Pages
HOMEWORK ASSIGNMENTS |
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Homework 2 (PDF) |
Homework 3 (PDF) |
Homework 4 (PDF) |
Note to OCW Users: MIT OpenCourseWare does not provide student access or discounts for MATLAB® software. It can be purchased from The MathWorks®. For more information about MATLAB Pricing and Licensing, contact The MathWorks directly.
Instructor Key
DF = Prof. Dan Frey
GS = Prof. Gilbert Strang
Session Key
L = Lecture
R = Recitation
E = Quiz or Exam
SES # | TOPICS | INSTRUCTORS | KEY DATES |
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L1 | Course introduction: What are differential equations and linear algebra? How do engineers use them? Four examples of first order differential equations. | DF |
Problem Set 1 Assigned |
L2 | First-order equations: Constant source, step function (Heaviside), delta (Dirac), exponentials, real and complex sinusoids. | GS | |
R1 | Recitation | ||
L3 | First-order equations (continued): Separable equations, exact equations. | GS |
Problem Set 1 Due Problem Set 2 Assigned |
L4 | Second-order equations: Second derivatives in engineering, complex numbers, constant coefficient equations. | GS | |
E1 | Quiz 1 | ||
L5 | Second-order equations (continued): Forced oscillations, examples in electrical and mechanical systems, Laplace transforms. | GS | Problem Set 3 Assigned |
L6 | Laplace transforms (continued): Graphical Methods: Direction fields, nonlinear equations, sources, sinks, saddles, and spirals. | DF | Problem Set 2 Due |
R2 | Recitation | ||
L7 | Graphical and numerical methods: Linearization, stability, Euler’s method. | DF | |
L8 | Linear systems of equations: Gaussian elimination, matrix multiplication. | DF | |
R3 | Recitation | ||
L9 | Linear systems of equations: Matrix inverse. Existence and uniqueness of solutions. Column, row, null space. | DF |
Problem Set 3 Due Problem Set 4 Assigned |
L10 | Linear systems of equations (continued): Mechanical engineering examples. | DF | |
R4 | Recitation | ||
E2 | Quiz 2: Oral exams scheduled throughout the week. | ||
L11 | Eigenvalues and eigenvectors: The eigenvalue problem. Diagonalization, exponentiation of a matrix. | DF |
Problem Set 4 Due Problem Set 5 Assigned |
L12 | Least squares and projection: Positive definite matrices. Singular value decomposition. | DF | Problem Set 5 Due |
L13 | Review session for the final exam. | DF | |
R5 | Review session for the final exam (continued). | ||
E3 | Final Exam |
Quiz And Exam Resources |
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Quiz 1 Resources |
Quiz 1 Announcement and Study Guide (PDF) |
Practice Quiz 1 (PDF) |
Quiz 1 (PDF) |
Quiz 2 Resources |
Quiz 2 Announcement and Study Guide (PDF) |
Quiz 2 (PDF) |
Final Exam Resources |
Final Exam Announcement and Study Guide (PDF) |
Practice Final Exam (PDF) |
Solution Code for the Practice Final Exam (M) |
These are additional MATLAB® scripts that were used in the course.
MATLAB® Scripts |
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A fish example |
A script that will make a matrix representation from a simpler representation of connectivity and angles |
An example of eigenvalues and eigenvectors |
An example of eigenvalues and eigenvectors |
An example of eigenvalues and eigenvectors |
Note to OCW Users: MIT OpenCourseWare does not provide student access or discounts for MATLAB software. It can be purchased from The MathWorks®. For more information about MATLAB Pricing and Licensing, contact The MathWorks directly.
Professor Strang’s Related Courses on OCW
Professor Strang’s foundational course 18.06 Linear Algebra has long been one of the most popular courses on OCW. It has received more than 10 million visits since its first publication in 2002. Professor Strang also has a website dedicated to his linear algebra teaching.
A new version of Professor Strang’s classic Linear Algebra was released in 2011 in the innovative OCW Scholar format designed for independent learners. 18.06SC Linear Algebra includes 35 lecture videos and 36 short (and highly-praised) problem-solving help videos by teaching assistants.
Professor Strang has continued to offer new insights into key mathematics subjects. In 2014, he published the new textbook Differential Equations and Linear Algebra. In 2016, that textbook was developed into a series of 55 short videos supported by MathWorks, with parallel videos about numerical solutions by Dr. Cleve Moler, the creator of MATLAB®. The textbook and video lectures help students in a basic ordinary differential equations course. This new series, Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, is also available on the MathWorks website.
In 2017, Professor Strang launched a new undergraduate course at MIT: 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning. Published on the OCW site in 2019, the course uses linear algebra concepts for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization and, above all, a full explanation of deep learning.
Professor Raj Rao was the inspiration for the course 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, and co-taught it with Professor Strang in the first year it was offered at MIT. Since then, he has developed a very successful course on computational linear algebra and machine learning at the University of Michigan. The website, Mynerva, describes this course, the online textbook, and future plans. Professor Rao’s textbook is complementary to Professor Strang’s new book, Linear Algebra for Everyone.
Professor Strang’s Related Textbooks, Notes, and Videos
Strang, Gilbert. Introduction to Linear Algebra. 5th ed. 2016. Wellesley-Cambridge Press. ISBN: 9780980232776.
Strang, Gilbert. Linear Algebra and Learning from Data. 2019. Wellesley-Cambridge Press. ISBN: 9780692196380.
Strang, Gilbert. Linear Algebra for Everyone. 2020. Wellesley-Cambridge Press. ISBN: 9781733146630.
Strang, Gilbert. Differential Equations and Linear Algebra. 2014. Wellesley-Cambridge Press.ISBN: 9780980232790.
ZoomNotes for Linear Algebra (PDF). Professor Strang created these notes in 2020 and 2021 when many MIT classes were moved online (using Zoom) due to the COVID-19 Pandemic. He hopes that faculty who are planning a linear algebra course and students who are reading for themselves will see these notes.
Guest speaker at the weekly OLSUME (Online Seminar on Undergraduate Mathematics Education) on the topic Linear Algebra and Deep Learning (MP4)
“Linear Algebra, Teaching, and MIT OpenCourseWare” (YouTube) on Lex Fridman Podcast
Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Recitations: 1 session / week, 1.5 hours / session
Course Description
This course is about the mathematics that is most widely used in the mechanical engineering core subjects: An introduction to linear algebra and ordinary differential equations (ODEs), including general numerical approaches to solving systems of equations. Topics covered include linear systems of equations, existence and uniqueness of solutions, Gaussian elimination. Initial value problems, 1st and 2nd order systems, forward and backward Euler, and the Runge-Kutta method (RK4). The course also covers eigenproblems: eigenvalues and eigenvectors, including complex numbers, functions, vectors and matrices. MATLAB® is used in this course.
Note to OCW Users: MIT OpenCourseWare does not provide student access or discounts for MATLAB software. It can be purchased from The MathWorks®. For more information about MATLAB Pricing and Licensing, contact The MathWorks directly.
Textbook
Strang, Gilbert. Differential Equations and Linear Algebra. Wellesley-Cambridge. 2014.
See the website for Prof. Strang’s textbook for more information, including samples of the text. In 2016, the textbook was developed into a series of 55 short videos, Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler.
Grading
Students can calculate their grade with this MATLAB script. (M)
ACTIVITIES | PERCENTAGES |
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Homework (5 assignments, 6% each) | 30% |
Quizzes (2 quizzes, 20% each) | 40% |
Final Exam | 30% |
Late Policy
It is expected that completed assignments will be submitted on the due date and time noted on the assignment. The usual policy for late assignments is that a letter grade is lost per day late. If no arrangement is made ahead of time, that is going to be adhered to strictly. The teaching staff is well aware of the multiple time demands on students. In the case of unusual circumstances or unavoidable conflicts, students discussed the details with the professor and explored alternatives before the due date.