course overview — 18.06: (applied) linear algebraweb.mit.edu/18.06/www/spring17/overview.pdf ·...
TRANSCRIPT
—courseoverview—18.06:(Applied)LinearAlgebra
Prof.StevenG.Johnson,MITAppliedMathSpring2017
hFp://web.mit.edu/18.06
Textbook:Strang,Introduc0ontoLinearAlgebra,5thediNon +supplementarynotes
Helpwanted:
arrive10minutesearlyandgetpaid$10toerasetheboards
(You,too,canbeablackboard
monitor,the“eraserofthewriNngs.”…shout-outtoTerryPratcheFfans…)
AdministraNveDetails
LecturesMWF11,10-250TuesdayrecitaNons—useStellartoswitchsecNonsWeeklypsets,dueWednesday11aminyourrecitaNonbox.
—noextensionsormakeup,butlowestpsetscorewillbedropped—pset1ispostedonStellar
Grading:homework15%,3exams45%(3/3,4/10,&5/5in50-340),finalexam40%CollaboraNonpolicy:talktoanyoneyouwant,readanythingyouwant,but:
•MakeaneffortonaproblembeforecollaboraNng.•WriteupyoursoluNonsindependently(from“blanksheetofpaper”).•Listyourcollaboratorsandexternalsources(notcoursematerials).
SyllabusandCalendar• SignificantoverlapwithStrang’sOCWvideolectures:theseareauseful
supplementbutnotareplacementforaFendinglecture.• Exam1:Friday3/3.EliminaNon,LUfactorizaNon,nullspacesandothersubspaces,
basesanddimensions,vectorspaces.(Book:1–3.5.)• Exam2:Monday4/10.Orthogonality,projecNons,least-squares,QR,Gram-
Schmidt,orthogonalfuncNons,complexity.(Book:1–4,10.5,11.1).• Exam3:Friday5/5.Eigenvectors,determinants,similarmaNces,Markovmatrices,
ODEs,symmetricmatrices,definitematrices,matricesfromgraphsandengineering.(Book:1–7,10.1–3.)
• Othertopics:defecNvematrices,SVDandprincipal-componentsanalysis,sparsematricesanditeraNvemethods,complexmatrices,symmetriclinearoperatorsonfuncNons.
• Finalexam:alloftheabove.
Whatis18.06about?2x1 + 4x2 − 2x3 = 24x1 + 9x2 −3x3 = 8−2x1 −3x2 + 7x3 =10
Highschool:3“linear”equaNons(only±and×constants)in3unknowns
Method:eliminateunknownsoneataNme.
Equivalentmatrixproblem
Ax=b
Axisa“linearoperaNon:”A(x+y)=Ax+AyA(3x)=3Ax
2 4 −24 9 −3−2 −3 7
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
x1x2x3
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟=
2810
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
A x b
take“dotproducts”ofrows×columns
matrixofcoefficients
vectorofunknowns
vectorofright-handsides
Whatis18.06about?
LinearsystemofequaNons,inmatrixform
Ax=b
2 4 −24 9 −3−2 −3 7
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
x1x2x3
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟=
2810
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
A x b
Willwelearnfastermethodstosolvethis?No.(ExceptifAisspecial.)Thestandard“GaussianeliminaNon”(and“LUfactorizaNon”)matrixmethodsarejustaslightlymoreorganizedversionofthehigh-schoolalgebraeliminaNontechnique.WillwegetbeFeratdoingthesecalculaNonsbyhand?Maybe,butwhocares?Nowadays,allimportantmatrixcalculaNonsaredonebycomputers.Willwelearnmoreaboutthecomputeralgorithms?AliFle.Butmostlythetechniquesfor“serious”numericallinear-algebraaretopicsforanothercourse(e.g.18.335).
Howdowethinkaboutlinearsystems?(imaginesomeonegivesyoua106×106matrix)
• Alltheformulasfor2×2and3×3matriceswouldfitononepieceofpaper.Theyaren’tthereasonwhylinearalgebraisimportant(asaclassorafieldofstudy).
• Largeproblemsaresolvedbycomputers,butmustbeunderstoodbyhumanbeings.(Andweneedtogivecomputerstherighttasks!)
• Breakupmatricesintosimplerpieces– Factorizematricesintoproductsofsimplermatrices:A=LU(triangular:Gauss),A=QR(orthogonal/
triangular),A=XΛX–1(diagonal:eigenvecs/vals),A=UΣV*(orthogonal/diagonal:SVD)
– Submatrices(matricesofmatrices).
• Breakupvectorsintosimplerpieces:subspacesandbasischoices.
• AlgebraicmanipulaNonstoturnharder/unfamiliarproblems(e.g.minimizaNonor
differenNalequaNons)intosimpler/familiarones:algebraonwholematricesatonce
Don’texpectalotof“turnthecrank”problemsonpsetsorexamsoftheform
“solvethissystemofequaNons.”
…wewillturnitupside-down,giveyoutheanswerandaskthequesNon,askaboutproperNesofthesoluNonfromparNal
informaNon,…thegeneralgoalistorequireyoutounderstandthecrankratherthanjustturnit.
(Examsmightbeabitharderthaninprevious18.06semesters,butgradecutoffswillbe
adjustedaccordingly.)
Wheredobigmatricescomefrom?
Lotsofexamplesinmanyfields,buthereareacouplethatarerelaNvelyeasytounderstand…
Engineering&ScienNficModeling[18.303,18.330,6.336,6.339,…]
hFp://gmsh.info/ UnknownfuncNons(fluidflow,mechanicalstress,electromagneNcfields,…)approximatedbyvaluesonadiscretemesh/gride.g.100x100x100grid
=106unknowns!
hFp://www.electronics-eeNmes.com/ comsol.com
Dataanalysis
Imageprocessing:imagesarejustmatricesofnumbers
(red/green/blueintensity)
[Mathw
orks]
[wikipedia]
Regression:(curvefi�ng)
Google“pagerank”problem(alsoforgenenetworksetc.)Determinethe“mostimportant”webpagesjustfromhowtheylink.matrix=(#webpages)×(#webpages)(entry=1iftheylink,0otherwise)
Notjustmatricesofnumbers• TherearelotsofsurprisingandimportantgeneralizaNonsoftheideasinlinearalgebra.
• Insteadofvectorswithafinitenumberofunknowns,similarideasapplytofuncNonswithaninfinitenumberofunknowns.
• InsteadofmatricesmulNplyingvectors,wecanthinkaboutlinearoperatorsonfuncNons
“A” “x” “b”linear
operator∇2
unknownfuncNonu(x,y,z)
right-handsidef(x,y,z)
∇2u = fPoisson’sequaNon(e.g.18.303)
18.06vs.18.700
“applied”vs.“pure”mathfewproofsvs.formalproofsexpected(deducepaFerns(definiNonstofromexamples,lemmastotheoremsinformalarguments)…traininginproofwriNng)
moreapplicaNonsvs.moretheoremsmoreconcretevs.moreabstractsomecomputersvs.onlypencil-and-paper
18.06vs.18.700
somepuppyvs.nopuppies
“Cookie”9-montholdLabradoodlepuppy(Letmeknowprivatelyifyoudon’twanttobeinroomwithadog,noquesNonsasked.)
Computerso�ware
[image:ViralShah]
Lotsofchoices:
julialang.org
Thissemester:arelaNvelynewlanguagethatscalesbeFertorealproblems.
Noprogrammingrequiredfor18.06,justa“glorifiedcalculator”toturnthecrank.Useitonline:loginatjuliabox.com
see“Julia”linkonStellarOpNonaltutorial:Friday5pm32-123