362 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Iff
1
,f
2
,...,f
m
arereal-valuedfunctionsdefinedonasetDinR
n
,then
FD
2
6
6
6
4
f
1
f
2
:
:
:
f
m
3
7
7
7
5
assignstoeveryXinDanm-vector
F.X/D
2
6
6
6
4
f
1
.X/
f
2
.X/
:
:
:
f
m
.X/
3
7
7
7
5
:
Recallthatf
1
,f
2
,...,f
m
arethecomponentfunctions,orsimplycomponents,ofF. We
write
FWR
n
!R
m
toindicatethatthedomainofFisinR
n
andtherangeofFisinR
m
.WealsosaythatFisa
transformationfromR
n
toR
m
.IfmD1,weidentifyFwithitssinglecomponentfunction
f
1
andregarditasareal-valuedfunction.
Example6.1.1
ThetransformationFWR
2
!R
3
definedby
F.x;y/D
2
4
2xC3y
xC4y
xy
3
5
hascomponentfunctions
f
1
.x;y/D2xC3y; f
2
.x;y/DxC4y; f
3
.x;y/Dxy:
LinearTransformations
ThesimplestinterestingtransformationsfromR
n
toR
m
arethelineartransformations,
definedasfollows
Definition6.1.1
AtransformationLWR
n
!R
m
definedonallofR
n
islinearif
L.XCY/DL.X/CL.Y/
forallXandYinR
n
and
L.aX/DaL.X/
forallXinRandrealnumbersa.
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Section6.1
LinearTransformationsandMatrices
363
Theorem6.1.2
AtransformationLWR
n
!R
m
definedonallofR
n
islinearifand
onlyif
L.X/D
2
6
6
6
4
a
11
x
1
Ca
12
x
2
CCa
1n
x
n
a
21
x
1
Ca
22
x
2
CCa
2n
x
n
:
:
:
a
m1
x
1
Ca
m2
x
2
CCa
mn
x
n
3
7
7
7
5
;
(6.1.1)
wherethea
ij
’sareconstants:
Proof
Ifcanbeseenbyinduction(Exercise6.1.1)thatifLislinear,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
/
(6.1.2)
foranyvectorsX
1
,X
2
,...,X
k
andrealnumbersa
1
,a
2
,...,a
k
. AnyXinR
n
canbe
writtenas
XD
2
6
6
6
4
x
1
x
2
:
:
:
x
n
3
7
7
7
5
Dx
1
2
6
6
6
4
1
0
:
:
:
0
3
7
7
7
5
Cx
2
2
6
6
6
4
0
1
:
:
:
0
3
7
7
7
5
CCx
n
2
6
6
6
4
0
0
:
:
:
1
3
7
7
7
5
Dx
1
E
1
Cx
2
E
2
CCx
n
E
n
:
Applying(6.1.2)withkDn,X
i
DE
i
,anda
i
Dx
i
yields
L.X/Dx
1
L.E
1
/Cx
2
L.E
2
/CCx
n
L.E
n
/:
(6.1.3)
Nowdenote
L.E
j
/D
2
6
6
6
4
a
1j
a
2j
:
:
:
a
mj
3
7
7
7
5
;
so(6.1.3)becomes
L.X/Dx
1
2
6
6
6
4
a
11
a
21
:
:
:
a
m1
3
7
7
7
5
Cx
2
2
6
6
6
4
a
12
a
22
:
:
:
a
m2
3
7
7
7
5
CCx
n
2
6
6
6
4
a
1n
a
2n
:
:
:
a
mn
3
7
7
7
5
;
whichisequivalentto(6.1.1). ThisprovesthatifLislinear,thenLhastheform(6.1.1).
Weleavetheproofoftheconversetoyou(Exercise6.1.2).
Wecalltherectangulararray
AD
2
6
6
6
4
a
11
a
12
 a
1n
a
21
a
21
 a
2n
:
:
:
:
:
:
:
:
:
:
:
:
a
m1
a
m2
 a
mn
3
7
7
7
5
(6.1.4)
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364 Chapter6
Vector-ValuedFunctionsofSeveralVariables
thematrixofthelineartransformation(6.1.1). Thenumbera
ij
intheithrowandjth
columnofAiscalledthe.i;j/thentryofA. WesaythatAisanmnmatrix,sinceA
hasmrowsandncolumns.Wewillsometimesabbreviate(6.1.4)as
ADŒa
ij
:
Example6.1.2
ThetransformationFofExample6.1.1islinear.ThematrixofFis
2
4
2
3
1
4
1 1
3
5
:
Wewillnowrecallthematrixoperationsthatweneedtostudythedifferentialcalculus
oftransformations.
Definition6.1.3
(a)
IfcisarealnumberandADŒa
ij
isanmnmatrix,thencAisthemnmatrix
definedby
cADŒca
ij
I
thatis,cAisobtainedbymultiplyingeveryentryofAbyc.
(b)
IfADŒa
ij
andBDŒb
ij
aremnmatrices,thenthesumACBisthemn
matrix
ACBDŒa
ij
Cb
ij
I
thatis,thesumoftwomnmatricesisobtainedbyaddingcorrespondingentries.
Thesumoftwomatricesisnotdefinedunlesstheyhavethesamenumberofrowsand
thesamenumberofcolumns.
(c)
IfADŒa
ij
isanmpmatrixandBDŒb
ij
isapnmatrix,thentheproduct
CDABisthemnmatrixwith
c
ij
Da
i1
b
1j
Ca
i2
b
2j
CCa
ip
b
pj
D
p
X
kD1
a
ik
b
kj
; 1im; ; 1j j n:
Thus,the.i;j/thentryofABisobtainedbymultiplyingeachentryintheithrowof
AbythecorrespondingentryinthejthcolumnofBandaddingtheproducts.This
definitionrequiresthatAhavethesamenumberofcolumnsasBhasrows.Otherwise,
ABisundefined.
Example6.1.3
Let
AD
2
4
2 1 2
1 0 3
0 1 0
3
5
; BD
2
4
0 1 1
1 0 2
3 0 1
3
5
;
and
CD
2
4
5 0
1 2
3 0 3 1
1 0 1 1
3
5
:
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Section6.1
LinearTransformationsandMatrices
365
Then
2AD
2
4
2.2/ 2.1/ 2.2/
2.1/ 2.0/ 2.3/
2.0/ 2.1/ 2.0/
3
5
D
2
4
4 2 4
2 0 6
0 2 0
3
5
and
ACBD
2
4
2C0 1C1 2C1
11 0C0 3C2
0C3 1C0 0C1
3
5
D
2
4
2 2 3
2 0 5
3 1 1
3
5
:
The(2,3)entryintheproductACisobtainedbymultiplyingtheentriesofthesecond
rowofAbythoseofthethirdcolumnofCandaddingtheproducts:thus,the(2,3)entry
ofACis
.1/.1/C.0/.3/C.3/.1/D4:
ThefullproductACis
2
4
2 1 2
1 0 3
0 1 0
3
5
2
4
5 0
1 2
3 0 3 1
1 0 1 1
3
5
D
2
4
15 0 3 7
2 0 4 1
3 0 3 1
3
5
:
NoticethatACC,BCC,CA,andCBareundefined.
Weleavetheproofsofnextthreetheoremstoyou(Exercises6.1.76.1.9)
Theorem6.1.4
IfA;B;andCaremnmatrices;then
.ACB/CCDAC.BCC/:
Theorem6.1.5
IfAandBaremnmatricesandrandsarerealnumbers;then
(a)
r.sA/D.rs/AI
(b)
.rCs/ADrACsAI
(c)
r.ACB/DrACrB:
Theorem6.1.6
IfA;B;andCaremp;pq;andqnmatrices;respectively;
then.AB/CDA.BC/:
ThenexttheoremshowswhyDefinition6.1.3isappropriate.Weleavetheprooftoyou
(Exercise6.1.11).
Theorem6.1.7
(a)
Ifweregardthevector
XD
2
6
6
6
4
x
1
x
2
:
:
:
x
n
3
7
7
7
5
asann1matrix;thenthelineartransformation(6.1.1)canbewrittenas
L.X/DAX:
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366 Chapter6
Vector-ValuedFunctionsofSeveralVariables
(b)
IfL
1
andL
2
arelineartransformationsfromR
n
toR
m
withmatricesA
1
andA
2
respectively;thenc
1
L
1
Cc
2
L
2
is thelinear transformationfromR
n
toR
m
with
matrixc
1
A
1
Cc
2
A
2
:
(c)
IfL
1
WR
n
!R
p
andL
2
WR
p
!R
m
arelineartransformationswithmatricesA
1
andA
2
;respectively;thenthecompositefunctionL
3
DL
2
ıL
1
;definedby
L
3
.X/DL
2
.L
1
.X//;
isthelineartransformationfromR
n
toR
m
withmatrixA
2
A
1
:
Example6.1.4
If
L
1
.X/D
2
4
2xC3y
3xC2y
xC y
3
5
and L
2
.X/D
2
4
xy
4xCy
x
3
5
;
then
A
1
D
2
4
2 3
3 2
1 1
3
5
and A
2
D
2
4
1 1
4
1
1
0
3
5
:
Thelineartransformation
LD2L
1
CL
2
isdefinedby
L.X/D2L
1
.X/CL
2
.X/
D2
2
4
2xC3y
3xC2y
xC y
3
5
C
2
4
xy
4xCy
x
3
5
D
2
4
3xC5y
10xC5y
xC2y
3
5
:
ThematrixofLis
AD
2
4
3 5
10 5
1 2
3
5
D2A
1
CA
2
:
Example6.1.5
Let
L
1
.X/D
xC2y
3xC4y
WR
2
!R
2
;
and
L
2
.U/D
2
4
uC v
u2v
3uC v
3
5
WR
2
!R
3
:
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Section6.1
LinearTransformationsandMatrices
367
ThenL
3
DL
2
ıL
1
WR
2
!R
3
isgivenby
L
3
.X/DL
2
..L
1
.X//D
2
4
.xC2y/C .3xC4y/
.xC2y/2.3xC4y/
3.xC2y/C .3xC4y/
3
5
D
2
4
4xC 6y
7x10y
6xC10y
3
5
:
ThematricesofL
1
andL
2
are
A
1
D
1 2
3 4
and
A
2
D
2
4
1
1
1 2
3
1
3
5
;
respectively.ThematrixofL
3
is
CD
2
4
4
6
7 10
6
10
3
5
DA
2
A
1
:
Example6.1.6
ThelineartransformationsofExample6.1.5canbewrittenas
L
1
.X/D
1 2
3 4

x
y
; L
2
.U/D
2
4
1
1
1 2
3
1
3
5
u
v
;
and
L
3
.X/D
2
4
4
6
7 10
6
10
3
5
x
y
:
ANewNotationforthe Differential
Ifareal-valuedfunctionf WR
n
!RisdifferentiableatX
0
,then
d
X
0
f Df
x
1
.X
0
/dx
1
Cf
x
2
.X
0
/dx
2
CCf
x
n
.X
0
/dx
n
:
Thiscanbewrittenasamatrixproduct
d
X
0
f DŒf
x
1
.X
0
/ f
x
2
.X
0
/  f
x
n
.X
0
/
2
6
6
6
4
dx
1
dx
2
:
:
:
dx
n
3
7
7
7
5
:
(6.1.5)
Wedefinethedifferentialmatrixoff atX
0
by
f
0
.X
0
/DŒf
x
1
.X
0
/ f
x
2
.X
0
/  f
x
n
.X
0
/
(6.1.6)
andthedifferentiallineartransformationby
dXD
2
6
6
6
4
dx
1
dx
2
:
:
:
dx
n
3
7
7
7
5
:
368 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Then(6.1.5)canberewrittenas
d
X
0
f Df
0
.X
0
/dX:
(6.1.7)
Thisisanalogoustothecorrespondingformulaforfunctionsofonevariable(Exam-
ple5.3.7),andshowsthatthedifferentialmatrixf0.X
0
/isanaturalgeneralizationofthe
derivative.Withthisnewnotationwecanexpressthedefiningpropertyofthedifferential
inawaysimilartotheformthatappliesfornD1:
lim
X!X
0
f.X/f.X
0
/f0.X
0
/.XX
0
/
jXX
0
j
D0;
whereX
0
D.x
10
;x
20
;:::;x
n0
/andf
0
.X
0
/.XX
0
/isthematrixproduct
Œf
x
1
.X
0
/ f
x
2
.X
0
/  f
x
n
.X
0
/
2
6
6
6
4
x
1
x
10
x
2
x
20
:
:
:
x
n
x
n0
3
7
7
7
5
:
Asbefore,weomittheX
0
in(6.1.6)and(6.1.7)whenitisnotnecessarytoemphasize
thespecificpoint;thus,wewrite
f
0
D
f
x
1
f
x
2
 f
x
n
and
df Df
0
dX:
Example6.1.7
If
f.x;y;´/D4x
2
3
;
then
f
0
.x;y;´/DŒ8xy´
3
4x
2
´
3
12x
2
2
:
Inparticular,ifX
0
D.1;1;2/,then
f
0
.X
0
/DŒ64 32 48;
so
d
X
0
f Df
0
.X
0
/dXDŒ64 32 48
2
4
dx
dy
3
5
D64dxC32dy48d´:
TheNormofaMatrix
Wewillneedthefollowingdefinitioninthenextsection.
Definition6.1.8
Thenorm;kAk;ofanmnmatrixADŒa
ij
isthesmallestnumber
suchthat
jAXjkAkjXj
forallXinR
n
:
Section6.1
LinearTransformationsandMatrices
369
Tojustifythisdefinition,wemustshowthatkAkexists. ThecomponentsofYDAX
are
y
i
Da
i1
x
1
Ca
i2
x
2
CCa
in
x
n
; 1im:
BySchwarz’sinequality,
y
2
i
.a
2
i1
Ca
2
i2
CCa
2
in
/jXj
2
:
Summingthisover1imyields
jYj
2
0
@
Xm
iD1
Xn
jD1
a
2
ij
1
A
jXj
2
:
Therefore,theset
BD
˚
K
ˇ
ˇ
jAXjKjXj forallXinR
n
isnonempty.SinceBisboundedbelowbyzero,Bhasaninfimum˛.If>0,then˛C
isinBbecauseifnot,thennonumberlessthan˛CcouldbeinB.Then˛Cwouldbe
alowerboundforB,contradictingthedefinitionof˛.Hence,
jAXj.˛C/jXj; X2R
n
:
Sinceisanarbitrarypositivenumber,thisimpliesthat
jAXj˛jXj; X2R
n
;
so˛2B.SincenosmallernumberisinB,weconcludethatkAkD˛.
InourapplicationswewillnothavetoactuallycomputethenormofamatrixA;rather,
itwillbesufficienttoknowthatthenormexists(finite).
SquareMatrices
LineartransformationsfromR
n
toR
n
willbeimportantwhenwediscusstheinversefunc-
tiontheoreminSection6.3andchangeofvariablesinmultipleintegralsinSection7.3.
Thematrixofsuchatransformationissquare;thatis,ithasthesamenumberofrowsand
columns.
Weassumethatyouknowthedefinitionofthedeterminant
det.A/D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
a
11
a
12
 a
1n
a
21
a
22
 a
2n
:
:
:
:
:
:
:
:
:
:
:
:
a
n1
a
n2
 a
nn
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ofannnmatrix
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
:
370 Chapter6
Vector-ValuedFunctionsofSeveralVariables
Thetranspose,A
t
,ofamatrixA(squareornot)isthematrixobtainedbyinterchanging
therowsandcolumnsofA;thus,if
AD
2
4
1 2
3
3 1
4
0 1 2
3
5
; then A
t
D
2
4
1 3
0
2 1
1
3 4 2
3
5
:
Asquarematrixanditstransposehavethesamedeterminant;thus,
det.A
t
/Ddet.A/:
Wetakethenexttheoremfromlinearalgebraasgiven.
Theorem6.1.9
IfAandBarennmatrices;then
det.AB/Ddet.A/det.B/:
Theentriesa
ii
,1i n,ofannnmatrixAareonthemaindiagonalofA.Thenn
matrixwithonesonthemaindiagonalandzeroselsewhereiscalledtheidentitymatrixand
isdenotedbyI;thus,ifnD3,
ID
2
4
1 0 0
0 1 0
0 0 1
3
5
:
WecallItheidentitymatrixbecauseAIDAandIADAifAisanynnmatrix.We
saythatannnmatrixAisnonsingularifthereisannnmatrixA
1
,theinverseof A,
suchthatAA1 DA1ADI.Otherwise,wesaythatAissingular
Ourmainobjectiveis toshowthatannnmatrixAisnonsingularifandonlyif
det.A/¤0.Wewillalsofindaformulafortheinverse.
Definition6.1.10
LetADŒa
ij
beannnmatrix;withn2:Thecofactorofan
entrya
ij
is
c
ij
D.1/
iCj
det.A
ij
/;
whereA
ij
isthe.n1/.n1/matrixobtainedbydeletingtheithrowandjthcolumn
ofA:TheadjointofA;denotedbyadj.A/;isthennmatrixwhose.i;j/thentryisc
ji
:
Example6.1.8
Thecofactorsof
AD
2
4
4
2 1
3 1 2
0
1 2
3
5
Section6.1
LinearTransformationsandMatrices
371
are
c
11
D
ˇ
ˇ
ˇ
ˇ
1 2
1 2
ˇ
ˇ
ˇ
ˇ
D4; c
12
D
ˇ
ˇ
ˇ
ˇ
3 2
0 2
ˇ
ˇ
ˇ
ˇ
D6; c
13
D
ˇ
ˇ
ˇ
ˇ
3 1
0
1
ˇ
ˇ
ˇ
ˇ
D
3;
c
21
D
ˇ
ˇ
ˇ
ˇ
2 1
1 2
ˇ
ˇ
ˇ
ˇ
D3; c
22
D
ˇ
ˇ
ˇ
ˇ
4 1
0 2
ˇ
ˇ
ˇ
ˇ
D
8; c
23
D
ˇ
ˇ
ˇ
ˇ
4 2
0 1
ˇ
ˇ
ˇ
ˇ
D
4;
c
31
D
ˇ
ˇ
ˇ
ˇ
2 1
1 2
ˇ
ˇ
ˇ
ˇ
D
5; c
32
D
ˇ
ˇ
ˇ
ˇ
4 1
3 2
ˇ
ˇ
ˇ
ˇ
D5; c
33
D
ˇ
ˇ
ˇ
ˇ
4
2
3 1
ˇ
ˇ
ˇ
ˇ
D10;
so
adj.A/D
2
4
4 3
5
6
8
5
3 4 10
3
5
:
Noticethatadj.A/isthetransposeofthematrix
2
4
4 6
3
3
8
4
5 5 10
3
5
obtainedbyreplacingeachentryofAbyitscofactor.
Foraproofofthefollowingtheorem,seeanyelementarylinearalgebratext.
Theorem6.1.11
LetAbeannnmatrix:
(a)
ThesumoftheproductsoftheentriesofarowofAandtheircofactorsequalsdet.A/;
whilethesumoftheproductsoftheentriesofarowofAandthecofactorsofthe
entriesofadifferentrowequalszeroIthatis;
Xn
kD1
a
ik
c
jk
D
det.A/; iDj;
0;
i¤j:
(6.1.8)
(b)
ThesumoftheproductsoftheentriesofacolumnofAandtheircofactorsequals
det.A/;whilethesumoftheproductsoftheentriesofacolumnofAandthecofactors
oftheentriesofadifferentcolumnequalszeroIthatis;
Xn
kD1
c
ki
a
kj
D
det.A/; iDj;
0;
i¤j:
(6.1.9)
Ifwecomputedet.A/fromtheformula
det.A/D
Xn
kD1
a
ik
c
ik
;
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