Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

1 of 45

IntroductionMatrixology (Linear Algebra)mdashEpisode 124

MATH 122 Fall 2016

Prof Peter Dodds

Dept of Mathematics amp Statistics | Vermont Complex Systems CenterVermont Advanced Computing Core | University of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 30 License

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

2 of 45

These slides are brought to you by

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

3 of 45

Outline

Exciting Admin

Importance

Usages

Key problems

Three ways of looking

Colbert on Equations

References

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

4 of 45

Basics

Instructor Prof Peter Dodds Lecture room and meeting timesPerkins 107Tuesday and Thursday 1005 am to 1120 am Office Farrell Hall second floor Trinity Campus E-mail peterdoddsuvmedu Course websitehttpwwwuvmedu pdoddsteachingcourses2016-08UVM-122 Textbook ldquoIntroduction to Linear Algebrardquo (3rd or4th or 5th edition) by Gilbert Strang (published byWellesley-Cambridge Press)

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

5 of 45

Our Textbook of Excellence

4th Edition V 3rd Edition V Unhelpful 2 ldquoIntroduction to Linear Algebrardquoby Gil Strang Textbook websitehttpmathmitedulinearalgebra MIT Open Courseware site for 1806(=Linear Algebra)httpocwmitedulinear-algebra-spring-2010

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

6 of 45

Yesness

Money quote from George Cobbrsquos review ofStrangrsquos bookDo you want a book written by a mathematician with alifetime experience using linear algebra to understandimportant authentic applied problems a formerpresident of the Society for Industrial and AppliedMathematics hellip

or do you want a book shaped mainly by the[a]esthetics of pure mathematicians with only a weaktheoretical connection to how linear algebra is used inthe natural and social sciences George Cobb Robert L Rooke Professor of

Mathematics and Statistics Mount Holyoke College Full review here [amazon]

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

7 of 45

Gil Strang Exalted Friend of the Matrix Professor of Mathematics at MIT since 1962 Many awardsincluding MAAHaimo Award forDistinguishedCollege orUniversity Teachingof Mathematics Rhodes Scholar Legend

More on Laplacian matrices graphs and othermadnesses here (Strangrsquos Wikipedia page is here

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

8 of 45

Admin

Potential paper products1 Outline

Papers to read1 ldquoThe Fundamental Theorem of Linear Algebrardquo [2]

2 ldquoToo Much Calculusrdquo [3]

Office hours 1000 to 1155 am WednesdaysFarrell Hall second floor Trinity Campus

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

9 of 45

Team MatrixologyWe may try out Slack Place for discussions about all things PoCS

including assignments and projects Once invited please sign up herehttpteam-matrixologyslackcom Very good Install Slack app on laptops tabletsphone Everyone will behave wonderfully

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

10 of 45

Grading breakdown

1 Levels (40) Ten one-week assignments Lowest assignment score will be dropped The last assignment cannot be dropped Each assignment will have a random bonus pointquestion which has nothing to do with linearalgebra

2 Challenge Levels (30) Three 75 minutes tests distributed throughout thecourse all of equal weighting

3 Final Boss Level (20) le Three hours of joyful celebration Thursday December 15 130 pm to 415 pm inPerkins 107

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

11 of 45

Grading breakdown

4 Mini-levels (10) Most meeting times will end with a 10 to 15minute mini-level There will be around 20 mini-levels

5 Homework (0)mdashProblems assigned online fromthe textbook Doing these exercises will be mostbeneficial and will increase happiness

6 General existencemdashit is extremely desirable thatstudents attend class and class presence will betaken into account if a grade is borderline

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

12 of 45

Questions are worth 3 points according to thefollowing scale 3 = correct or very nearly so 2 = acceptable but needs some revisions 1 = needs major revisions 0 = way off

Schedule The course will mainly cover chapters 2 through 6of the textbook (You should know all about Chapter 1)

Week (dates) Tuesday Thursday1 (830 and 901) = = + Level 12 (906 and 908) = = + Level 23 (913 and 915) = = + Level 34 (920 and 922) = and review Challenge Level 15 (927 and 929) Big picture Big picture + Level 46 (1004 and 1006) Big picture Big picture + Level 57 (1011 and 1013) Big picture Big picture + Level 68 (1018 and 1020) Big picture Challenge Level 29 (1025 and 1027) Normal equation Gram-Schmidt Process +

Level 710 (1101 and 1103) Eigenstuff Eigenstuff + Level 811 (1108 and 1110) Determinants Determinants + Level 912 (1115 and 1117) Eigenstuff textitChallenge Level 313 (1122 and 1124) Thanksgiving Thanksgiving14 (1129 and 1201) Positive Definite Matrices

+ Level 10SVD

15 (1206) SVD mdash

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

14 of 45

Important dates1 Classes run from Tuesday August 30 to Tuesday

December 62 AddDrop Audit PassNo Pass

deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31

(Sadness)4 Reading and Exam periodmdashSaturday December

10 to Friday December 16

More stuffDo check your zoo account for updates regarding thecourse

Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

14 of 45

Important dates1 Classes run from Tuesday August 30 to Tuesday

December 62 AddDrop Audit PassNo Pass

deadlinemdashMonday September 123 Last day to withdrawmdashMonday October 31

(Sadness)4 Reading and Exam periodmdashSaturday December

10 to Friday December 16

More stuffDo check your zoo account for updates regarding thecourse

Academic assistance Anyone who requires assistancein any way (as per the ACCESS program or due toathletic endeavors) please see or contact me as soonas possible

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

15 of 45

More stuff

Being good people1 In class there will be no electronic gadgetry no cell

phones no beeping no text messaging etc Youreally just need your brain some paper and awriting implement here (okay and Matlab)

2 Second I encourage you to email me questionsideas comments etc about the class but requestthat you please do so in a respectful fashion

3 Finally as in all UVM classes Academic honestywill be expected and departures will be dealt withappropriately See httpwwwuvmeducses forguidelines

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

16 of 45

Even more stuffLate policy Unless in the case of an emergency (a realone) or if an absence has been predeclared and amake-up version sorted out assignments that are notturned in on time or tests that are not attended will begiven 0

Computing Approximately 2 out of 10 questions perassignment will be Matlab based

Note for assignment problems written details ofcalculations will be required

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

17 of 45

Why are we doing thisBig deal Linear Algebra is a body of mathematics thatdeals with discrete problems

Many things are discrete Information (0rsquos amp 1rsquos letters words) People (sociology) Networks (the Web people again food webs ) Sounds (musical notes)

Even more

If real data iscontinuous wealmost alwaysdiscretize it(0rsquos and 1rsquos)

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

18 of 45

Why are we doing thisLinear Algebra is used in many fields to solveproblems

Engineering Computer Science Physics

Biology Ecology Economics Science of theSociotechnocene

Big exampleGooglersquos Pagerank

Some truth Linear Algebra is as important as Calculushellip Calculus equiv the blue pillhellip

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

19 of 45

Why are we doing this

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

20 of 45

You are now choosing the red pill

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

21 of 45

The Truth

Calculus is the Serpentrsquos Mathematics

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

22 of 45

The Platypus of Truth

Platypuses are masters of Linear Algebra

The Actual Truth

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

24 of 45

Matrices as gadgets

A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

24 of 45

Matrices as gadgets

A matrix transforms a vector into a new vector primethrough matrix multiplication (whatever that is)prime = We can use matrices to Grow vectors Shrink vectors Rotate vectors Flip vectors Do all these things in different directions Reveal the true ur-dystopian reality

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

25 of 45

Digital photographs are matrices

Usually three matrices RGB color model

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

27 of 45

Best fit line (least squares)

100

101

102

103

104

105

106

107

10minus2

10minus1

100

101

102

103

104

Mass (g)

Bas

al m

etab

olis

m (

W) Linearalgebra doesthisbeautifully Calculusversion isclunkyAnd evil

From ldquoRe-examination of the lsquo34rsquo law of metabolismrdquo [1]Dodds Rothman and WeitzJournal of Theoretical Biology 209 9ndash27 2001

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

28 of 45

The many delights of Eigenthings

Using Linear Algebra wersquoll somehow connect

FibonacciNumbers GoldenRatio Spirals Sunflowerspine cones HarvardSquare

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

29 of 45

This is a math course

httpwwwpimpartworkscomartworkrandomsteveoWax-On-Wax-Off Itrsquos all connected ldquoMore laterrdquo

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

30 of 45

Three key problems of Linear Algebra

1 Given a matrix and a vector find such that = 2 Eigenvalue problem Given find and such

that = 3 Coupled linear differential equations

dd() = ()

Our focus will be largely on 1 partly on 2

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

31 of 45

Major course objective

To deeply understand the equation = theFundamental Theorem of Linear Algebra and thefollowing picture

Column Space

Rn

Null Space

~xn

~xr

A ~xn = ~0

A~xr = ~p

A~xlowast = ~p

d = n minus r

d = r

~xlowast = ~xr + ~xn

~p

~b

~e

Avi = σiui

Space

Left Null

Rm

~0

~0

d = m minus r

d = r

Row Space

v1

vr+1 vn

ur

u1

ur+1um

vr

What is going on here We have 24 episodes to findout

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

32 of 45

The fourfold ways of = case example big picture

solutions = = [ 1 00 1 ] 1 always

= gt [ 1 0 K10 1 K2 ] infin always

gt = ⎡⎢⎣ 1 00 10 0 ⎤⎥⎦ 0 or 1

gt gt ⎡⎢⎢⎣1 0 reg10 1 reg20 0 00 0 0 ⎤⎥⎥⎦ 0 or infin

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

33 of 45

Our new BFF = Broadly speaking = translates as follows represents reality (eg music structure) contains building blocks (eg notes shapes) specifies how we combine our building blocks to

make (as best we can)How can we disentangle an orchestrarsquos sound

Radiolabrsquos amazing pieceA 4-Track Mind

What about pictures waves signals

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

34 of 45

Is this your left nullspace

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

35 of 45

Linear Algebra complimentsputdowns Wow you have such a tinyhuge [delete asapplicable] left nullspace

See also The Dunning-Kruger effect

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

36 of 45

Our friend = What does knowing give usIf we can represent reality as a superposition (orcombination or sum) of simple elements we can domany things Compress information See how we can alter information (filtering) Find a systemrsquos simplest representation Find a systemrsquos most important elements See how to adjust a system in a principled way

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

37 of 45

Three ways to understand = Way 1 The Row Picture Way 2 The Column Picture Way 3 The Matrix Picture

Example minus1 + 2 = 121 + 2 = 4 Call this a 2 by 2 system of equations 2 equations with 2 unknowns Standard method of simultaneous equationssolve above by adding and subtracting multiplesof equations to each other = Row Picture

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

38 of 45

Three ways to understand = Row Picturemdashwhat we are doing (a) Finding intersection of two lines (b) Finding the values of 1 and 2 for which both

equations are satisfied (truehappy) A splendid and deep connection(a) Geometry (b) Algebra

Three possible kinds of solution1 Lines intersect at one point mdashOne unique solution2 Lines are parallel and disjoint mdashNo solutions3 Lines are the same mdashInfinitely many solutions

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

39 of 45

Three ways to understand = The column pictureSee minus1 + 2 = 121 + 2 = 4as 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] General problem 1 1 + 2 2 = Column vectors are our lsquobuilding blocksrsquo Key idea try to lsquoreachrsquo by combining (summing)multiples of column vectors 1 and 2

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

40 of 45

Three ways to understand = We love the column picture Intuitive Generalizes easily to many dimensions

Three possible kinds of solution1 1 not parallel 2 1 solution2 1 parallel to 2 but not parallel to No solutions3 1 2 and all parallel infinitely many solutions

(assuming neither 1 or 1 are 0)

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

41 of 45

Three ways to understand = Difficulties Do we give up if = has no solution No We can still find the that gets us as close to

as possible Method of approximationmdashvery important We may not have the right building blocks but wecan do our best

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

42 of 45

Three ways to understand = The Matrix PictureNow see 1 [ minus12 ] + 2 [ 11 ] = [ 14 ] as = ∶ [ minus1 12 1 ] [ 12 ] = [ 14 ]

is now an operator transforms into Roughly speaking does two things to 1 RotationFlipping2 Dilation (stretchingcontraction)

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

43 of 45

The Matrix Picture

Key idea in linear algebra Decomposition or factorization of matrices Matrices can often be written as products or sumsof simpler matrices = = = Σ = sum

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

44 of 45

More Truth about Mathematics

The Colbert Report on Math (February 7 2006)

Episode 124

Introduction

Exciting Admin

Importance

Usages

Key problems

Three ways oflooking

Colbert onEquations

References

45 of 45

References I

[1] P S Dodds D H Rothman and J S WeitzRe-examination of the ldquo34-lawrdquo of metabolismJournal of Theoretical Biology 2099ndash27 2001pdf

[2] G StrangThe fundamental theorem of linear algebraThe American Mathematical Monthly100(9)848ndash855 1993 pdf

[3] G StrangToo much calculus 2002SIAM Linear Algebra Activity Group Newsletterpdf