Thursday, January 26, 2012

OPEN LEARNING OCW 2012: OCW publishes Linear Algebra, the first of the 2012 OCW Scholar courses

OCW has released a new version of Linear Algebra, one of its most visited courses, in the innovative OCW Scholar format designed for independent learners.

Taught by Professor Gilbert Strang, 18.06SC Linear Algebra addresses systems of linear equations and the properties of matrices. The concepts of linear algebra are used to solve problems in physics, economics, engineering, and other disciplines.

18.06SC is the first of six OCW Scholar courses planned for release by the end of February.

> Go to 18.06SC Linear Algebra
> Read the complete press release

OCW Scholar

Linear Algebra

Multi-colored cubes arranged to create the illusion of hexagons.
This optical illusion was seen on a restaurant floor in Paris, and coded in MATLAB® by Shev Macnamara. Each color can become the tops of the cubes if you look at them correctly.


Prof. Gilbert Strang



Course Description

This course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines such as physics, economics and social sciences, natural sciences, and engineering.

Course Format

Click to get started. This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:
  • A complete set of Lecture Videos by Professor Gilbert Strang.
  • Summary Notes for all videos along with suggested readings in Prof. Strang's textbook.
  • Problem Solving Videos on every topic taught by an experienced MIT Recitation Instructor.
  • Problem Sets to do on your own with Solutions to check your answers against the when you're done.
  • A selection of Java® Demonstrations to illustrate key concepts.
  • A full set of Exams with Solutions, including review material to help you prepare.
  • An Online Study Group at OpenStudy, where you can connect with other independent learners.

About OCW Scholar

OCW Scholar courses are designed specifically for OCW’s single largest audience: independent learners. These courses are substantially more complete than typical OCW courses, and include new custom-created content as well as materials repurposed from previously published courses. Learn more about OCW Scholar.


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Course Overview

This course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines. Linear algebra is a branch of mathematics that studies systems of linear equations and the properties of matrices. The concepts of linear algebra are extremely useful in physics, economics and social sciences, natural sciences, and engineering. Due to its broad range of applications, linear algebra is one of the most widely taught subjects in college-level mathematics (and increasingly in high school).


18.02 Multivariable Calculus is a formal prerequisite for MIT students wishing to enroll in 18.06 Linear Algebra, but knowledge of calculus is not required to learn the subject.
To succeed in this course you will need to be comfortable with vectors, matrices, and three-dimensional coordinate systems. This material is presented in the first few lectures of 18.02 Multivariable Calculus, and again here.
The basic operations of linear algebra are those you learned in grade school – addition and multiplication to produce "linear combinations." But with vectors, we move into four-dimensional space and n-dimensional space!

Course Goals

After successfully completing the course, you will have a good understanding of the following topics and their applications:
  • Systems of linear equations
  • Row reduction and echelon forms
  • Matrix operations, including inverses
  • Block matrices
  • Linear dependence and independence
  • Subspaces and bases and dimensions
  • Orthogonal bases and orthogonal projections
  • Gram-Schmidt process
  • Linear models and least-squares problems
  • Determinants and their properties
  • Cramer's Rule
  • Eigenvalues and eigenvectors
  • Diagonalization of a matrix
  • Symmetric matrices
  • Positive definite matrices
  • Similar matrices
  • Linear transformations
  • Singular Value Decomposition


This course, designed for independent study, has been organized to follow the sequence of topics covered in an MIT course on Linear Algebra. The content is organized into three major units:
  • Ax = b and the Four Subspaces
  • Least Squares, Determinants and Eigenvalues
  • Positive Definite Matrices and Applications
Each unit has been further divided into a sequence of sessions that cover an amount you might expect to complete in one sitting. Each session has a video lecture on the topic, accompanied by a lecture summary. For further study, there are suggested readings in Professor Strang's textbook:
Amazon logo Strang, Gilbert. Introduction to Linear Algebra. 4th ed. Wellesley, MA: Wellesley-Cambridge Press, February 2009. ISBN: 9780980232714.
Click on the navigation links in the left column to display the sessions in the three units.

To help guide your learning, you will see how problem solving is taught by an experienced MIT Recitation Instructor.
Finally, within each unit you will be presented with sets of problems at strategic points, so you can test your understanding of the material.

MIT expects its students to spend about 150 hours on this course. More than half of that time is spent preparing for class and doing assignments. It's difficult to estimate how long it will take you to complete the course, but you can probably expect to spend an hour or more working through each individual session.

Meet the Team

This OCW Scholar course was developed by:
  • Gilbert Strang, Professor of Mathematics, Massachusetts Institute of Technology
With technical and writing assistance from:
  • Heidi Burgiel, Professor of Mathematics and Computer Science, Bridgewater State University
The Help Session Videos were developed by:
  • Martina Balagovic
  • Linan Chen
  • Benjamin Harris
  • Ana Rita Pires
  • David Shirokoff
  • Nikola Kamburov
To learn more about each of the TA's, visit the Meet the TAs page.

Meet the TAs

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A woman in a black coat with a blue necklace stands in front of a pond. Martina Balagovic
Hi, I am Martina. By now, I taught linear algebra classes in three universities and in many different flavors. I love algebra, and this is probably my favorite class to teach. It gives students the chance to soar into rigorous, abstract algebraic thinking, while being firmly grounded by its many examples. This way one can reason abstractly and then check one's intuition and reasoning on small, very concrete examples. At the same time, one can explore numerous applications in engineering and computer science. I hope the examples we prepared for you will be useful!
Martina Balagovic grew up in Zagreb, Croatia. She got her undergraduate degree from the University of Zagreb. She was a graduate student at MIT from 2007, earning a Ph.D. degree in 2011. She is currently a postdoc at the University of York in the UK. She is fascinated by algebra and does research in representation theory.

A woman with black hair in a black sweater stands in front of a blackboard. She has her hand on her hip. Linan Chen
Hello, my name is Linan and I'm glad to meet you here. When I was a graduate student at MIT, I had this great opportunity to make some video sessions on solving linear algebra problems for Course 18.06. So, here I am, with the exercises I choose for you. I would like to show you how simple pictures of lines intersecting can give rise to matrices, and how matrices can help with problems from other subjects such as solving differential equations or calculating volumes. Besides, I believe you also notice from these videos that drawing the related picture can often speed up the problem solving process, and it's important that you practice the skill of drawing such pictures. So, have fun with these exercises, and I hope you find them good supplements to the lectures.
Linan Chen comes from Shenyang, a city in the northeast of China. After completing her B.A. in Mathematics from Tsinghua University, she continued her graduate study at MIT where she obtained her Ph.D. in Mathematics in 2011. Linan has been an instructor for various math courses, and for her teaching effort, she was awarded the Charles and Holly Housman Award for Excellence in Teaching from the Department of Mathematics at MIT in 2011.
Linan is currently a Postdoctoral Fellow at McGill University, and her main research interest lies in probability theory, as well as its applications in other disciplines.

A headshot of a young man with red hair and glasses.
Benjamin Harris
Hey, I'm Ben and I taught recitations of 18.06: Linear Algebra at MIT in the spring of 2009 and the spring of 2010. In 18.06, I enjoyed helping students take apart matrices, whether it is with the PLU-factorization, the QR-decomposition, or eigenvalues and eigenvectors. Matrices and linear transformations occur naturally throughout science and math. I hope Gilbert Strang's lectures and our recitation videos help you to pull apart these mathematical objects.
Benjamin Harris is currently a postdoctoral researcher at Louisiana State University. He obtained his Ph.D. in Mathematics from MIT in 2011. His research concerns Lie groups and their representations. More specifically, he is interested in wave front cycles of tempered representations, Fourier transforms of nilpotent coadjoint orbits, irreducible characters, and branching laws for discrete series.

A young woman smiles brightly at the camera. She is sitting in front of some green plants and is wearing a bright red and white shirt.Ana Rita Pires
My name is Ana. In 2011 when I started as a postdoc at Cornell University, my first teaching assignment was...Linear Algebra!  I have taught 18.06 several times here at MIT, and Linear Algebra might be my favorite subject to teach, because it has so many neat ideas and interesting applications, all the while being fairly self-contained.
Ana grew up in Portugal and got her undergraduate degree there.  She came to MIT in 2005 as a graduate student in mathematics and started her research on symplectic geometry.

A young man with black hair is standing beside a chalkboard with equations on it. He is wearing a sweater with grey and yellow horizontal stripes. His hands are in his pockets.David Shirokoff
Hi, I'm Dave and welcome to 18.06.  This course is a great backbone math class for any engineering or science student. For instance, many of the concepts covered will reappear in upper year courses and provide a stepping stone towards understanding "how things work". I hope you enjoy the problems in the videos, and that they help provide further insight into the material!
David Shirokoff grew up in Welland (ON) Canada and completed his undergrad at the University of Toronto. He recently finished his Ph.D. at MIT in applied math under the supervision of Ruben Rosales. His research interests are in applied differential equations, dynamical systems and numerical methods.

A young man with black hair and a striped shirt stands in the foreground.Nikola Kamburov
Hi, I am Nikola. I hope you find these recitation videos, supplementing Prof. Strang's superb linear algebra lectures, helpful and illuminating. Good luck with your studies!
Nikola Kamburov is a Ph.D. student in the Department of Mathematics at MIT. He grew up in Bulgaria and he earned his bachelor's degree in mathematics from Princeton University. His research focuses on nonlinear partial differential equations.



Unit I: Ax = b and the Four Subspaces

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Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang
The big picture of linear algebra: Four Fundamental Subspaces.
Mathematics is a tool for describing the world around us. Linear equations give some of the simplest descriptions, and systems of linear equations are made by combining several descriptions.
In this unit we write systems of linear equations in the matrix form Ax = b. We explore how the properties of A and b determine the solutions x (if any exist) and pay particular attention to the solutions to Ax = 0. For a given matrix A we ask which b can be written in the form Ax.

Unit II: Least Squares, Determinants and Eigenvalues

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Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang
A graph and its edge-node incidence matrix.
Each component of a vector in Rn indicates a distance along one of the coordinate axes. This practice of dissecting a vector into directional components is an important one. In particular, it leads to the "least squares" method of fitting curves to collections of data. This unit also introduces matrix eigenvalues and eigenvectors. Many calculations become simpler when working with a basis of eigenvectors.
The determinant of a matrix is a number characterizing that matrix. This value is useful for determining whether a matrix is singular, computing its inverse, and more.

Unit III: Positive Definite Matrices and Applications

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Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang
Positive definite and semidefinite: graphs of x'Ax.
In this unit we discuss matrices with special properties – symmetric, possibly complex, and positive definite. The central topic of this unit is converting matrices to nice form (diagonal or nearly-diagonal) through multiplication by other matrices. Generally, this process requires some knowledge of the eigenvectors and eigenvalues of the matrix.

Further Study

The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout this unit.

Final Course Review

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Session Overview

Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang Even the "easy" material from the first third of the class is useful in answering these surprisingly complicated questions! Your understanding of linear algebra has been built up using the four subspaces, the eigenvalues, and all the special types of matrices: symmetric, orthogonal, projections, permutations (and you can add more).

Session Activities

Lecture Video and Summary

Problem Solving Video

Final Exam

Click for previous page.

Session Overview

Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang Nine questions in a three-hour closed-book exam would be typical for this course at MIT. We try to cover all the way from Ax=0 (the null space and the special solutions) to projections, determinants, eigenvalues, and even a touch of singular values from the eigenvalues of ATA. That is the good matrix of linear algebra: square, symmetric, and positive definite or at least semidefinite.

Check Yourself

Exams and Solutions

Download Course Materials

This package contains the same content as the online version of the course.
For help downloading and using course materials, read our frequently asked questions.
Strang, Gilbert. 18.06SC Linear Algebra, Fall 2011. (Massachusetts Institute of Technology: MIT OpenCourseWare), (Accessed 26 Jan, 2012). License: Creative Commons BY-NC-SA

For more information about using these materials and the Creative Commons license, see our Terms of Use.

MIT OpenCourseWare Publishes Linear Algebra in Innovative OCW Scholar Format

One of OCW's most popular courses, Linear Algebra, is now available in a version designed to support independent learning.
CAMBRIDGE, MA, January 25, 2012 — MIT’s OpenCourseWare has released a new version of Linear Algebra, one of its most visited courses, in the innovative OCW Scholar format designed for independent learners. Taught by Professor Gilbert Strang, 18.06SC Linear Algebra addresses systems of linear equations and the properties of matrices. The concepts of linear algebra are used to solve problems in physics, economics, engineering, and other disciplines. 18.06SC is the first of six OCW Scholar courses planned for release by the end of February.
Linear Algebra was one of the original 50 courses published on the MIT OpenCourseWare proof-of-concept site launched in 2002. Over the past ten years this course has received a total of 3.1 million visits from educators and learners around the world. Professor Strang, who is one of the most widely known mathematicians in the world, hopes that the new, robust version—with its problem solving videos—will help students everywhere.
“I'm very proud of this new version of 18.06,” said Professor Strang. “OCW has reached out to millions of educators and learners around the globe. With this new approach, even more people can see the beauty and usefulness of Linear Algebra.” In September, Strang was named the first MathWorks Professor of Mathematics, assuming a professorship recently endowed by MathWorks, the maker of mathematical software.
OCW Scholar courses represent a new approach to OCW publication. MIT professors and students work closely with the OCW team to restructure the learning experience for independent learners, who typically have few additional resources available to them. The courses offer more materials than typical OCW courses and include new custom-created content. The OCW Scholar version of Linear Algebra includes videos of all the course lectures supplemented by lecture summaries and by 36 short videos showing how to solve specific problems.
The first five of a planned twenty OCW Scholar courses were launched by MIT OpenCourseWare in January 2011, and have collectively received more than 800,000 visits in less than a year. The initial OCW Scholar courses included Classical Mechanics, Electricity and Magnetism, Solid State Chemistry, Single Variable Calculus, and Multivariable Calculus. Linear Algebra is the first of seven OCW Scholar courses that will be published in 2012. Other upcoming OCW Scholar courses include Principles of Microeconomics, Differential Equations, Introduction to Psychology, Fundamentals of Biology, Introduction to Electrical Engineering and Computer Science I, and Introduction to Computer Science and Programming. OCW Scholar courses are published on the OCW site with the support of the Stanton Foundation.

About MIT OpenCourseWare

MIT OpenCourseWare makes the materials used in the teaching of substantially all of MIT's undergraduate and graduate courses—more than 2,100 in all—available on the Web, free of charge, to any user in the world. OCW receives an average of 1.75 million web site visits per month from more than 215 countries and territories worldwide. To date, more than 100 million individuals have accessed OCW materials. MIT OpenCourseWare is supported by donations from site visitors, grants and corporate sponsorship.

About Gilbert Strang

Gilbert Strang attended MIT as an undergraduate and was a Rhodes Scholar at Balliol College, Oxford. He received his Ph.D. from UCLA, and since then he has taught at MIT. He is a Fellow of the American Academy of Arts and Sciences and was elected to the National Academy of Sciences. He is the MathWorks Professor of Mathematics at MIT and an Honorary Fellow of Balliol College, and has published eight textbooks. He was the President of SIAM during 1999 and 2000, and Chair of the Joint Policy Board for Mathematics. He has received numerous awards and prizes, including the von Neumann Medal, the Henrici Prize, first Su Buchin Prize, and the Haimo Prize.

About the Stanton Foundation

The Stanton Foundation was created by Frank Stanton, who is widely regarded as one of the greatest executives in the history of electronic communications. During his 25 years as president of CBS, he turned a lesser-known radio network into a broadcasting powerhouse. Stanton made many historic contributions to the industry and to the society it served. In 1960, he initiated the first televised presidential debates—the famous Nixon-Kennedy "Great Debates"—which required a special Act of Congress before they could proceed. He also spearheaded the creation of the first coast-to-coast broadcasting system, allowing CBS to become the first network to present a news event live across the continental United States, a speech by President Truman at the opening of the Japanese Peace Conference in San Francisco. Frank Stanton was the commencement speaker at MIT in 1961.
Stephen Carson
External Relations Director
MIT OpenCourseWare

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