa`[m>Fe`[mF^fm?(F/F[oF[oF/Fe]dmC(>FafmFe`[m@$FcfmFcin?(F/F[oF[oF/F\a[mC&
>FafmFe`[m@$FcfmFcinFda[m>Fe`[mF^fm?(F/F[oF[oF/F_^dmC&Fe_em>Fe`[mF^fm>FafmFe`[m
@$FcfmFcin@$F]a[mC%>FboFc]clF[b[m>Fe`[mF^fm>FcdnFe`[m>FganFbo@$Fb_dm>FcvFbb[m@&
FcdalC%?(FboF[oF[oFg_dmF_s@%Fi_dm>Fe`[mFhaflF\`dm>FdzF``dm>FganF\jmFc[nC(>FganF
\jm@%Ff`dm>Fdb[mFbjmFi`dmF\admF_adm>FdzF^fm?(F/F[oF[oF/Fa\dmC$Fe[il>FdzF^fm>Fcv
Fbb[m>F`[lFiadm?(FboF[oF[oFcvF_s?(F[uF[oF[oFi^yF_s>F^bdmFec[m>F`gnF`bdm>F[xF]e[
mFjdm?(FboF[oF[oFf[\lF_s>Fe`[mFebdm>FdjpFgo?(FboF[oF[oFf[\lF_s>Fdjp-F_bn6$FdjpF
_idm>FgdnF`[l>FjdnFibdm>FialFf^\m?(FboF[oF[oFi^yF_s>Fbb\mF]cgl>F]_pF_cdm>FauFba
l?(FboF[oF[oFi^yF_s>FauFdcdm>F^cnFjcdm>Fa`rF^gel-Fjfm6#%enThe~linear~combinatio
n~problem~yields~the~augmented~matrixGFihcl-Fjfm6#%6whose~echelon~form~isG>FhvF
eedmFf]fmF_in>F]^[mFcfdm@%/F]^[mFbalC'>FetFffdm>FboFjfdm?(F/F[oF[oF/F\gdm>FboFg
[cl?(F[uF[oF[oFboF_s?(Fh`nF[oF[oFhfdmF_s>FegdmFfgdm@%FhgdmC'FbhdmFehdm>Fi`lFgo?
(FboF[oF[oFi^yF_s>Fi`lF]idmF`idm@$FdffmC'-Fjfm6%FgidmF\hdmFiidmFehdm>Fi`lFgo?(F
boF[oF[oFi^yF_s>Fi`lF]idmF`idmC'>F_`sFejdm-Fjfm6#%XThe~vector~is~a~linear~combi
nation~of~the~given~vectorsG-Fjfm6#%Aand~it~can~be~written~as~followsG?(FboF[oF
[oFi^yF_s@'Ff[em>FauFi[emF]\em>FauF_\em>FauFd\emFh\emF/6$FafmF]^[mF/F/,
I)gaussmanRF/6CF^hmF_hmFbgoFfdmF^hqF_hqF`hqFhhqFdiqFaiqFjhqFhiqF[iqF\iqFghnF`hn
FihqF]jqF`iqFbiqFciqF\jqF^fvFiiqFjiqF^iqFbhqF_iqFeiqF\hqF[jqF_hnFggoF/F/C4>Fa`r
F\_s>F^cnF``s>FdzFbjm>FevF_]^m@$/-Fi[\lF\]lFbalC(-F^em6#%WIs~the~matrix~in~row~
echelon~form?~Answer~yes;~or~no;|+G>FhvF^fm>FafmFhv@$F]dsFj[r@%Fg[[l-Fjfm6#%NNo
,~the~matrix~is~already~in~row~echelon~formG-Fjfm6#%GYes,~the~matrix~is~in~row~
echelon~formG-F_o6#%5End~of~the~procedureGFfbs>FcvFibs-Fjfm6#%8The~original~mat
rix~is:G-Fjfm6#Fcj[mFjdm>FetF[o>FfcnFbal?(F/F[oF[oF/30FetFbal0FetF]jmC)-F^em6#%
MPlease~enter~a~row~operation~as~in~the~menu|+G>FcvF^fm>FafmFcv@$F]dsF^ds@-5/Fc
vFgfm/FcvFefmC$>FetFbal>FfcnFbal5/FcvFjds/FcvF\esC%-%,gaussnostepGF\]l>FetF]jm>
FfcnFbal5/FcvFies/FcvF[fsC$@(/Fd]lF[o>Fdz-Fafs6%FdzF[xFial/Fd]lF]jm>Fdz-F[er6%F
dzF]aqFci_l/Fd]lFi_n?(FboF[oF[oFhc[mF_s>&Fdz6$F]_pFbo&F`\qF`gm@%/FfcnFbalC$-Fjf
m6$FdgsFdz>FfcnF\^qC%FffxFifxFjdm-F:6$-F^hsFixF_hsC%?(FboF[oF[oFevF_s@$50-Ffhs6
$FcdgmF]dclFbal0-Ffhs6$FcdgmFhddmFbalC%@$Fhdgm>Fi`lFidgm@$F[egm>Fi`lF\egm>F]aqF
bo@%Fa[`lC*>FialF]aq>Facn-FjisFix?(FboF[oF[oFevF_sC$@$/F]dclFacn>F[xFbo@$/Fhddm
Facn>F[xFbo>FdzF^cgm-Fjfm6&%1Interchange~rowsGF[xFgjsFial-Fjfm6$FjjsFdzFjdm>Fd]
lF[oC$?(FboF[oF[oFevF_s@$50-Ffhs6$FhegmF]dclFbal0-Ffhs6$FhegmFhddmFbal>F_`sFbo@
%/F_`sF]aqC'>Fdz-F[er6%FdzF]aqFi`lFh\`l-Fjfm6$F[]tFdzFjdm>Fd]lF]jmFg]t>FfcnFbal
-F:6$FcdgmF]^tC)>FgdnFbal>FjdnFbal?(FboF[oF[oFevF_sC$@$F[ggm@&F\ggm>F]_pFboF_gg
m>F]_pFbo@$Fgdgm@%/FgdnFbalC$>FgdnFbo@&Fhdgm>FcdnFidgmF[egm>FcdnF\egmC$>FjdnFbo
@&Fhdgm>F`[lFidgmF[egm>F`[lF\egm>F`\q-Ff_t6$FdzF]_p@'3F_afl/FgdnF]_pC'?(FboF[oF
[oFhc[mF_s>Fgcgm&-Fhjn6#,&*&FcdnF[o-Ff_t6$FdzFgdnF[oF[o*&F`[lF[o-Ff_t6$FdzFjdnF
[oF[oF`gm-Fjfm6(F]atFjdnFhdrF`[lF^atFgdn-Fjfm6#F\btFi`gmFjdm3Fe_[m/FjdnF]_pC'?(
FboF[oF[oFhc[mF_s>FgcgmF^jgm-Fjfm6(F]atFgdnFhdrFcdnF^atFjdnFjjgmFi`gmFjdm-Fjfm6
#%fnWARNING!~This~is~not~an~elementary~row~operation.~Try~againG>Fd]lFi_n>FfcnF
balC$Fgbt>FfcnFbal>F`dqFhj[m@$/-Fb^o6$-Fhjn6#,&F`dqF[oFdzF^vF[oFbalC%@%2Fhc[mF_
]^m>F[uFhc[m>F[uF_]^m?(F/F[oF[oF//&Fdz6$F[uF[uFbal>F[uFhjal@$3/Fi\hmF[oF`agmC$-
Fjfm6$FadtFdz>FetF]jm@$3/FetFbalF`agm-Fjfm6$FhdtFdzFjdm@$FedvFga`lFjdm-Fjfm6#%b
oThis~is~the~end~of~the~Interactive~Gauss~procedure.~Try~other~examples~GF/Fagm
F/F/,
I1type/CM_MathFuncRFO6%F]c`lF,%#kkG6%%'systemG%)rememberGFeclF/C$@$F]d_lC%>FboF
cd_l>F[u-F\y6$/Fh`nFbo-.F5Fbdo@%54F[u-F:6$Fbo%)mathfuncG-F_o6#F_sFjbalFjbalF/F/
F/6",
IAcontext/cfparse/removeduplicatesR6#'%&clistGFg_n6'%%LocCG%&lastxGF^hmF\hqFc[^
lFg[^lFh[^lC+@$2-F`v6#FK""#-F_oFb`hm>8'Fa`hm>F]]^l-Fd_]l6#;"""Ff`hm>F_\^lFhd[l?
(8&F[ahmF[ahmFf`hmF_s@%/&F\^^l6#F^ahmF_\^l>&F]]^lFbahmF\pC$>FdahmFaahm>F_\^lFaa
hm>8(7#-F^x6$Fdahm/F^ahmFj`hm?(Fh[^lF[ahmF[ahmFh[^l32""!-F`v6#Fiahm/&Fiahm6#F[a
hmQ0<SeparatorLine>Ff_^l>Fiahm&Fiahm6#;Fc`hm!""?(Fh[^lF[ahmF[ahmFh[^l3F`bhm/&Fi
ahm6#F\chmFgbhm>Fiahm&Fiahm6#;F[ahm!"#FiahmFh[^lFh[^lFh[^lFh[^l,
I&menu1RF/6/F\jqF\hqF`hqFc^[mF`hnFghn%%det1GF[balF^hmF_hmFg][mFajcmF^hoFgdmF/C;
>Fh`nFbjm>FboF][n>F[uFe`jl>F`[l-Fhjm6%FboF[oFbal>FcdnF]e[m>Fcdn-Fg\r6%Fj^r;F[oF
[uFbggl>Ffcn-Fhjn6#-Fajr6$-Fg\r6%Fh`nF]fclFfdhmFcdn-Fjfm6#%:The~coefficient~mat
rix~isG-FjfmFbdo-Fjfm6$%Aand~the~reduced~row~echelon~formG-F\^yFbdo-F^em6#%\oIs
~the~coefficient~matrix~row~equivalent~to~the~identity~matrix?|+G-