—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:(curvefing)

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.)

Computersoware

[image:ViralShah]

Lotsofchoices:

julialang.org

Thissemester:arelaNvelynewlanguagethatscalesbeFertorealproblems.

Noprogrammingrequiredfor18.06,justa“glorifiedcalculator”toturnthecrank.Useitonline:loginatjuliabox.com

see“Julia”linkonStellarOpNonaltutorial:Friday5pm32-123