— courseoverview—18.06:LinearAlgebra

Prof.StevenG.Johnson,MITAppliedMathFall2017

http://web.mit.edu/18.06

Textbook:Strang,IntroductiontoLinearAlgebra,5th edition+supplementarynotes

Helpwanted:

arrive10minutesearlyandgetpaid$10toerasetheboards

(Youcanforeverafterputblackboardmonitor,onyourresumé

…shout-outtoTerryPratchettfans…)

AdministrativeDetails

LecturesMWF11,54-100Tuesdayrecitations— useStellartoswitchsections

Weeklypsets,dueWednesday11aminyourrecitationbox.— noextensions ormakeup,butlowestpset scorewillbedropped— pset 1tobepostedshortly

Grading:homework15%,3exams45%(9/25,10/30,&11/27in50-340),&finalexam40%

Collaborationpolicy:talktoanyoneyouwant,readanythingyouwant,but:•Makeaneffortonaproblembeforecollaborating.•Writeupyoursolutions independently (from“blanksheetofpaper”).•Listyourcollaboratorsandexternal sources (notcoursematerials).

SyllabusandCalendar• SignificantoverlapwithStrang’s OCWvideo lectures: theseareauseful

supplement butnotareplacement forattending lecture.Likelytopics:• Exam1:Monday9/25.Elimination, LUfactorization,nullspaces andother

subspaces, basesanddimensions, vectorspaces,complexity. (Book:1–3.5,11.1)• Exam2:Monday10/30.Orthogonality,projections, least-squares, QR,Gram–

Schmidt,orthogonal functions (Book:1–4,10.5).• Exam3:Monday11/27. Eigenvectors,determinants, similarmatices,Markov

matrices,ODEs,symmetricmatrices, definitematrices,matrices fromgraphsandengineering. (Book:1–7,10.1–3.)

• Othertopics:defectivematrices, SVDandprincipal-components analysis, sparsematrices anditerativemethods,complexmatrices, symmetric linearoperatorsonfunctions.

• Finalexam:alloftheabove.

Whatis18.06about?2x1 + 4x2 − 2x3 = 24x1 + 9x2 −3x3 = 8−2x1 −3x2 + 7x3 =10

Highschool:3“linear”equations(only± and× constants)in3unknowns

Method:eliminateunknownsoneatatime.

Equivalentmatrix problem

Ax=b

Axisa“linearoperation:”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?

Linearsystemofequations,in matrix form

Ax=b

2 4 −24 9 −3−2 −3 7

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

x1x2x3

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟=

2810

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

A x b

Willwelearnfastermethods tosolvethis? No.(Except ifAisspecial.) Thestandard“Gaussianelimination” (and“LUfactorization”)matrixmethodsare justaslightlymoreorganized versionofthehigh-schoolalgebraelimination technique.

Willwegetbetteratdoingthesecalculations byhand?Maybe,butwhocares?Nowadays,allimportantmatrixcalculations aredonebycomputers.

Willwelearnmoreaboutthecomputeralgorithms? Alittle. Butmostlythetechniquesfor“serious”numerical linear-algebra aretopicsforanothercourse(e.g.18.335).

Howdowethink aboutlinearsystems?(imaginesomeonegivesyoua106×106 matrix)

• Alltheformulasfor2×2and3×3matriceswouldfitononepieceofpaper.Theyaren’tthereasonwhylinearalgebrais important(asaclass orafieldofstudy).

• Largeproblemsaresolvedbycomputers,butmustbeunderstoodbyhumanbeings. (Andweneedtogivecomputerstherighttasks!)

• Understandnon-squareproblems:#equations >#unknownsorviceversa

• Breakupmatrices intosimpler pieces– Factorizematricesintoproductsofsimplermatrices:A=LU(triangular:Gauss),A=QR

(orthogonal/triangular),A=XΛX–1 (diagonal:eigenvecs/vals),A=UΣV*(orthogonal/diagonal:SVD)

– Submatrices (matricesofmatrices).

• Breakupvectorsintosimplerpieces: subspaces andbasischoices.

• Algebraicmanipulations toturnharder/unfamiliar problems(e.g.minimization ordifferentialequations) intosimpler/familiar ones:algebraonwholematrices atonce

Don’texpectalotof“turnthecrank”problemsonpsets orexamsoftheform

“solvethissystemofequations.”

…wewillturnitupside-down,giveyoutheanswerandaskthequestion,askaboutpropertiesofthesolutionfrompartial

information,…thegeneralgoalistorequireyoutounderstandthecrankratherthanjustturnit.

(Examsmightbeabitharderthaninprevious18.06semesters,butgradecutoffswillbe

adjustedaccordingly.)

Wheredobigmatricescomefrom?

Lotsofexamplesinmanyfields,buthereareacouplethatarerelativelyeasytounderstand…

Engineering&ScientificModeling[18.303,18.330,6.336,6.339,…]

http://gmsh.info/ Unknownfunctions(fluidflow,mechanicalstress,electromagnetic fields,…)approximated byvaluesonadiscretemesh/grid

e.g.100x100x100grid=106 unknowns!

http://www.electronics-eetimes.com/ comsol.com

Dataanalysis

Imageprocessing:imagesarejustmatrices ofnumbers

(red/green/blue intensity)

[Mathw

orks]

[wikipedia ]

Regression:(curvefitting)

Google“pagerank”problem(alsoforgenenetworksetc.)

Determine the“mostimportant”webpages justfromhowtheylink.

matrix=(#webpages)× (#webpages)(entry=1iftheylink,0otherwise)

Notjustmatricesofnumbers• Therearelotsofsurprisingandimportantgeneralizations oftheideasinlinearalgebra.

• Insteadofvectors withafinitenumberofunknowns,similarideasapplytofunctionswithaninfinitenumberofunknowns.

• Insteadofmatrices multiplyingvectors,wecanthinkaboutlinearoperatorsonfunctions

“A” “x” “b”linear

operator∇2

unknownfunctionu(x,y,z)

right-handsidef(x,y,z)

∇2u = fPoisson’sequation(e.g.18.303)

18.06 vs.18.700

“applied”vs.“pure”math

fewproofsvs.formalproofsexpected(deducepatterns (definitions tofromexamples, lemmas totheorems

informalarguments) …traininginproofwriting)

moreapplications vs.moretheoremsmoreconcrete vs.moreabstract

somecomputersvs.onlypencil-and-paper

Computersoftware

[image:ViralShah]

Lotsofchoices:

julialang.org

Thissemester: arelativelynewlanguagethatscalesbetter torealproblems.

Noprogrammingrequiredfor18.06,justa“glorified calculator”toturnthecrank.

Use itonline: loginatjuliabox.comsee“Julia” linkonStellar

Optionaltutorial:Monday5pm32-xxx