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IB Further Math - Linear Algebra Summary

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A summary of the results included in the syllabus of IB Further Mathematics, Topic 1, Linear Algebra
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 Linear Algebra - Summary of Results  Compare results with syllabus outline (numbers refer to syllabus - Topic 1) 1. Matrices and operations with them, including use of GDC.  Pay particular attention and think through the different ways of representing matrix multiplication (see Howard Anton, p.29 - 34  ). They will be helpful for theoretical interpretation of matrices, systems of linear equations and vector spaces.  By columns , see formula (6) on p.31. This is the most helpful and important one in the topic! It will be used (among other uses) also in the Eigenvalues  part. Mat rix product  AX  as a linear combination of the columns of matrix  A, (see below (10) and Example 8 from p.32.)
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  • 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.


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