Course 18.06: Linear Algebra (Fall 2005)

Department of Mathematics
Massachusetts Institute of Technology

General Information

Lecturer: Gilbert Strang, room 2-240, e-mail gs@math.mit.edu

Lectures: MWF 3-4   room 54-100

Course Administrator: Denis Chebikin, room 2-333, phone 3-7826, e-mail chebikin@math.mit.edu.

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 information:   (ps, pdf).

Syllabus:   (ps, pdf).

Basic MATLAB info:

• MATLAB on Athena
• Short MATLAB Tutorial. In pdf.

Goals of the Linear Algebra Course (html).

A Factorization Review (ps, pdf)

Glossary for Linear Algebra   (ps, pdf)

Linear Algebra in a Nutshell   (ps, pdf)

Problem Sets

Problem Set #1 (ps, pdf)

Problem Set #2 (ps, pdf)

Problem Set #3 (ps, pdf)

Review Problems for Exam #1 (ps, pdf) - these are NOT to be handed in!

Problem Set #4 (ps, pdf)

Problem Set #5 (ps, pdf)

Problem Set #6 (ps, pdf)

Review Problems for Exam #2 (ps, pdf) - these are NOT to be handed in!

Problem Set #7 (ps, pdf)

Problem Set #8 (ps, pdf)

Problem Set #9 (ps, pdf)

Problem Set #10 (ps, pdf) - NOTE the due date!

Practice Final Exam (ps, pdf)

Answers and Solutions to Selected Problems

Problem Set #1 (ps, pdf)

Problem Set #2 (ps, pdf)

Problem Set #3 (ps, pdf)

Exam #1 (ps, pdf)

Problem Set #4 (ps, pdf)

Problem Set #5 (ps, pdf)

Problem Set #6 (ps, pdf)

Exam #2 (ps, pdf)

Problem Set #7 (ps, pdf)

Problem Set #8 (ps, pdf)

Problem Set #9 (ps, pdf)

Problem Set #10 (ps, pdf)

Exam #3 (ps, pdf)

Practice Final Exam (ps, pdf)

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

Demos

READ THIS ! There are new eigenvalue applets WITH SOUND (use Flashplayer)

• eigen_sound    This 3-minute demo shows eigenvectors of 2 by 2 matrices
• power_method    Powers AⁿV lead toward the top eigenvalue/eigenvector

      eigen_sound is also broken into 7 independent pieces

MINI-LECTURES ON EIGENVALUES (with voice explanation)

• Full Lecture (all eight together)

• det(A-\lambda I)=0
• Eigenvectors and Trace
• Powers
• Diagonalization
• Differential Equations
• Symmetry
• Positive Definite
• SVD

• Eigenvalues
• Power method
• SVD (Singular Value Decomposition)
• Gaussian Elimination
• Determinants
• Gram-Schmidt = Orthogonalization
• Inner Product of Functions
• Sum of Fourier Series
• Gibbs Phenomenon
• Aliasing
• Column Spaces
• Least Squares

Other Information

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 3 by 3 Symmetric Matrices   (ps, pdf)

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

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

Additional MATLAB info:

• Added on 11/19/2003: Here 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.
• Added on 11/3/2003: MATLAB diary from review lecture on 11/3/2003
• MATLAB Teaching Codes
• Best Guide to MATLAB
• Cool MATLAB demos by The Mathworks.
• MATLAB Recitation Demos from 1997

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 = 0 represents 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 form a 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 = x₁ a₁ + ... + x_n a_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 (new article by Alan Edelman and Gilbert Strang): (ps , pdf )

An Essay by Professor Strang: Too Much Calculus: (ps , pdf )

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

Exams from Previous Years

Spring '05: Exam 1: (ps, pdf). Solutions: (ps, pdf).

Spring '05: Exam 2: (ps, pdf). Solutions: (ps, pdf).

Spring '05: Exam 3: (ps, pdf). Solutions: (ps, pdf).

Spring '05: Final Exam: (ps, pdf) Solutions: (ps, pdf)

Spring '04: Exam 1: (ps, pdf).

Spring '04: Exam 2: (ps, pdf) Solutions: (ps, pdf).

Spring '04: Exam 3: (ps, pdf) Solutions: (ps, pdf).

Spring '04: Final Exam: (ps, pdf) Solutions: (ps, pdf).

Spring '03: Exam 1, March 3, 2003: (ps, pdf). Solutions: (ps, pdf).

Spring '03: Exam 2, April 9, 2003: (ps, pdf). Solutions: (ps, pdf).

Fall '03: Final: (ps, pdf).

Fall '03: Exam 1: (ps, pdf). Solutions: (ps, pdf).

Fall 2002 Exams

Spring 2002 Exams

Fall 2001 Exams

Spring 2001 Exams

Fall 2000 Exams

Spring 2000 Exams

Fall 1999 Exams

Fall 1998 Exams

Spring 1998 Exams

Fall 1997 Exams

Spring 1997 Exams

Fall 1996 Exams

Spring 1996 Exams

More Practice Exams...

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