LinearAlgebraSummaryofResults Compareresultswithsyllabusoutline(numbersrefertosyllabusTopic1)1.Matricesandoperationswiththem,includinguseofGDC.
Payparticularattentionandthinkthroughthedifferentwaysofrepresentingmatrixmultiplication(seeHowardAnton,p.2934).Theywillbehelpfulfortheoreticalinterpretationofmatrices,systemsoflinearequationsandvectorspaces.
Bycolumns,seeformula(6)onp.31.Thisisthemosthelpfulandimportantone
inthetopic!Itwillbeused(amongotheruses)alsointheEigenvaluespart.
MatrixproductAXasalinearcombinationofthecolumnsofmatrixA,(seebelow(10)andExample8fromp.32.)
Matricesasrepresentationoflineartransformations(seeparagraph1.8,p.76)
2. Matricesinsystemsoflinearequations Matrixofcoefficients,augmentedmatrixofthesystem. MethodofGaussianelimination,rowechelonform,reducedrowechelonform
useofGDCtofindthem.Useofelementarymatrices,includingwithGDC. Numberofsolutionsfromrowechelonformsolutionofsystemswithinfinitely
manysolutions. Homogeneoussystem andthesetofitssolutions(thenullspaceorthexA = 0
kernelspaceofmatrixA).LetAbeamatrixofdimension .m n isasubspaceinthedomain(A) ull(A) er(A) x |Ax }N = N = K = { Rn = 0
ofthelineartransformationrepresentedbymatrixA. representstheranknullitytheoremimN(A) ullity(A) ank(A)d = n = n r
(box1.6ofsyllabus).Itisalsoequaltothenumberoffreevariablesinthesolutionofthehomogeneoussystem.
ThenullspaceN(A)andtherowspace are(A) pan(a , , .., )R = S 1 a2 . am orthogonalcomplements.
Solutionofthesystem .xA = b Thesystemisconsistent(hassolution)if,andonlyif,
(A) ange(A) pan(a , , .., ) x|x a a .. a }b C = R = S 1 a2 . an = { = m1 1 + m2 2 + . + mn n i.e.bisalinearcombinationofthecolumnvectorsofA.
Thesolutionofthesystem canberepresentedasxA = b (box1.7syllabus).particularsolution" (A)x = " + N
Example1:3. Someimportantspecialcases ConsiderasquaresystemoflinearequationsAX=B,whereAisa matrix.Thenthen n
followingstatementsareequivalent(partofthemTheorem1.6.4,p.63)a) Aisinvertible(hasinverse,isnotsingular,)b) etA =d / 0 c) ThehomogeneoussystemAx=0hasonlytrivialsolution.d) ThereducedrowechelonformofAistheidentitymatrixIn.e) Aisexpressibleasaproductofelementarymatrices.f) ThesystemAx=bisconsistent(hassolution)forevery matrixb(righthandn 1
sideofthesystem).g) ThesystemAx=bhasexactlyonesolutionforevery matrixb(righthandn 1
sideofthesystem).
h) ThelineartransformationTdefinedbythematrixAisbijective(onetoone).i) TherangeofTisRn.Thekernel(orthenullspace,thesetofallvectorsmapped
tothezerovector)consistsonlyofthe0vector,i.e.is{0}.j) Thesetofthecolumnvectors(andtherowvectors)ofthematrixAislinearly
independent.k) Thesetofthecolumnvectors(andtherowvectors)ofthematrixAisabasisin
Rn.
ConsidernowasquarematrixAwhichisnotinvertible(discussthepossibilitiesandformulateasimilarsequenceofequivalentstatementsasabove)
a) Aisnotinvertible(doesnothaveinverse,issingular,)b) etAd = 0 c) ThehomogeneoussystemAx=0hasnontrivialsolutions,i.e.hasinfinitelymany
solutions.d) ThesystemAx=bisconsistent(hassolution)onlywhentherighthandsidevector(n 1
matrixb.)isfromtherangeofthetransformationdefinedbyA,i.e.bisfromC(A)thecolumnspaceofthematrixA.
e) ThereducedrowechelonformofAwillhaveatleastonerowofzeroesi.eitwillhave
theform (withpossiblyrearrangementofthecolumns).ThentherankofthematrixAisr
anditsreducedrowechelonformfromtheGDCis .Theinformationthatwecanreadfromtherrefis:
Column3ofthematrixisalinearcombinationofcolumns1and2,moreprecisely(checkit!) .Inotherwordscolumns1and2forma 6c 8cc3 = . 1 . 2 basisforthecolumnspaceC(A)andthethirdcolumnhascoordinates 6, .8)( . T
inthisbases. Thesystemofthelinearequationsisconsistent(hassolution),asitsrighthand
sidethevector isinthecolumnspaceC(A).Indeedtherreftells , 4, 2)b = ( 1 T usthat(checkit!) ..4c .2cb = 1 1 + 1 2
Thegeneralsolutionofthesystemis .Itisx, , ) 1.4, .2, ) (.6, .8, 1)( y z T = ( 1 0 T + t T thesumofthesolutionofthehomogeneoussystem(orthenullspaceofthematrixA) ,andtheparticularsolution(any(A) t(.6, .8, 1) , t }N = { T R particularsolution),inourexample ofthesystem .Thisis1.4, 1.2, 0)( T xA = b ageneralresult(theorem)trueforallsystemsoflinearequations.
4. Matricesasrepresentationsoflineartransformations
Definitionofvectorspace.SpecificexamplesRn,Cn,P2,M22,etc.
Themostimportantis .Rn Wecanvisualize,orinterpretgeometricallyresultsin orthecoordinateplaneR2
.the3 imensionalspaceR3 d Whatarethepossiblesubspacesin ?R2 Whatarethepossiblesubspacesin ?R3
Lineardependenceandbasis.(Seeparagraph4.3and4.4inbook).
Example3:
Tocheckthatasystemofvectorsisindependent1.Determinewhetherthevectorsarelinearlydependentin iftheyarenotfindanexpressionR3 forthelinearcombinationequaltozero.
a) , 0, 4) , (5, , 2) , (1, , )( 3 T 1 T 1 3 T
MakethematrixAconsistingofthegivenvectorsasrespectivecolumns,i.e.
Thequestionaboutthelineardependenceofasystemofvectorsisequivalentto:
findingoutwhetherthehomogeneoussystem hasnontrivialsolutions.AnonxA = 0 trivialsolutionwillprovidethelinearcombinationofvectorsequaltozeroandwillmakethevectorslinearlynonindependent.
Anotherpossibleformulationis ?Ifyes,thenthesystemofvectorsisetA =d / 0 linearlyindependent(andformsbasesinthecorresponding(sub)space.
InthecaseofarectangularmatrixthequestioncanberewordedaswhethertherankofthematrixAisfull(equaltotheminimumdimensionofthematrix).
Inalldifferentrewordingsofthequestion,asusual,ourmaintoolistherrefofA.Inthiscase
,(i.e. ,or ),thereforethesystemofvectorsisindependentandref(A)r = I etA =d / 0 ank(A)r = 3 formsabasisin .R3 2. Considernowthesystemofvectorsfrom givenasR4
.Thecorrespondingmatrixof(2, , 3, 0) , , 1, 0, ) , (7, , 6, 6) }{ 1 T ( 1 2 T 5 T
columnvectorsis anditsreducedrowechelonformis
.Fromtherrefwededucethat:
TherankofAis2,thisissmallerthan3(thenumberofvectorsinthesystem),thereforethesystemofvectorsislinearlydependent.Indeedtherreftellsusthat .c cc3 = 2 1 3 2
Thusthereareonly2independentvectorsinthesystem andtheyformabasisc , c }{ 1 2 fora2dimensionalsubspace,Sin (ortheyspana2dimensionalsubspaceS)namelyR4
m(2, , 3, 0) ( , 1, 0, ) m, }S = { 1 T + n 1 2 T n R
b) Changeofbases(notincludedinsyllabusexplicitly,couldbeasked?)Example4.Giventhevector inthestandardbasisof ,finditscoordinates6, , 3)v = ( 1 T {e , , }R3 1 e2 e3 inthenewbases ,whereb , , }{ 1 b2 b3
eb1 = e1 + e2 + 2 3 eb2 = 2 1 e2
.b3 = e1 + e2 + e3
Thequestioncanberewordedasoneaboutasystemoflinearequations,(asalmostalways!)i.e.wearesearchingforscalars suchthat ,where, ,m n t b b b (m, , )v = m 1 + n 2 + t 3 = B n t
T
thematrix iscalledthetransitionmatrix frombasis totheB = T be b , , }{ 1 b2 b3 standardbasis .Inthetransitionmatrixeachcolumnrepresentsthecoordinatesof{e , , } 1 e2 e3 correspondingvectoroftheoldbasisthroughthenewone.Itiseasytoseethatthesolutionofthesystemis ,sothecoordinatesofvinthenewbasisarem, n, t) v 1, , )( T = B1 = ( 3 1 T 1, , )( 3 1 T
(Checkit!).Notethatthematrix .B1 = T eb
Lineartransformationsandmatrices Definitionandpropertiesoflineartransformation(operator)
Tofindthematrixrepresentingalineartransformation
Example:Considerthetransformation Reflectionintheline (thisistheMy=x y = x geometricdescriptionofthetransformation)
TofindthematrixArepresentingthistransformation: Findtheimagesofthecoordinatevectorsunderthetransformation
and .ThematrixofthetransformationAisformedbytheimagevectorsascorrespondingcolumnvectors,i.e. .e |e )A = ( 1 2
So andthetransformationisrepresentedbythematrixasfollows
(thisisthematrixrepresentationofthetransformation).
Whatcanyoureadfromthematrixofatransformation?
Theimagesofthecoordinatevectors. Thescalefactorbywhichtheareaoftheoriginalistransformed.
rea etA reaA image = d A original Compositionoftransformations(transformationaftertransformation)
Likeinthecaseoffunctionsweconsidertransformationsthataretheresultofothertransformationsperformedoneaftertheother.LetthetransformationTArepresentedbythematrixA,beperformedafterthetransformationTB representedbythematrixB.Thenthecompositetransformation followedbyTB
or after isdenotedby andisrepresentedbythematrixTA TA TB TA TB .Notetheorder!(fromrighttoleft)BA
Famoustransformationsin wearesupposedtorecognizeandknowtheR2
matricesofthefamoustransformationsintheplane
Geometry Determinant Matrix
Rotationbyangleclockwise
etRd = 1
Reflectioninlinean.xy = t
etM d = 1
Projectiononlinean.xy = t
etPd = 0
Projectiononlinexy = m
etPd m = 0
Enlargementw.r.t.O(0,0),scalefactork
etEd k = k2
KernelandRangeofLinearTransformations
Example:Findthekernelofthetransformation projectionontotheline .T = Pm xy = m Geometricallythekernelconsistsofallvectorsfrom whoseimageisthezerovector,ortheR2 origin .Thusthekernelconsistsofallvectorsonthelinepassingthroughtheoriginand(0, 0)O perpendiculartothelineontowhichweproject.
er(P ) (x, ) |y /m)x}K m = { y T = ( 1 Therangeofthetransformationcoincideswiththecolumnspaceofthematrixandconsistsofthevectorsonthelineontowhichweproject.
ange(P ) (P ) (x, ) |y x}R m = C m = { y T = m
EigenvaluesandEigenvectorsEigenvalueisaGermanword,notevennearlypronouncedasifitwasanEnglishone.Itstandsforownvalue.Eigenvectorsformamatrix,whichdiagonalizesthecoefficientmatrix.Theeigenvalueproblem:Foranysquare matrixA,findthevaluesofthescalar suchthatthesystemofequationsn n
x xA = hasnonzerosolutions.
Thesevaluesof arecalledeigenvaluesofthematrixA. Thesolutionsofthesystemofequations(infinitelymany!WHY?)corresponding
tothevaluesof ,arecalledeigenvectorsofthematrixA. Thesystem canberewrittenas .Thereforethesystemisax xA = x x A I)xA = ( = 0 homogeneousoneandwillhavenonzerosolutionsif,andonlyif
et(A I)d = 0
ThisiscalledthecharacteristicequationofmatrixA.Itisapolynomialequationofdegreen.Wearerequiredtosolveonlycaseswhen withreal,distinctroots.n = 2
Example
Giventhematrix .Thecharacteristicpolynomial is()P
Thecharacteristicequationis anditssolutions,theeigenvaluesofthematrix,are()P = 0
.or 1 = 5 2 = 2 Propertiesofcharacteristicequationandeigenvaluesproveandthinkthrough!
.Inotherwords,thesumofeigenvaluesequalsthesumofa race(A)1 + 2 = 11 + a22 = T thediagonalelementsofthematrix(thetraceofthematrix).Trueforalln,weneedtoknowandunderstanditforn=2.
et(A)1 2 = d ThematrixAsatisfiesitscharacteristicequation,i.e. (A) A 0IP = A2 3 1 = 0
Thecorrespondingsolutionsare: and .Sothe/4x, wheny = 3 = 5 , when y = x = 2 eigenvectorsare,e.g. and .4, ) when( 3 T = 5 1, ) when ( 1 T = 2 ThematrixPcontainingtheeigenvectorswilldiagonalizethematrixA.Finishtheexample!..Somesynonymsglossary
Topic(context) Systemsoflinearequations
Matrices Lineartransformations
Solution(s)of xA = 0 Nullspace (A)N KernelofthelineartransformationrepresentedbythematrixA,Ker(A)
Setofvectorsbforwhich hasxA = b solution(isconsistent)
ColumnspaceofmatrixA, (A)C
RangeofthelineartransformationrepresentedbymatrixA
Linearindependenceofasystemofvectors
SystemAx=0hasonlytrivialsolution.
MatrixAofthevectorsascolumnshasfullrank.
