﻿

# asp.net mvc 4 and the web api pdf free download : Asp.net merge pdf files SDK application service wpf windows azure dnn mechanics2-part1290

MathematicalPrependix
17
Read the components from these equations. E.g.
a
x
x
=
d
2
x=dt
2
. Forcylindrical l coordinates
the appropriate basis vectors conform to this system, with ^
r
pointing away y from the e origin and
^
perpendiculartothat.
~r
=
z^z
+
r^r
,and plane polarsimply omits the
z
. Nowtheunitvectors s are
functionsofposition,implyingthatasaparticlemovestheseunitvectorswillchange,andyouhaveto
usetheproductruletodierentiatetheterms. ^
z
isconstantsoitcausesnotrouble.
~v
_
z^z
+_
r^r
+
r
_
^r
The one newfeatureis the third term, andforthatyou need to notice that ^
r
is a a functionofthe
coordinate
,thoughnotof
z
or
r
. Toevaluatethisderivative,usethechainrule.
d^r
dt
=
d^r
d
d
dt
Therstofthesederivatives,
d
^
r=d
entiatingvectorsthathaveaconstantmagnitude: thederivativeisperpendiculartotheoriginalvector.
Toshowthis,let
~u
beany vectorofconstantmagnitude. Thatis,
~u
.
~u
=
C
. Dierentiatethiswith
respecttoanything.
d
dt
~u
.
~u
=
dC
dt
=0=2
~u
.
d~u
dt
(0
:
38)
That’sallyouneed,becauseitsaysthatthederivativeiseitherzeroorisperpendicularto
~u
asclaimed.
d
^
r=d
isperpendicularto ^
r
. Itisinthe
^
direction. Now,what t isits magnitude? ? Asketch
^
directionandnot
along
^
?"
^
r
(
)
^
r
(
+
)
^
r
Fig.0.11
Thethreevectors ^
r
(
),^
r
(
+
),and^
r
formanisoscelestriangle. Constructthebisector
ofthevertexangle,andyouimmediatelyseethatthelengthof^
r
is
^
r
=2
.
1
.
sin(
=
2)
As
!0,thesinebehavesas
=
2itself,sothequotient
^
r
=
!1.
d
^
r
d
=
^
;
andsimilarly
d
^
d
^
r
(0
:
39)
Nowbacktovelocityandacceleration.
~v
=
d~r=dt
_
z^z
+_
r^r
+
r
_
^r=z_^z+_r^r+r
_
^
(0
:
40)
Anotherderivative:
~a
=
d~v=dt
=
z
^
z
+
r
^
r
+_
r
_
^
r
+_
r
_
^
+
r
^
+
r
_
_
^
=
z^z
+
r^r
+_
r
d^r
d
_
+_
r
_
^
+
r
^
+
r
_
d
^
d
_
=
z
^
z
+^
r
r
r
_
2
+
^
r
+2_
r
_
(0
:
41)
Asp.net merge pdf files - Merge, append PDF files in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Provide C# Demo Codes for Merging and Appending PDF Document
pdf merger online; acrobat combine pdf
Asp.net merge pdf files - VB.NET PDF File Merge Library: Merge, append PDF files in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
VB.NET Guide and Sample Codes to Merge PDF Documents in .NET Project
attach pdf to mail merge; break pdf into multiple files
MathematicalPrependix
18
Hereyouneed
d
^
=d
,anotherderivativeofaunitvector,soit’sperpendicularto
^
. Howbigisit?
Youcandothesamesortofgeometryaswith^
r
,ornoticethat
^r
.
^
=0  !
d
d
^r
.
^
=0=
d^r
d
.
^
+^
r
.
d
^
d
=1+^
r
.
d
^
d
andthisestablishesthesignandmagnitudeof
d
^
=d
asinEq.(0.39).Increasingtheangle
alittle
bitrotatesanobjectalittlebitcounterclockwise.Thatrotates ^
r
toward
^
.Itrotates
^
toward ^
r
.
Thatthebasisvectorsvarywithpositionandarenotparalleltoeachotherasyoumovearound
thecoordinatesystemisafamiliarideainanothercontext: geography.
^
Up,
^
South,and
^
Eastarenot
parallelvectorsasyoumovearoundtheEarth.
Forsphericalcoordinatesthederivationscanbedonealongthesamelines,butwithalotmore
algebra. Itisnotworth h the troubletogothrough it,and youdon’tneed theresults asoften. . The
r
fromthecylindricalone|they’respelled
thesame).
~r
=
r
^
r
~v
=
_
~r
_
r^r
+
r
_
^
+
r
_
sin
^
~a
=
r
r
_
2
sin
2
r
_
2
^
r
+
r
+2_
r
_
r
_
2
sin
cos
^
+
r
sin
+2_
r
_
sin
+2
r
_
_
cos
^
(0
:
42)
Ingeographicalterms,
^
r
c
Up
^
S
c
outh
^
E
b
ast
Example
Circularmotionisafamiliarexamplefromintroductorycourses. If
z
=0and
r
=aconstant,the
equations(0.40)and(0.41)are
~v
=
r
_
^
and
~a
^
rr
_
2
+
^
r
^
r
v
2
r
+
^
dv
dt
(0
:
43)
Thelastformcomesfromusingthemagnitudeoftherstoftheseequationsfor
~v
,thatis
v
=
r
_
,and
r
_
2
=
v
2
=r
). Thetangential
componentis
r
=
d
(
r
_
)
=dt
=
dv=dt
Now waita minute! Ifyou u believe this manipulation,look
againmorecritically.Isthemotioncounterclockwiseorclockwiseanddoesitmatter?Is
_
positiveor
negative? Isthemagnitudeofavectorpositive? ? Whenyouexpressavectorintermsofcomponents,
thecorrectequationisnot
v
=
r
_
,but
v
=
r
_
,statingthatthephi-componentofthevelocityis
r
_
.
Gobackandmodifytheseequationsappropriately.
Example
x
=
x
0
,aconstant,
y
=
v
0
t
shouldhaveconstantvelocityandzeroacceleration,butthat’snotso
obviousifyouseeitinpolarcoordinates.
r
=
p
x
2
+
y
2
=
q
x
2
0
+
v
2
0
t
2
and
=tan
1
(
y=x
)=tan
1
v
0
t
x
0
~v
=^
r
_
r
+
^
r
_
=^
r
v
2
0
t
p
x
2
0
+
v
2
0
t
2
+
^
q
x
2
0
+
v
2
0
t
2
1
1+
v
0
t=x
0
2
.
v
0
x
0
=^
r
v
2
0
t
p
x
2
0
+
v
2
0
t
2
+
^
v
0
x
0
p
x
2
0
+
v
2
0
t
2
(0
:
44)
VB.NET PDF- HTML5 PDF Viewer for VB.NET Project
PDF; Merge PDF Files; Split PDF Document; Remove Password from PDF; Change PDF Permission Settings. FREE TRIAL: HOW TO: XDoc.HTML5 Viewer for C#▶: C# ASP.NET:
add pdf files together; acrobat combine pdf files
Online Merge PDF files. Best free online merge PDF tool.
Thus, C#.NET PDF document merge library control can Download and try RasterEdge.XDoc. PDF for .NET and imaging solutions, available for ASP.NET AJAX, Silverlight
reader merge pdf; .net merge pdf files
MathematicalPrependix
19
Youcan see that this has the correctbehavior r at
t
= 0 and as
t
! 1. . Does s ithavethe correct
magnitude?Andwhatisitsderivative?
0.7ComplexAlgebra
Thissectionappearsinchapters3,4,6,andimplicitlymanyotherplaces.
Thereare some standard manipulationswith complexarithmetic thattake some practice. . Even n the
basic +, ,,andarenotexactly y whatyoulearnedinthird grade,so I’llstartwiththose. . The
standardcommutative,associative,anddistributivelawsapplytotherstthree,so
(7+2
i
)(6+3
i
)=(6+3
i
)(7+2
i
)=36+33
i
(1+2
i
)
(3+4
i
)(5+6
i
)
=
(1+2
i
)(3+4
i
)
(5+6
i
)=(1+2
i
)( 9+38
i
)= 85+20
i
(1+2
i
)
(3+4
i
)+(5+6
i
)
=(1+2
i
)(3+4
i
)+(1+2
i
)(5+6
i
)=
(1+2
i
)(8+10
i
)= 12+26
i
As fordivision,itisnomorecommutativeherethanitis forrealnumbers,butasimpletrick
allowsyoutosimplifysomeexpressions. Thecomplexconjugateofanumberisthenumberfoundby
changingthesignoftheimaginarypart.
z
=5+7
i
=)
z
=5 7
i
The
notation is acommononeforthisoperation,though
z
is anothernotationthatmany prefer.
Whatistheproductofanumberanditscomplexconjugate?
z
=5+7
i; z
=5 7
i
=)
z
z
=(5 7
i
)(5+7
i
)=25+49+35
i
35
i
=74
z
z
is always realand positive: : (
a
+
ib
)(
a
ib
) =
a
2
+
b
2
is the squareofthe magnitude ofthe
complexnumber,thesquareof
p
a
2
+
b
2
.Howdoyouusethistomanipulatedivision?Rationalizethe
denominatorofaquotient.
1+2
i
3+4
i
=
(1+2
i
)(3 4
i
)
(3+4
i
)(3 4
i
)
=
11+2
i
25
(0
:
45)
Multiplyinganumberbyitscomplexconjugateresultsinareal,soyoucanmultiplythenumeratorand
denominatorofaquotientbythe complex conjugate ofthedenominatorin orderto bringtheresult
Example
Afewcasesofsuchmanipulation,simplifyingcomplexexpressions:
3 4
i
i
=
(3 4
i
)(2+
i
)
(2
i
)(2+
i
)
=
10 5
i
5
=2
i:
(3
i
+1)
2
1
i
+
3
i
2+
i
=( 8+6
i
)
(2+
i
)+3
i
(2
i
)
(2
i
)(2+
i
)
=( 8+6
i
)
5+7
i
5
=
2 26
i
5
:
i
3
+
i
10
+
i
i
2
+
i
137
+1
=
i
)+( 1)+
i
( 1)+(
i
)+(1)
=
1
i
=
i:
Whatisthegeometricinterpretationof
i
?Itisafactoritrotatesyouby90
.
z
=1+3
i
iz
i
2
z
i
3
z
iz
=
i
(1+3
i
)= 3+
i
i
2
z
=
i
( 3+
i
)= 1 3
i
i
3
z
=
i
( 1 3
i
)=3
i
i
4
z
=
z
VB.NET PDF Convert to HTML SDK: Convert PDF to html files in vb.
Embed converted html files in html page or iframe. Export PDF form data to html form in .NET WinForms and ASP.NET. Turn PDF images to HTML images in VB.NET.
pdf merge documents; reader combine pdf pages
C# HTML5 PDF Viewer SDK to view PDF document online in C#.NET
PDF; Merge PDF Files; Split PDF Document; Remove Password from PDF; Change PDF Permission Settings. FREE TRIAL: HOW TO: XDoc.HTML5 Viewer for C#▶: C# ASP.NET:
break a pdf into multiple files; add multiple pdf files into one online
MathematicalPrependix
20
Whatis
i
n
? Eachmultiplicationby
i
rotatesyouby90
inthecomplex plane,so
i
4
=1,and
i
217
=
i
4
.
54+1
=
i
.
Variousrootsof1orof 1orof
i
appearcommonly,andyouneedtheexponentialrepresentation,
Euler’sformula,tondthem.Thisis
x
+
iy
=
r
cos
+
ir
sin
=
re
i
x
y
(0
:
46)
Youcanderivethisequationfromtheseries(0.1). Put
i
intotheseriesfortheexponentialandcollect
therealandimaginarypieces,doneinsection3.2. Theresultis
e
i
=cos
+
i
sin
.
Specialcasesofthisequationsay
e
2
i
=1
;
e
i
= 1
;
e
i=
2
=
i;
e
2
ni
=1
Therearethreecuberootsofone,andallthatyouneedtondthemistheprecedingline.
1
1
=
3
=
e
2
ni
1
=
3
Take
n
tobeasuccessionofintegers
n
=0  ! ! 1
1
=
3
=1
n
=1  !
e
2
i
1
=
3
=
e
2
i=
3
=cos2
=
3+
i
sin2
=
3=( 1+
i
p
3)
=
2
n
=2  !
e
4
i
1
=
3
=
e
4
i=
3
=cos4
=
3+
i
sin4
=
3=( 1
i
p
3)
=
2
Ifyoukeep goingto
n
=3
;
4
;
etc.orusenegativeintegers,you simply repeatthese threevalues. . A
pictureoftherootsshows themequallyspaced aroundtheunitcircle,exactly asdictatedbyEuler’s
equation,andthesamesortofpictureappearsforhigherrootstoo.
e
2
i=
3
e
2
i=
8
e
10
i=
8
The polarform of f complex x numbers uses the exponential l representation, , and here are some
examplesthatusethismanipulation.
p
i
=
e
i=
2
1
=
2
=
e
i=
4
=
1+
i
p
2
:
i
1+
i
3
=
p
2
e
i=
4
p
2
e
i=
4
!
3
=
e
i=
2
3
=
e
3
i=
2
=
i:
2
i
1+
i
p
3
25
=
2
e
i=
2
2
1
2
+
i
1
2
p
3
!
25
=
2
e
i=
2
2
ei=
3
!
25
=
e
i=
6
25
=
e
i
(4+1
=
2)
=
i
VB.NET PDF File Split Library: Split, seperate PDF into multiple
Split PDF file into two or multiple files in ASP.NET webpage online. Support to break a large PDF file into smaller files in .NET WinForms.
c# pdf merge; how to combine pdf files
C# HTML5 Viewer: Load, View, Convert, Annotate and Edit PDF
HTML5 Viewer for C# .NET. Related Resources. To view, convert, edit, process, protect, sign PDF files, please refer to XDoc.PDF SDK for .NET overview.
MathematicalPrependix
21
Anotherapplication of Euler’s formula a is s to ordinary trigonometry. . What t happens when you
multiplytwocomplexnumbersexpressedinpolarform?
z
1
z
2
=
r
1
e
i
1
r
2
e
i
2
=
r
1
r
2
e
i
(
1
+
2
)
(0
:
47)
Fromthisyoucanimmediatelydeducesomeofthecommontrigonometricidentities.UseEuler’s
formulaintheprecedingequationandwriteoutthetwosides.
r
1
(cos
1
+
i
sin
1
)
r
2
(cos
2
+
i
sin
2
)=
r
1
r
2
cos(
1
+
2
)+
i
sin(
1
+
2
)
Thefactors
r
1
and
r
2
cancel. Nowmultiplythetwobinomialsontheleftandmatchtherealandthe
imaginarypartstothecorrespondingtermsontheright. Theresultisthepairofequations
cos(
1
+
2
)=cos
1
cos
2
sin
1
sin
2
sin(
1
+
2
)=cos
1
sin
2
+sin
1
cos
2
(0
:
48)
and you have amuch simplerthanusualderivation of these common identities. . Youcandosimilar
manipulationsforothertrigonometricidentities,andinsomecasesyouwillencounterrelationsforwhich
there’sreallynootherwaytogettheresult. Thatiswhyyouwillndthatinphysicsapplicationswhere
you mightuse sines orcosines (oscillations,waves)noone uses anythingbut complexexponentials.
Getusedtoit.
Theimportantapplicationsofcomplexnumbersinthistextappearwhenyouwanttodierentiate
complexfunctions,especiallytheexponential.
d
dx
e
ix
=
ie
ix
=
d
dx
cos
x
+
i
sin
x
= sin
x
+
i
cos
x
andyoucaneasilyseethatthesecondandthefourthformsagree.Doanotherderivativeandyouget
d
2
dx
2
e
ix
=
i
2
e
ix
e
ix
sothisfunction
e
ix
satisestheharmonicoscillatorequation,thesubjectofchapterthree.
Therearesomepracticeexercises oncomplexalgebraattheendofthischapter,butformore
examplesseechapterthreeofMathematicalTools,mentionedinthebibliographyonpageiii.
0.8Separationofvariables
Thissectionappearsinchapters2,3,4,andinanotherversion,inchapter7.
The subject of dierential equations is large enough that you can make aprofession of it and still
not exhaust the e subject, but t in this text, when you solve dierential equations, there are justtwo
methodsthatshowupwithanyregularity.\Separationofvariables"isone.\Linearconstantcoecient
equations"is the other(nextsection). . Afterthatthere e are afewequations such asEq.(6.10) that
~
F
=
m~a
isthissemester’sdierentialequation. Thersttoolinyourkitisseparationofvariables,and
itiseasiesttounderstandifyoustartwithanexampleortwo. Let
c
beaconstant.
dx
dt
=
c
2
+
x
2
!
dx
c
2+
x
2
=
dt
!
Z
dx
c
2+
x
2
=
Z
dt
VB.NET Create PDF from Word Library to convert docx, doc to PDF in
PDF; Merge PDF Files; Split PDF Document; Remove Password from PDF; Change PDF Permission Settings. FREE TRIAL: HOW TO: XDoc.HTML5 Viewer for C#▶: C# ASP.NET:
c# merge pdf; combine pdf online
C# PDF Convert to SVG SDK: Convert PDF to SVG files in C#.net, ASP
Instantly convert all PDF document pages to SVG image files in C#.NET class application. Perform high-fidelity PDF to SVG conversion in both ASP.NET web and
split pdf into multiple files; add pdf pages together
MathematicalPrependix
22
Therstoftheseisthedierentialequationtobesolved. Itisarstorderequation,meaningthatit
isarelationbetweenthefunction
x
andonlytherstderivative
dx=dt
. Therearetwovariableshere,
theindependentvariable
t
,andthedependentvariable
x
.Youcan’tsimplyintegratethiswithrespect
to
t
becausetherightsideisafunctionof
x
,andthatisan(unknown)functionofthevariable
t
. To
separatevariablesputallthe
x
’sononesideoftheequationandallthe
t
’sontheother. Thesecond
equationdoesthis.Itisnowsetupforintegration.
Nowdotheintegral,atrigsubstitutionworks:
x
=
c
tan
.
dx
=
c
sec
2
d
!
Z
c
sec
2
d
c
2
+
c
2
tan
2
=
Z
1
c
d
=
1
c
=
1
c
tan
1
x
c
=
t
+
D
andthesolutionis
x
(
t
)=
c
tan
c
(
t
+
D
). Withaninitialconditionssuchas
x
(0)=
x
0
youhave
x
(0)=
x
0
=
c
tan(
cD
)  !
D
=
1
c
tan
1
x
0
=c
!
x
(
t
)=
c
tan
ct
+tan
1
(
x
0
=c
)
Checkthelastexpression:
x
(0)=
c
tan
tan
1
(
x
0
=c
)
=
x
0
mistake. Astimeincreases,
x
(
t
)increases,so(
c
2
+
x
2
)increases,so
dx=dt
increases,sotheslopeof
thecurve
x
versus
t
getsbiggerandbigger|that’showthetangentof
t
behaves.
Thismethodlookslikesuchaspecialone;thecombinationoffactorsthatwillletyoudothis
seemssoimprobablethatitcan’tworkveryoften.True. But,ithappensinenoughimportantspecial
1:
dN=dt
N;
2:
d
2
x
dt
2
!
2
x;
3:
t
dx
dt
=
x
+
;
4:
t
dx
dt
+
tx
=
x
(0
:
49)
Equations1,3,and4areseparable,butnot2,thoughinchapters2and3youwillseesomemanipu-
lationsthatwilldigaseparableequationoutofeventhatone.
Wait,couldn’tyoumanipulatethe second ofthese tobe
d2x
x
!
2
dt
2
and integrate? ? No!
There’snosuchmathematicsasthis,sodon’ttry.
Forotherexamplesofthismethod,lookatEqs.(2.13),(2.17),(2.23),(3.55).
0.9Constant CoecientODEs
Thissortofdierentialequationshowsupofteninthiscourse,startinginchaptertwo,andcommonly
afterthat. Itlookslike
3
d
2
x
dt
2
4
dx
dt
+7
x
=0
or
d
3
x
dt
3
+
d
2
x
dt
2
+
dx
dt
+
x
=
A
cos
!t
Thedependentvariablecanhaveanynumberofderivatives,butitappearsjusttotherstpower,no
x
2
or
x
dx
dt
orsin(
kx
).Thatmakestheseequationlinear.Thatthecoecientofthe
x
’sareconstantsmake
theseconstantcoecientlinearequations.Thattherstonehasonlytermsin
x
oritsderivativesmakes
ithomogeneous andthatthesecondonehasanextratermwithno
x
atallmakesitinhomogeneous.
Theprecisedenitionofhomogeneousisthatifyoumultiplythevariable
x
byaconstant
,thenthe
wholeexpressionismultipliedbysomepowerof
,i.e.
n
. Here
n
=1.
Therstcase,thelinearconstantcoecienthomogeneousone,hasasimplesolution. Allyou
havetonoticeisthatthederivativeofanexponentialisanexponential,andtryasolution
x
(
t
)=
Ae
t
.
3
d
2
x
dt
2
4
dx
dt
+7
x
=0  !3
A
2
e
t
4
Ae
t
+7
Ae
t
=0
Ae
t
[3
2
4
+7]=0
MathematicalPrependix
23
Sinceneither
A
northeexponentialarezero,thatleaves3
2
4
+7=0,apolynomialequationwith
tworoots,givingtwosolutionstotheequation. Becauseyouaretryingtoundotwoderivativestoget
x
youwillsomehowgettwoarbitraryconstants. Thekeypropertyoflinearhomogeneousequationsis
thatthesumoftwosolutionsisasolution,sothefullsolutiontothisequationis
A
1
e
1
t
+
A
2
e
2
t
;
where
1
;
2
=
2
i
p
17

3
Howdoyouhandletheinhomogeneouscaseexampleabove? Anexponentialwon’tworkhere.
Youwillnotget
A
cos
!t
outofitinordertomatchtheright-handside. Thesumoftwosolutionsis
nolongerasolution. But,thereisonesimplication: Ifyou(temporarily)throwawaytheinhomoge-
neousterm(
A
cos
!t
),youcansolvetheremaininghomogeneouspartoftheequationwithasimple
exponential. O.k.yougetacubicequation,butit’sonly y apolynomialequationsotherearewaysto
handleit. Thispartialsolutionwillhavethreearbitraryconstants. . Nowifsomehowyoucanndany
d
3
x
hom
dt
3
+
d
2
x
hom
dt
2
+
dx
hom
dt
+
x
hom
=0
;
withthreearbitraryconstants
d
3
x
inh
dt
3
+
d
2
x
inh
dt
2
+
dx
inh
dt
+
x
inh
=
A
cos
!t;
withnone
Then
x
(
t
)=
x
inh
(
t
)+
x
hom
(
t
). Howdoyouverifythis? Plugintotheoriginalequationandwatchit
work.
Fortheproblemsyouencounterin this book,ndingthespecial, inhomogeneoussolution will
notbedicult,andlateryouwillseesomegeneralmethodsforndingsuchsolutionsevenwhenitis
dicult.
0.10Matrices
Thissectionappearsinchapters4,8,10.
Justasyouhavecomponentsofvectorswithrespecttoabasisyou will havecomponents ofcertain
typesofvector-valued functions. . You u have(
v
x
;v
y
;v
z
)or(
v
r
;v
;v
) with threecomponents fora
vector. Animportantsortoffunction(alinear,vector-valuedfunctionofavectorvariable)appearsin
describingtheangularmomentumofarigidbody. Italsoappearsindescribingdielectricpropertiesof
acrystal. Andindescribingrotationsofvectors. And
:::
. Anyway,ittoohascomponents(ninethis
time)andtheseformmatrices.Thedevelopmentoftheseideas,showingthereasonfortheodd-looking
rulesthatmatricesobey,canwaituntilthey’reneededinsection 8.2. Forthemomentthiswillbea
summaryofsomeruleswithoutanydiscussionofthereasonsthattheyarethewaytheyare.
Forthemomentthenamatrixisasquarearrayofnumbers. Theycanberectangulartoo,but
a b
c d
+
e f
g h
=
a
+
e b
+
f
c
+
g d
+
h
(0
:
50)
andofcoursesubtractionjustchangesallthe+signsto . . Whatmatrixplaystheroleofzerosothat
Isaidthatthereareninecomponentsandtheseobjectshaveonlyfour. Ifyouknoweverything
step from one dimensionto two is thebigone. . Afterthatthesteptothree e dimensions oreven
N
dimensionsisrelatively small. . Besides,it’seasiertowritetheseandtheytakeonly y about 8/
27
ofthe
arithmetictomanipulatethem.
MathematicalPrependix
24
Multiplicationobeys
a b
c d

e f
g h
=
ae
+
bg af
+
bh
ce
+
dg cf
+
dh
(0
:
51)
Yourunacrossthe rows of therst matrix and down thecolumns ofthe second matrix in orderto
formultiplication. Whatisit? Whatentriesinthe e rstfactorof(0.51)make theproductequalthe
arrayof
e;f;g;h
,therebyreproducingthesecondfactor? Forthetopleftentryoftheproduct,
ae
+
bg
=
e
forall
e
andforall
g
=)
a
=1
; b
=0
Thismakesthetoprightentryworktoo. Similarlyforthebottomentriesyouneedtohave
c
=0and
d
=1.Thatmakestheidentitymatrix
(
I
)=
1 0
0 1
The orderofmultiplication matters,and multiplication isnotcommutative. . You u can howevercheck
thespecialcaseshowingthatthisidentitymatrixfunctionsjustaswellastherighthandfactorasit
doesontheleft.
Theinverseofamatrixisthatmatrixsuchthattheproductwiththeoriginalistheidentity.Set
therightsideofEq.(0.51)totheidentitymatrixandsolvethefourequationsinthefourunknowns
a;b;c;d
andnotmine.
e f
g h
1
=
1
eh
fg
h
f
g
e
(0
:
52)
Multiply this by the originalmatrix and verify that you gettheidentity. . Itworksineitherorder,so
checkitbothways.
Thereisno common notationforamatrix as there isforvectors. . Inthe e lattercaseyou see
boldfacetypeoranarroworsometimesasquigglyunderline,butformatricestherearenostandards.
Sometimesaboldfacesansseriffontischosenforthispurpose,anditservesaswellasanythingelse
sothat’swhatIwillusehere.
A=
a b
c d
;
B=
e f
g h
;
then
AB=C=
ae
+
bg af
+
bh
ce
+
dg cf
+
dh
TheinversematrixasinEq.(0.52)obeys
BB
1
=B
1
B=I
ThestatementthatmatrixmultiplicationisnotcommutativeisAB6=BA.Youdohavetheassociative
lawthough: A(BC)=(AB)C.Alsothedistributivelaw: A(B+C)=AB+AC.
Simultaneousequations
Matricesappearinmanyinterestingandelegantcontexts. Theyalsoappearinmundanesettings,but
thesearenolessimportant.Howdoyousolvetwolinearequationsintwounknowns?
ax
+
by
=
p;
cx
+
dy
=
q:
multiplytherstby
d
andthesecondby
b
,thensubtract
dax
+
dby
=
dp;
bcx
+
bdy
=
bq
!
dax
bcx
=
dp
bq
!
x
=
dp
bq
da
bc
multiplytherstby
c
andthesecondby
a
,thensubtract
(0
:
53)
cax
+
cby
=
cp;
acx
+
=
aq
!
cby
=
cp
aq
!
y
=
cp
aq
bc
MathematicalPrependix
25
Thisismatrixinversionindisguise.
a b
c d

x
y
=
p
q
or
Mx=p
MultiplybothsidesofthismatrixequationbytheinverseofMfromEq.(0.52).
M
1
Mx=x=M
1
p=
1
bc
d
b
c a

p
q
=
x
y
Thisisexactlythesameastheprecedingexplicitsolutionfor
x
and
y
.Infact,thatexplicitsolutionis
howtheinversematrixisderived,sothiscomparisonisreallycircular.
Doesthisalwayswork?No.Youcan’tdividebyzero,andinEq.(0.53)Iignoredthatimportant
point.
dax
bcx
=
dp
bq
! (
da
bc
)
x
=
dp
bq
also
(
da
bc
)
y
=
aq
cp
(0
:
54)
Whatif
da
bc
=0?thentherightsidesoftheequationsmustbezero,otherwisethereisnosolution.
Youcanhaveasolutionif
p
and
q
arebothzeroorifaparticularcombinationof
p
,
q
,andtheelements
ofthematrixconspiretomaketherightsidezero.
bc
=determinantofthematrix
Thedeterminantdeterminesthenatureofthesolutions(deterministicallyofcourse).
1. Ifthedeterminantisnon-zerothenthesolutionexistsandisunique.
2. Ifthedeterminantiszeroand
p
or
q
isnon-zerothereisnosolutionunlessspecial
circumstancesoccur;thenthereareaninnitenumberofsolutions.
3. Ifthedeterminantiszeroandboth
p
and
q
arezerothereareaninnitenumberof
solutions.
Case#1isroutine. Yousolvesimultaneousequationsandyouexpecttondasolution. . The
secondcase isexceptional,anditcanbeusedtodeterminepropertiesoftheright-handside. . Itwill
showupindisguiseinsections7.10and10.7. Thethirdcaseisthemostcommonforthepurposesof
thisbook.Itmeansthatthetwoequationsyouaresolvingare
ax
+
by
=0
;
cx
+
dy
=0
but
bc
=0
(0
:
55)
Ifforexample
d
and
b
are6=0,multiplytherstoftheseby
d
:
dax
+
dby
=0.Now
=
bc
,sothis
equationisthesameas
bcx
+
bdy
=0or
cx
+
dy
=0. Thatmeansyoureallyhaveoneequationfor
thetwounknowns,nottwo. Thatinturnmeansthatyouhaveaninnitenumberofsolutions
x
and
y
x
and
y
byany constantandyouhave
another.Youcanunderstandthismostsimplybyagraphicalinterpretation.
ax
+
by
=0 isastraightlinethroughtheorigin.
andthisgraphrepresentsaninnitenumberofpossiblesolutions.
And howdoyou write \boldfacesans serif" on paper? ? Perhapsby y using\Blackboard Bold"
style:
ABCDEFGHIJKLMNOPQRSTUVWXYZ
.Thisisawaytofakeboldfacetypeinwriting.
IndexNotation
A
ij
isthesetofelementsofthematrixA. Theindices
i
and
j
runfromonetowhateverthesizeof
thematrixis(twointheseexamples).Therstindexspeciestherowandthesecondthecolumn.
A
row
;
column
=
A
ij
!
A
11
A
12
A
21
A
22
rstrow
secondrow