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372 Chapter6
Vector-ValuedFunctionsofSeveralVariables
wesaythatweareexpandingthedeterminantincofactorsofitsithrow. Sincewecan
choosei arbitrarilyfromf1;:::;ng,therearenwaystodothis. . Ifwecomputedet.A/
fromtheformula
det.A/D
Xn
kD1
a
kj
c
kj
;
wesaythatweareexpandingthedeterminantincofactorsofitsjthcolumn.Therearealso
nwaystodothis.
Inparticular,wenotethatdet.I/D1foralln1.
Theorem6.1.12
LetAbeannnmatrix:Ifdet.A/ D 0;thenAissingular:If
det.A/¤0;thenAisnonsingular;andAhastheuniqueinverse
A
1
D
1
det.A/
adj.A/:
(6.1.10)
Proof
Ifdet.A/ D 0, , thendet.AB/ D 0foranynnmatrix, , byTheorem6.1.9.
Therefore,sincedet.I/D1,thereisnomatrixnnmatrixBsuchthatABDI;thatis,A
issingularifdet.A/D0.Nowsupposethatdet.A/¤0.Since(6.1.8)impliesthat
Aadj.A/Ddet.A/I
and(6.1.9)impliesthat
adj.A/ADdet.A/I;
dividingbothsidesofthesetwoequationsbydet.A/showsthatifA
1
isasdefinedin
(6.1.10),thenAA
1
DA
1
ADI.Therefore,A
1
isaninverseofA.Toseethatitisthe
onlyinverse,supposethatBisannnmatrixsuchthatABDI.ThenA
1
.AB/DA
1
,
so.A
1
A/BDA
1
.SinceAA
1
DIandIBDB,itfollowsthatBDA
1
.
Example6.1.9
InExample6.1.8wefoundthattheadjointof
AD
2
4
4
2 1
3 1 2
0
1 2
3
5
is
adj.A/D
2
4
4 3
5
6
8
5
3 4 10
3
5
:
Wecancomputedet.A/byfindinganydiagonalentryofAadj.A/. (Why?) ) Thisyields
det.A/D25.(Verify.)Therefore,
A
1
D
1
25
2
4
4 3
5
6
8
5
3 4 10
3
5
:
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Section6.1
LinearTransformationsandMatrices
373
Nowconsidertheequation
AXDY
(6.1.11)
with
AD
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
22
 a
2n
:
:
:
:
:
:
:
:
:
:
:
:
a
n1
a
n2
 a
nn
3
7
7
7
5
; XD
2
6
6
6
4
x
1
x
2
:
:
:
x
n
3
7
7
7
5
; and
YD
2
6
6
6
4
y
1
y
2
:
:
:
y
n
3
7
7
7
5
:
HereAandYaregiven,andtheproblemistofindX.
Theorem6.1.13
Thesystem(6.1.11)hasasolutionXforanygivenYifandonlyif
Aisnonsingular:Inthiscase;thesolutionisuniqueandisgivenbyXDA
1
Y.
Proof
SupposethatAisnonsingular,andletXDA
1
Y.Then
AXDA.A
1
Y/D.AA
1
/YDIYDYI
thatis,Xisasolutionof(6.1.11). ToseethatXistheonlysolutionof(6.1.11),suppose
thatAX
1
DY.ThenAX
1
DAX,so
A
1
.AX/DA
1
.AX
1
/
and
.A
1
A/XD.A
1
A/X
1
;
whichisequivalenttoIXDIX
1
,orXDX
1
.
Conversely,supposethat(6.1.11)hasasolutionforeveryY,andletX
i
satisfyAX
i
D
E
i
,1in.Let
BDŒX
1
X
2
X
n
I
thatis,X
1
,X
2
,...,X
n
arethecolumnsofB.Then
ABDŒAX
1
AX
2
AX
n
DŒE
1
E
2
E
n
DI:
ToshowthatBDA
1
,wemuststillshowthatBADI.Wefirstnotethat,sinceABDI
anddet.BA/Ddet.AB/D1(Theorem6.1.9),BAisnonsingular(Theorem6.1.12).Now
notethat
.BA/.BA/DB.AB/A/DBIAI
thatis,
.BA/.BA/D.BA/:
MultiplyingbothsidesofthisequationontheleftbyBA/
1
yieldsBADI.
Thefollowingtheoremgives ausefulformulaforthecomponents ofthesolutionof
(6.1.11).
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374 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Theorem6.1.14(Cramer’sRule)
IfA D D Œa
ij
isnonsingular;thenthesolu-
tionofthesystem
a
11
x
1
Ca
12
x
2
CCa
1n
x
n
Dy
1
a
21
x
1
Ca
22
x
2
CCa
2n
x
n
Dy
2
:
:
:
a
n1
x
1
Ca
n2
x
2
CCa
nn
x
n
Dy
n
.or;inmatrixform;AXDY/isgivenby
x
i
D
D
i
det.A/
; 1in;
whereD
i
isthedeterminantofthematrixobtainedbyreplacingtheithcolumnofAwith
YIthus;
D
1
D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
y
1
a
12
 a
1n
y
2
a
22
::: a
2n
:
:
:
:
:
:
:
:
:
:
:
:
y
n
a
n2
 a
nn
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
; D
2
D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
a
11
y
1
a
13
 a
1n
a
21
y
2
a
23
 a
2n
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
a
n1
y
n
a
n3
 a
nn
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
; ;
D
n
D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
a
11
 a
1;n1
y
1
a
21
 a
2;n1
y
2
:
:
:
:
:
:
:
:
:
:
:
:
a
n1
 a
n;n1
y
n
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
:
Proof
FromTheorems6.1.12and6.1.13,thesolutionofAXDYis
2
6
6
6
4
x
1
x
2
:
:
:
x
n
3
7
7
7
5
DA
1
YD
1
det.A/
2
6
6
6
4
c
11
c
21
 c
n1
c
12
c
22
 c
n2
 
:
:
:

c
1n
c
2n
 c
nn
3
7
7
7
5
2
6
6
6
4
y
1
y
2
:
:
:
y
n
3
7
7
7
5
D
2
6
6
6
4
c
11
y
1
Cc
21
y
2
CCc
n1
y
n
c
12
y
1
Cc
22
y
2
CCc
n2
y
n
:
:
:
c
1n
y
1
Cc
2n
y
2
CCc
nn
y
n
3
7
7
7
5
:
But
c
11
y
1
Cc
21
y
2
CCc
n1
y
n
D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
y
1
a
12
 a
1n
y
2
a
22
::: a
2n
:
:
:
:
:
:
:
:
:
:
:
:
y
n
a
n2
 a
nn
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
;
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Section6.1
LinearTransformationsandMatrices
375
ascanbeseenbyexpandingthedeterminantontherightincofactorsofitsfirstcolumn.
Similarly,
c
12
y
1
Cc
22
y
2
CCc
n2
y
n
D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
a
11
y
1
a
13
 a
1n
a
21
y
2
a
23
 a
2n
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
a
n1
y
n
a
n3
 a
nn
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
;
ascanbeseenbyexpandingthedeterminantontherightincofactorsofitssecondcolumn.
Continuinginthiswaycompletestheproof.
Example6.1.10
Thematrixofthesystem
4xC2yC´D1
3xyC2´D2
yC2´D0
is
AD
2
4
4
2 1
3 1 2
0
1 2
3
5
:
Expandingdet.A/incofactorsofitsfirstrowyields
det.A/D4
ˇ
ˇ
ˇ
ˇ
1 2
1 2
ˇ
ˇ
ˇ
ˇ
2
ˇ
ˇ
ˇ
ˇ
3 2
0 2
ˇ
ˇ
ˇ
ˇ
C1
ˇ
ˇ
ˇ
ˇ
3 1
0
1
ˇ
ˇ
ˇ
ˇ
D4.4/2.6/C1.3/D25:
UsingCramer’sruletosolvethesystemyields
xD
1
25
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
1
2 1
2 1 2
0
1 2
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
D
2
5
; yD
1
25
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
4 1 1
3 2 2
0 0 2
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
D
2
5
;
´D
1
25
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
4
2 1
3 1 2
0
1 0
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
D
1
5
:
Asystemofnequationsinnunknowns
a
11
x
1
Ca
12
x
2
CCa
1n
x
n
D0
a
21
x
1
Ca
22
x
2
CCa
2n
x
n
D0
:
:
:
a
n1
x
1
Ca
n2
x
2
CCa
nn
x
n
D0
(6.1.12)
(or,inmatrixform,AX D D 0)ishomogeneous. ItisobviousthatX
0
D 0satisfiesthis
system.Wecallthisthetrivialsolutionof(6.1.12).Anyothersolutionsof(6.1.12),ifthey
exist,arenontrivial.
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376 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Wewillneedthefollowingtheorems. Theproofsmaybefoundinanylinearalgebra
text.
Theorem6.1.15
Thehomogeneoussystem(6.1.12)ofnequationsinnunknownshas
anontrivialsolutionifandonlyifdet.A/D0:
Theorem6.1.16
IfA
1
;A
2
;...;A
k
arenonsingularnnmatrices;thensoisA
1
A
2
A
k
;
and
.A
1
A
2
A
k
/
1
DA
1
k
A
1
k1
A
1
1
:
6.1Exercises
1.
Prove:IfLWR
n
!R
m
isalineartransformation,then
L.a
1
X
1
Ca
2
X
2
CCa
k
X
k
/Da
1
L.X
1
/Ca
2
L.X
2
/CCa
k
L.X
k
/
ifX
1
;X
2
;:::;X
k
areinRanda
1
,a
2
,...,a
k
arerealnumbers.
2.
ProvethatthetransformationLdefinedbyEqn.(6.1.1)islinear.
3.
FindthematrixofL.
(a)
L.X/D
2
4
3xC4yC6´
2x47C2´
7xC2yC3´
3
5
(b)
L.X/D
2
6
6
4
2x
1
C4x
2
3x
1
2x
2
7x
1
4x
2
6x
1
C x
2
3
7
7
5
4.
FindcA.
(a)
cD4;AD
2
4
2 2 4
6
0 0 1
3
3 4 7 11
3
5
(b)
cD2;AD
2
4
1
3 0
0
1 2
1 1 3
3
5
5.
FindACB.
(a)
AD
2
4
1
2 3
1
1 4
0 1 4
3
5
; BD
2
4
1
0
3
5
6 7
0 1
2
3
5
(b)
AD
2
4
0 5
3 2
1 7
3
5
; BD
2
4
1 2
0 3
4 7
3
5
6.
FindAB.
(a)
AD
2
4
1
2 3
0
1 4
0 1 4
3
5
; BD
2
4
1 2
0 3
4 7
3
5
(b)
AD
5 3 2 1
6 7 4 1
; BD
2
6
6
4
1
3
4
7
3
7
7
5
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Section6.1
LinearTransformationsandMatrices
377
7.
ProveTheorem6.1.4.
8.
ProveTheorem6.1.5.
9.
ProveTheorem6.1.6.
10.
SupposethatACBandABarebothdefined.WhatcanbesaidaboutAandB?
11.
ProveTheorem6.1.7.
12.
FindthematrixofaL
1
CbL
2
.
(a)
L
1
.x;y;´/D
2
4
3xC2yC ´
xC4yC2´
3x4yC ´
3
5
,
L
2
.x;y;´/D
2
4
xCy ´
2xCyC3´
yC ´
3
5
; aD2; bD1
(b)
L
1
.x;y/D
2
4
2xC3y
x y
4xC y
3
5
; L
2
.x;y/D
2
4
3xy
xCy
xy
3
5
; aD4; bD
2
13.
FindthematricesofL
1
ıL
2
andL
2
ıL
1
,whereL
1
andL
2
areasinExercise6.1.12
(a)
.
14.
WritethetransformationsofExercise6.1.12intheformL.X/DAX.
15.
Findf
0
andf
0
.X
0
/.
(a)
f.x;y;´/D3x
2
y´, X
0
D.1;1;1/
(b)
f.x;y/Dsin.xCy/, X
0
D.=4;=4/
(c)
f.x;y;´/Dxyex´, X
0
D.1;2;0/
(d)
f.x;y;´/Dtan.xC2yC´/, X
0
D.=4;=8;=4/
(e)
f.X/DjXjWR
n
!R, X
0
D.1=
p
n;1=
p
n;:::;1=
p
n/
16.
LetADŒa
ij
beanmnmatrixand
Dmax
˚
ja
ij
j
ˇ
ˇ
1im;1i n
:
ShowthatkAk
p
mn.
17.
Prove:IfAhasatleastonenonzeroentry,thenkAk¤0.
18.
Prove:kACBkkAkCkBk.
19.
Prove:kABkkAkkBk.
20.
SolvebyCramer’srule.
(a)
xC yC2´D
1
2x yC ´D1
x2y3´D
2
(b)
xC y ´D 5
3x2yC2´D 0
4xC2y3´D14
378 Chapter6
Vector-ValuedFunctionsofSeveralVariables
(c)
xC2yC3´D5
x
 ´D1
xC yC2´D4
(d)
x yC ´2wD 1
2xC y3´C3wD 4
3xC2y
C wD13
2xC y ´
D 4
21.
FindA
1
bythemethodofTheorem6.1.12.
(a)
1 2
3
4
(b)
2
4
1 2
3
1 0 1
1 1
2
3
5
(c)
2
4
4
2 1
3 1 2
0
1 2
3
5
(d)
2
4
1 0 1
0 1 1
1 1 0
3
5
(e)
2
6
6
4
1 2
0 0
2 3
0 0
0 0
2 3
0 0 1 2
3
7
7
5
(f)
2
6
6
4
1 1
2 1
2 2 1
3
1 4
1
2
3 1
0
1
3
7
7
5
22.
For1   i;j   m, leta
ij
D a
ij
.X/beareal-valuedfunctioncontinuousona
compactsetKinRn.Supposethatthemmmatrix
A.X/DŒa
ij
.X/
isnonsingularforeachXinK,anddefinethemmmatrix
B.X;Y/DŒb
ij
.X;Y/
by
B.X;Y/DA
1
.X/A.Y/I:
Showthatforeach>0thereisaı>0suchthat
jb
ij
.X;Y/j<; 1i;jm;
ifX;Y2KandjXYj<ı.H
INT
:Showthatb
ij
iscontinuousontheset
˚
.X;Y/
ˇ
ˇ
X2K;Y2K
:
ThenassumethattheconclusionisfalseanduseExercise5.1.32toobtainacontradiction:
6.2 CONTINUITY AND DIFFERENTIABILITY OF TRANS-
FORMATIONS
Throughouttherestofthischapter,transformationsFandpointsXshouldbeconsideredas
writteninverticalformwhentheyoccurinconnectionwithmatrixoperations. However,
wewillwriteXD.x
1
;x
2
;:::;x
n
/whenXistheargumentofafunction.
Section6.2
ContinuityandDifferentiabilityofTransformations
379
Continuous Transformations
InSection5.2wedefinedavector-valuedfunction(transformation)tobecontinuousatX
0
ifeachofitscomponentfunctionsiscontinuousatX
0
.Weleaveittoyoutoshowthatthis
impliesthefollowingtheorem(Exercise1).
Theorem6.2.1
SupposethatX
0
isin;andalimitpointof;thedomainofFWR
n
!
Rm:ThenFiscontinuousatX
0
ifandonlyifforeach>0thereisaı>0suchthat
jF.X/F.X
0
/j< if jXX
0
j<ı and X2D
F
:
(6.2.1)
ThistheoremisthesameasTheorem5.2.7exceptthatthe“absolutevalue”in(6.2.1)
nowstandsfordistanceinRratherthanR.
IfCisaconstantvector,then“lim
X!X
0
F.X/DC”meansthat
lim
X!X
0
jF.X/CjD0:
Theorem6.2.1impliesthatFiscontinuousatX
0
ifandonlyif
lim
X!X
0
F.X/DF.X
0
/:
Example6.2.1
Thelineartransformation
L.X/D
2
4
xC yC´
2x3yC´
2xC y´
3
5
iscontinuousateveryX
0
inR
3
,since
L.X/L.X
0
/DL.XX
0
/D
2
4
.xx
0
/C .yy
0
/C.´´
0
/
2.xx
0
/3.yy
0
/C.´´
0
/
2.xx
0
/C .yy
0
/.´´
0
/
3
5
;
andapplyingSchwarz’sinequalitytoeachcomponentyields
jL.X/L.X
0
/j
2
.3C14C6/jXX
0
j
2
D23jXX
0
j
2
:
Therefore,
jL.X/L.X
0
/j< if jXX
0
j<
p
23
:
DifferentiableTransformations
InSection5.4wedefinedavector-valuedfunction(transformation)tobedifferentiableat
X
0
ifeachofitscomponentsisdifferentiableatX
0
(Definition5.4.1). Thenexttheorem
characterizesthispropertyinausefulway.
380 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Theorem6.2.2
AtransformationFD.f
1
;f
2
;:::;f
m
/definedinaneighborhoodof
X
0
2R
n
isdifferentiableatX
0
ifandonlyifthereisaconstantmnmatrixAsuchthat
lim
X!X
0
F.X/F.X
0
/A.XX
0
/
jXX
0
j
D0:
(6.2.2)
If(6.2.2)holds;thenAisgivenuniquelyby
AD
@f
i
.X
0
/
@x
j
D
2
6
6
6
6
6
6
6
6
6
4
@f
1
.X
0
/
@x
1
@f
1
.X
0
/
@x
2

@f
1
.X
0
/
@x
n
@f
2
.X
0
/
@x
1
@f
2
.X
0
/
@x
2

@f
2
.X
0
/
@x
n
:
:
:
:
:
:
:
:
:
:
:
:
@f
m
.X
0
/
@x
1
@f
m
.X
0
/
@x
2

@f
m
.X
0
/
@x
n
3
7
7
7
7
7
7
7
7
7
5
:
(6.2.3)
Proof
LetX
0
D .x
10
;x
20
;:::;x
n0
/. IfFisdifferentiableatX
0
,thensoaref
1
,f
2
,
...,f
m
(Definition5.4.1).Hence,
lim
X!X
0
f
i
.X/f
i
.X
0
/
Xn
jD1
@f
i
.X
0
/
@x
j
.x
j
x
j0
/
jXX
0
j
D0; 1im;
whichimplies(6.2.2)withAasin(6.2.3).
Nowsupposethat(6.2.2)holdswithADŒa
ij
. Sinceeachcomponentofthevectorin
(6.2.2)approacheszeroasXapproachesX
0
,itfollowsthat
lim
X!X
0
f
i
.X/f
i
.X
0
/
Xn
jD1
a
ij
.x
j
x
j0
/
jXX
0
j
D0; 1im;
soeachf
i
is differentiableat X
0
, andtherefore e soisF (Definition5.4.1). ByTheo-
rem5.3.6,
a
ij
D
@f
i
.X
0
/
@x
j
; 1im; 1j j n;
whichimplies(6.2.3).
AtransformationTWR!Roftheform
T.X/DUCA.XX
0
/;
whereUisaconstantvectorinR
m
,X
0
isaconstantvectorinR
n
,andAisaconstantmn
matrix,issaidtobeaffine.Theorem6.2.2saysthatifFisdifferentiableatX
0
,thenFcan
bewellapproximatedbyanaffinetransformation.
Section6.2
ContinuityandDifferentiabilityofTransformations
381
Example6.2.2
Thecomponentsofthetransformation
F.X/D
2
4
x
2
C2xyC´
x C2x´Cy
x
2
C y
2
2
3
5
aredifferentiableatX
0
D.1;0;2/. Evaluatingthepartialderivativesofthecomponents
thereyields
AD
2
4
2 2 1
5 1 2
2 0 4
3
5
:
(Verify).Therefore,Theorem6.2.2impliesthattheaffinetransformation
T.X/DF.X
0
/CA.XX
0
/
D
2
4
3
5
5
3
5
C
2
4
2 2 1
5 1 2
2 0 4
3
5
2
4
x1
y
´2
3
5
satisfies
lim
X!X
0
F.X/T.X/
jXX
0
j
D0:
DifferentialofaTransformation
IfFD.f
1
;f
2
;:::;f
m
/isdifferentiableatX
0
,wedefinethedifferentialofFatX
0
tobe
thelineartransformation
d
X
0
FD
2
6
6
6
4
d
X
0
f
1
d
X
0
f
2
:
:
:
d
X
0
f
m
3
7
7
7
5
:
(6.2.4)
WecallthematrixAin(6.2.3)thedifferentialmatrixofFatX
0
anddenoteitbyF
0
.X
0
/;
thus,
F
0
.X
0
/D
2
6
6
6
6
6
6
6
6
6
6
6
4
@f
1
.X
0
/
@x
1
@f
1
.X
0
/
@x
2

@f
1
.X
0
/
@x
n
@f
2
.X
0
/
@x
1
@f
2
.X
0
/
@x
2

@f
2
.X
0
/
@x
n
:
:
:
:
:
:
:
:
:
:
:
:
@f
m
.X
0
/
@x
1
@f
m
.X
0
/
@x
2

@f
m
.X
0
/
@x
n
3
7
7
7
7
7
7
7
7
7
7
7
5
:
(6.2.5)
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