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

---

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.

Instructors:

Prof. Gilbert Strang

Level:

Undergraduate

---

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

---

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.

Syllabus

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).

Prerequisites

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

Format

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:
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.

• VIEW ALL COURSES

• Course Home

Syllabus

Unit I: Ax = b and the Four Subspaces

• The Geometry of Linear Equations
• An Overview of Key Ideas
• Elimination with Matrices
• Multiplication and Inverse Matrices
• Factorization into A = LU
• Transposes, Permutations, Vector Spaces
• Column Space and Nullspace
• Solving Ax = 0: Pivot Variables, Special Solutions
• Solving Ax = b: Row Reduced Form R
• Independence, Basis and Dimension
• The Four Fundamental Subspaces
• Matrix Spaces; Rank 1; Small World Graphs
• Graphs, Networks, Incidence Matrices
• Exam 1 Review
• Exam 1

Unit II: Least Squares, Determinants and Eigenvalues

• Orthogonal Vectors and Subspaces
• Projections onto Subspaces
• Projection Matrices and Least Squares
• Orthogonal Matrices and Gram-Schmidt
• Properties of Determinants
• Determinant Formulas and Cofactors
• Cramer's Rule, Inverse Matrix and Volume
• Eigenvalues and Eigenvectors
• Diagonalization and Powers of A
• Differential Equations and exp(At)
• Markov Matrices; Fourier Series
• Exam 2 Review
• Exam 2

Unit III: Positive Definite Matrices and Applications

• Symmetric Matrices and Positive Definiteness
• Complex Matrices; Fast Fourier Transform (FFT)
• Positive Definite Matrices and Minima
• Similar Matrices and Jordan Form
• Singular Value Decomposition
• Linear Transformations and their Matrices
• Change of Basis; Image Compression
• Left and Right Inverses; Pseudoinverse
• Exam 3 Review
• Exam 3

• Final Course Review

• Final Exam

• Download Course Materials

Meet the TAs

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.

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.

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.

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.

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.

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

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

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

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.

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

Final Course Review

Session Overview

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

• Watch the video lecture Final Course Review (00:43:25)
• Read the accompanying lecture summary (PDF)
• Lecture Video Transcript (PDF)

Problem Solving Video

• Watch the recitation video on Final Exam Problem Solving (00:12:16)

Final Exam

Session Overview

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

• Exams (PDF)
• Solutions (PDF)

Download Course Materials

22.80 MB

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.

• RSS Feeds
• Privacy and Terms of Use
• Site Map
• Cite OCW Content

Strang, Gilbert. 18.06SC Linear Algebra, Fall 2011. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (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.
Contact:
Stephen Carson
External Relations Director
MIT OpenCourseWare
617-253-1250
scarson@mit.edu
http://ocw.mit.edu

OCW Stories

Do you have an OCW story? Share it with us.
Students | Educators | Self Learners | OCW Supporters

 

Students

Kunle Adejumo
Student
Nigeria

Kunle Adejumo is finishing his fourth year of engineering studies at Ahmadu Bello University in Zaria, Nigeria. By all rights, he should now be in his fifth and final year, but local strikes and instability in Nigeria have added almost a full year to his studies at Ahmadu Bello. Read more.

Clinton Blackburn
MIT Student
United States

In between studying for classes and working at the MIT Computing Help Desk, MIT senior Clinton Blackburn volunteers as an ambassador for OCW during his travels to Africa. Read more.

Maria Karamitsou
Student
Greece

A fifth-year civil engineering student at Aristotle University of Thessaloniki in Greece, Maria Karamitsou spent the summer of 2005 working at an institute of seismology, studying earthquake-resistant construction techniques. Read more.

Juan Lara
Student
Mexico

In 1998 Juan Eduardo Leal Lara's father, an industrial electronics engineer, was selected by the National Polytechnic Institute of Mexico to establish a Mechatronics Program for Mexican universities. When he told his then eight-year-old son about this combination of mechanical engineering, electronic engineering, and computer engineering, Lara's interest was immediate. Read more.

Mat Peterson
Student
United States

Working with the material on OCW gave Peterson confidence in his ability to study at the MIT level, but even more importantly, provided a window into an instructional approach that appealed to him. Read more.

^ Back to top

Educators

Richard Hall
Educator
Australia

Richard Hall received a Ph.D. in computer science from LaTrobe University in Melbourne, Australia, in 2002. Shortly thereafter, he found himself playing a different role in LaTrobe's laboratories and lecture halls: teaching courses in introductory information systems, beginning microprocessors, and advanced computer-aided software engineering. Read more.

Triatno Yudo Harjoko
Educator
Indonesia

Triatno Yudo Harjoko has a long and close association with the University of Indonesia in Depok, Indonesia. A graduate of the institution, Harjoko has also been a professor of architecture at the school since 1979, and is currently head of the architecture department. In addition, Harjoko has designed several of the university's buildings, including the expansive Faculty of Engineering complex, and the stunning University Mosque. Read more.

Shirley Harrell
Educator
United States

Shirley Harrell, an assistant professor in the School of Management at Cambridge College in Cambridge, MA, speaks with pride of the educational environment of her institution. As an open-enrollment school, she explains, Cambridge has "a gate at the end, not the beginning," and welcomes students from a wide range of backgrounds, experiences, and abilities. Read more.

Kian Wah Liew
Educator
Malaysia

A secondary school mathematics teacher in Kuala Lumpur, Malaysia, Kian Wah Liew introduces his 18-year-old students to a range of complex concepts, such as matrices, determinants, and differential equations. Though his lessons are conducted in Mandarin, the students use an English textbook - which tends to make Liew's life more complicated, since his students possess a wide range of abilities in both English and mathematics. Read more.

Amy Santee
Educator
United States

Homeschooling her daughters means that Amy Santee knows how to think outside the box. She takes them on field trips and finds a wide variety of outside activities for them like ballet and choir. Her freedom to step outside the classroom keeps the lessons fresh and exciting. "I realize that not all learning is fun," says Santee. "But I strive to make as much as possible enjoyable and educational at the same time." Read more.

Gilbert Strang
MIT Mathematics professor
United States

Gilbert Strang was an early supporter of OCW, envisioning the benefits of making MIT courses freely available and accessible online. "A big part of my life is to open mathematics to students everywhere," says Strang, MIT class of 1955 and a Mathematics professor at the Institute. Read more.

Robert Talbert
Educator
United States

When Robert Talbert, associate professor of mathematics and computing science at Franklin College, was looking for inspiration and resources for his CMP 150 Computer Tools for Problem Solving course, he turned to MIT OpenCourseWare's 6.00 Introduction to Computer Science and Programming course, taught by Professors Eric Grimson and John Guttag. Read more.

François Viruly
Educator
South Africa

As director of Viruly Consulting – a leading South African real estate analysis firm – property economist François Viruly has acquired extensive experience in the South African commercial real estate market over the past decade. Viruly is passionate about his field, and makes an effort to share his expertise with students at the University of the Witwatersrand in Johannesburg, as well as the University of Cape Town and the University of Pretoria. Read more.

Karen Willcox
MIT Aeronautics and Astronautics professor
United States

In her first year, Willcox was surprised and disappointed to find that many of her students' math skills were less proficient than she expected. She has been working ever since both to better understand this phenomenon and to counteract it. Read more.

 

^ Back to top

Self Learners

Harry Crissy
Self Learner
United States

As an expert in community development in South Carolina, Harry Crissy does much of his job on his feet. By combining his shoe-leather commitment to improving underserved areas with OCW courses in systems analysis, Crissy has helped entrepreneurs get projects off the ground and collaborated to transform a vacant building into a facility for a minority-based technical school's culinary program. Read more.

ENERSA
Self Learner
Haiti

Entrepreneurs Jean-Ronel Noel and Alex Georges are working to bring renewable energy to communities throughout Haiti. Through their company, Enersa they planned to create solar panels to serve the needs of their country, but in their research and development process, they required guidance in electrical engineering. Read more.

Wendy Ermold
Self Learner
United States

Physicist Wendy Ermold's career in as a researcher at University of Washington's Polar Science Center has taken her places some would never dream of going - at least not without a very heavy coat. Read more.

Harihar Subramanian
Self Learner
India

Harihar Subramanian is an applications engineer with the Bangalore design center of a California-based high technology start-up. He first learned of OCW as a student at Nanyang Technological University in Singapore. Since then, he has consulted OCW for information in a variety of areas such as business, physics, electronics, and essay writing. Read more.

 

^ Back to top

Supporters

Jon Gruber
MIT Class of 1964 and OCW supporter
United States

An enthusiasm for what's "cutting edge" in technology has molded Jon Gruber's life, and has now made its mark on MIT OpenCourseWare. Read more.

Gene Sprouse
MIT Class of 1963 and OCW supporter
United States

Gene Sprouse always had a passion for science and physics. Originally encouraged by his high school physics teacher, Sprouse nurtured his enthusiasm for physics through college and graduate school. Read more.