Course 18.06: Linear Algebra (Spring 2008)

Department of Mathematics
Massachusetts Institute of Technology

[info] [psets] [solns] [videos] [demos] [extras] [old exams and psets]

Announcements

(05/13) Ana will have additional office hours Friday 3 to 5 in her office. All sections are welcome.

(05/13) The TAs will be holding office hours all day Monday the 19th (before the test). We'll be in room 2-151 from 10 am to 8 pm. Drop by anytime if you have questions.

(05/13) Aaron will not be having office hours on Tuesday or Wednesday. He will have office hours Sunday 12-2:30.

(05/12) Here is an outline of topics covered on the final: pdf. In particular, sections 7.3, 7.4, 8.1, 8.5, and 10.3 are not on the final.

General Information

Lecturer: Gilbert Strang (gs@math.mit.edu), room 2-240

Lectures: MWF 11 room 54-100

Course Administrator: Brian Lehmann (lehmann@math.mit.edu), room 2-089, phone 2-1195
Office hour: Monday 3:00--4:00

Course Information (pdf) and Syllabus (pdf)

Textbook:	Introduction to Linear Algebra, 3rd Edition by Gilbert Strang published by Wellesley-Cambridge Press.
	The press website includes a review of the book, by Professor Herman Gollwitzer of Drexel University.

Course Management Website (requires MIT certificates)
You can see your grades and change recitations here. (Bookmark it!)

Recitations:	Make recitation changes online at the Course Management Website. If you are not yet assigned to a recitation, please choose one as soon as possible. Make sure you turn in homework to your assigned recitation.

#	Time	Room	Instructor	Office	Hour	Phone	E-mail: @math.mit.edu
Lecture	MWF 11	54-100	G. Strang	2-240	---	3-4383	gs
Rec. 1	M 2	2-131	A. Ritter	2-085	T 4:30--6:00	2-1192	afr@mit
2	M 2	4-149	A. Tievsky	2-492	T 4:00--5:00 W 10:00-11:00	3-4093	tievsky
3	M 3	2-131	A. Ritter	2-085	T 4:30--6:00	3-1192	afr@mit
4	M 3	2-132	A. Tievsky	2-492	T 4:00--5:00 W 10:00-11:00	3-4093	tievsky
5	T 11	2-132	J. Yin	2-333	T 1:15-2:15	3-7826	jbyin
6	T 11	8-205	A. Pires	2-251	T 4:30--5:30	3-7566	arita
7	T 12	2-132	J. Yin	2-333	T 1:15-2:15	3-7826	jbyin
8	T 12	8-205	A. Pires	2-251	T 4:30--5:30	3-7566	arita
9	T 12	26-142	P. Buchak	2-093	Th 12:30--1:30	2-1198	pmb
10	T 1	2-132	B. Lehmann	2-089	M 3:00--4:00 Th 1:30-2:30	2-1195	lehmann
11	T 1	26-142	P. Buchak	2-093	Th 12:30--1:30	2-1198	pmb
12	T 1	26-168	P. McNamara	2-314	M 4:00--5:00	4-1459	petermc
13	T 2	2-132	B. Lehmann	2-089	M 3:00--4:00 Th 1:30-2:30	2-1195	lehmann
14	T 2	26-168	P. McNamara	2-314	M 4:00--5:00	4-1459	petermc
Course administrator			B. Lehmann	2-089	M 3:00--4:00 Th 1:30-2:30	2-1195	lehmann

Basic MATLAB info:

MATLAB on Athena (html)

A Matlab Cheat-sheat (pdf)

Short MATLAB Tutorial (ps, pdf)

Additional MATLAB info:

MATLAB Teaching Codes

The best guide to MATLAB

Cool MATLAB demos by The Mathworks

MATLAB and Parallel Computing

Some helpful overviews of Linear Algebra:

Goals of the Linear Algebra Course (html)

Starting with Two Matrices (pdf)

A Factorization Review (ps, pdf)

Glossary for Linear Algebra (ps, pdf)

Linear Algebra in a Nutshell (ps, pdf)

Fourier Sine Series Examples (pdf)

Notes on function spaces, Hermitian operators, and Fourier series (pdf)

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Problem Sets

Problem Sets are due on Wednesdays before 4 pm in 2-106 (next to the undergraduate math office).

Problem Set 1 (pdf)

Problem Set 2 (pdf)

Problem Set 3 (pdf)

Exam 1 (pdf)

Problem Set 4 (pdf)

Problem Set 5 (pdf); Matlab code (fitlinesine.m)

Exam 2 (pdf)

Problem Set 6 (pdf)

Problem Set 7 (pdf)

Problem Set 8 (pdf)

Exam 3 (pdf)

Problem Set 9 (pdf)

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Solutions

Solutions to Problem Sets will be posted on Fridays.

Problem Set 1 (pdf)

Problem Set 2 (pdf)

Problem Set 3 (pdf)

Exam 1 (pdf)

Problem Set 4 (pdf)

Problem Set 5 (pdf)

Exam 2 (pdf)

Problem Set 6 (pdf)

Problem Set 7 (pdf)

Problem Set 8 (pdf)

Exam 3 (pdf)

Problem Set 9 (pdf)

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Videos

Videos of Professor Strang's Fall 1999 Lectures

To improve your video experience, we have made it possible for visitors to download the streaming video files. Here's the URL structure for a link to an MIT OCW video lecture delivered in a streaming format:

http://mfile.akamai.com/7870/rm/mitstorage.download.akamai.com/7870/18/18.06/videolectures/strang-1806-lec01-26aug1999-220k.rm

If you want to download the same file and play it off-line, use the following URL --
the only difference is in the first part of the URL:

http://ocw.mit.edu/ans7870/18/18.06/videolectures/strang-1806-lec01-26aug1999-220k.rm

This same basic approach will work for most (not all) of the MIT OCW streaming videos. Simply find the URL to the streaming media, and replace the first part of the URL:

http://mfile.akamai.com/7870/rm/mitstorage.download.akamai.com/7870     with     http://ocw.mit.edu/ans7870

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Demos

INTERACTIVE DEMO (with voiceover explanation) on Matrix Multiplication, A = LU, and PA = LU

Interactive demo (uses Flash Player)

EIGENVALUE DEMOS (with sound!) (use Flash Player)

2x2 Eigenvectors This 3-minute demo shows eigenvectors of 2 by 2 matrices
Watch the whole thing, or by individual parts: Part1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7

The Power Method Powers AⁿV lead toward the top eigenvalue/eigenvector

MINI-LECTURES ON EIGENVALUES (with voice explanation)

Full Lecture (all eight together)

Or to view individually (about 2 minutes each)

det(A-\lambdaI)=0

Eigenvectors and Trace

Powers

Diagonalization

Differential Equations

Symmetry

Positive Definite

SVD

JAVA DEMOS (these are interactive, without voice explanation)

Eigenvalues

Power method

SVD(Singular Value Decomposition)

Gaussian Elimination

Determinants

Gram-Schmidt= Orthogonalization

Inner Product of Functions

Sum of Fourier Series

Extras

The 3rd edition(2003) of the textbook is now available!
Instructors could write directly to gs@math.mit.edu to see the new book.
It has Worked Examples and many new features: Glossary, Conceptual
Questions and "Linear Algebra in a Nutshell" will be useful to everyone.
Glossary (ps, pdf)
Conceptual Questions for Review (ps, pdf)
Linear Algebra in a Nutshell (ps, pdf)

A Basis for 3by 3 Symmetric Matrices (ps, pdf)

Gram-Schmidt in 9 Lines of MATLAB (ps, pdf)

Gram-Schmidt orthogonalization -- a nice example (ps, pdf)

The SVD at work(ps, pdf): These are the pictures resulting from the best rank 1, rank 5, rank 10, rank 20 and rank 50 approximations to a 499 by 750 black-and-white intensity matrix. The approximations were obtained by keeping the k largest singular values in the SVD. The bottom right picture is the original one.

Question from Professor Ian Christie, West Virginia University:

Find unit vectors h(t) and m(t) in the direction of the hour and minute hands of a clock, where t denotes the elapsed time in hours. If t = 0represents noon then m(0) = h(0) = (0,1). At what time will the hands of the clock first be perpendicular? At what time after noon will the hands first forma straight line? In the dot product m(t) * h(t),remember that sin x sin y + cos x cos y = cos(x - y). Solution: (ps, pdf)

Multiplication by Columns! The multiplication Ax produces a combination
of the columns of A. If the vectors a₁, a₂, ... , a_n are those columns, then

Ax = x₁a₁ + ... + x_na_n = combination of columns (in the column space!)

A summary of how the properties of different matrices are reflected in the eigenvalues/eigenvectors: (ps, pdf).

Pascal Matrices (article by Alan Edelman and Gilbert Strang): (ps, pdf)

Too Much Calculus (an essay by Professor Strang): (ps, pdf)

Linear Algebra and Music (pdf) This fascinating article, with MATLAB codes for music and for telephone tones and for recovering answering machine information, was contributed by Derrick Smith of Laney College in Oakland. Thank you!!

INTERESTING DEMOS:

Gauss-Jordan demo (9/14/98)

LU demo(9/14/98)

The Media Lab's Eigenfaces Demo

Linear Algebra Records

Projections of famous and not so famous three and four dimensional solids

Interactive least squares fitting

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