Science & MathOpen learning
A Vision of Linear Algebra
This collection of videos presents Professor Strang’s updated vision of how linear algebra could be taught. It starts with six brief videos, recorded in 2020, containing many ideas and suggestions about the recommended order of topics in teaching and learning linear algebra. Topics include A New Way to Start Linear Algebra , The Column Space of a Matrix, The Big Picture of Linear Algebra, Orthogonal Vectors, Eigenvalues and Eigenvectors, and Singular Values and Singular Vectors. An additional brief video, recorded in 2021, Finding the Nullspace: Solving Ax = 0 by Elimination , computes the nullspace of any matrix A . In 2023, Professor Strang recorded a new one-hour video, Five Factorizations of a Matrix, providing an overall look at linear algebra by highlighting five different ways that a matrix gets factored. Two more videos were added in 2024: The Four Fundamental Subspaces and Least Squares and Elimination and Factorization A = CR
Dept 18RES.18-010+spring_2020
MathematicsOpen learning
Algebra I Student Notes
Algebra I is the first semester of a year-long introduction to modern algebra. Algebra is a fundamental subject, used in many advanced math courses and with applications in computer science, chemistry, etc. The focus of this class is studying groups, linear algebra, and geometry in different forms. These notes, which were created by students in a recent on-campus 18.701 Algebra I class, are offered here to supplement the materials included in OCW’s version of 18.701. They have not been checked for accuracy by the instructors of that class or by other MIT faculty members.
Dept 18RES.18-011+fall_2021
MathematicsOpen learning
Applied Geometric Algebra
László Tisza was Professor of Physics Emeritus at MIT, where he began teaching in 1941. This online publication is a reproduction of the original lecture notes for the course “Applied Geometric Algebra” taught by Professor Tisza in the Spring of 1976. Over the last 100 years, the mathematical tools employed by physicists have expanded considerably, from differential calculus, vector algebra and geometry, to advanced linear algebra, tensors, Hilbert space, spinors, Group theory and many others. These sophisticated tools provide powerful machinery for describing the physical world, however, their physical interpretation is often not intuitive. These course notes represent Prof. Tisza’s attempt at bringing conceptual clarity and unity to the application and interpretation of these advanced mathematical tools. In particular, there is an emphasis on the unifying role that Group theory plays in classical, relativistic, and quantum physics. Prof. Tisza revisits many elementary problems with an advanced treatment in order to help develop the geometrical intuition for the algebraic machinery that may carry over to more advanced problems. The lecture notes came to MIT OpenCourseWare by way of Samuel Gasster, ‘77 (Course 18), who had taken the course and kept a copy of the lecture notes for his own reference. He dedicated dozens of hours of his own time to convert the typewritten notes into LaTeX files and then publication-ready PDFs. You can read about his motivation for wanting to see these notes published in his Preface. Professor Tisza kindly gave his permission to make these notes available on MIT OpenCourseWare.
Dept 8RES.8-001+spring_2009
MathematicsOpen learning
Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra
Calculus Revisited is a series of videos and related resources that covers the materials normally found in freshman- and sophomore-level introductory mathematics courses. Complex Variables, Differential Equations, and Linear Algebra is the third course in the series, consisting of 20 Videos, 3 Study Guides, and a set of Supplementary Notes. Students should have mastered the first two courses in the series (Single Variable Calculus and Multivariable Calculus) before taking this course. The series was first released in 1972, but equally valuable today for students who are learning these topics for the first time. About the Instructor Herb Gross has taught math as senior lecturer at MIT and was the founding math department chair at Bunker Hill Community College. He is the developer of the Mathematics As A Second Language website, providing arithmetic and algebra materials to elementary and middle school teachers. Acknowledgements Funding for this resource was provided by the Gabriella and Paul Rosenbaum Foundation. Other Resources by Herb Gross Calculus Revisited: Single Variable Calculus Calculus Revisited: Multivariable Calculus
Dept 18RES.18-008+fall_2011
MathematicsOpen learning
Calculus Revisited: Multivariable Calculus
Calculus Revisited is a series of videos and related resources that covers the materials normally found in freshman- and sophomore-level introductory mathematics courses. Multivariable Calculus is the second course in the series, consisting of 26 videos, 4 Study Guides, and a set of Supplementary Notes. The series was first released in 1971 as a way for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for the first time. About the Instructor Herb Gross has taught math as senior lecturer at MIT and was the founding math department chair at Bunker Hill Community College. He is the developer of the Mathematics As A Second Language website, providing arithmetic and algebra materials to elementary and middle school teachers. Acknowledgements Funding for this resource was provided by the Gabriella and Paul Rosenbaum Foundation. Other Resources by Herb Gross Calculus Revisited: Single Variable Calculus Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra
Dept 18RES.18-007+fall_2011
MathematicsOpen learning
Calculus Revisited: Single Variable Calculus
Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshman-level introductory calculus course. The series was first released in 1970 as a way for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for the first time. About the Instructor Herb Gross has taught math as senior lecturer at MIT and was the founding math department chair at Bunker Hill Community College. He is the developer of the Mathematics As A Second Language website, providing arithmetic and algebra materials to elementary and middle school teachers. Acknowledgements Funding for this resource was provided by the Gabriella and Paul Rosenbaum Foundation. Other Resources by Herb Gross Calculus Revisited: Multivariable Calculus Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra
Dept 18RES.18-006+fall_2010
EngineeringOpen learning
Computational Science and Engineering I
This course provides the fundamental computational toolbox for solving science and engineering problems. Topics include review of linear algebra, applications to networks, structures, estimation, finite difference and finite element solutions of differential equations, Laplace’s equation and potential flow, boundary-value problems, Fourier series, the discrete Fourier transform, and convolution. We will also explore many topics in AI and machine learning throughout the course.
Dept 1818.085+summer_2020
MathematicsOpen learning
Computational Science and Engineering I
This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace’s equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications. Note: This course was previously called “Mathematical Methods for Engineers I.”
Dept 1818.085+fall_2008
EngineeringOpen learning
Engineering Math: Differential Equations and Linear Algebra
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.
Dept 22.087+fall_2014
Machine LearningOpen learning
Introduction to Deep Learning
This is MIT’s introductory course on deep learning methods with applications to computer vision, natural language processing, biology, and more! Students will gain foundational knowledge of deep learning algorithms and get practical experience in building neural networks in TensorFlow. Course concludes with a project proposal competition with feedback from staff and panel of industry sponsors. Prerequisites assume calculus (i.e. taking derivatives) and linear algebra (i.e. matrix multiplication), and we’ll try to explain everything else along the way! Experience in Python is helpful but not necessary.
Dept 66.S191+january-iap_2020
EngineeringOpen learning
Introduction to Neural Computation
This course introduces quantitative approaches to understanding brain and cognitive functions. Topics include mathematical description of neurons, the response of neurons to sensory stimuli, simple neuronal networks, statistical inference and decision making. It also covers foundational quantitative tools of data analysis in neuroscience: correlation, convolution, spectral analysis, principal components analysis, and mathematical concepts including simple differential equations and linear algebra.
Dept 99.40+spring_2018
EngineeringOpen learning
Introduction to Numerical Analysis
This course analyzed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations and direct and iterative methods in linear algebra.
Dept 1818.330+spring_2004
MathematicsOpen learning
Introduction to Numerical Analysis
This course analyzed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra.
Dept 1818.330+spring_2012
MathematicsOpen learning
Introduction to Numerical Methods
This course offers an advanced introduction to numerical analysis, with a focus on accuracy and efficiency of numerical algorithms. Topics include sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems), floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) are also surveyed.
Dept 6, 1818.335J+spring_2019
MathematicsOpen learning
Introduction to Stochastic Processes
This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix.
Dept 1818.445+spring_2015
EngineeringOpen learning
Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler
Learn Differential Equations: Up Close with _Gilbert Strang and_ Cleve Moler is an in-depth series of videos about differential equations and the MATLAB® ODE suite. These videos are suitable for students and life-long learners to enjoy. About the Instructors Gilbert Strang is the MathWorks Professor of Mathematics at MIT. His research focuses on mathematical analysis, linear algebra and PDEs. He has written textbooks on linear algebra, computational science, finite elements, wavelets, GPS, and calculus. Cleve Moler is chief mathematician, chairman, and cofounder of MathWorks. He was a professor of math and computer science for almost 20 years at the University of Michigan, Stanford University, and the University of New Mexico. These videos were produced by The MathWorks ® and are also available on The MathWorks website.
Dept 18RES.18-009+fall_2015
MathematicsOpen learning
Linear Algebra
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. It parallels the combination of theory and applications in Professor Strang’s textbook Introduction to Linear Algebra . 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 Linear Algebra . 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 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.
Dept 1818.06SC+fall_2011
MathematicsOpen learning
Linear Algebra
This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.
Dept 1818.06+spring_2010
MathematicsOpen learning
Linear Algebra
This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with 18.06 Linear Algebra , more emphasis is placed on theory and proofs.
Dept 1818.700+fall_2013
MathematicsOpen learning
Linear Algebra - Communications Intensive
This is a communication intensive supplement to Linear Algebra (18.06). The main emphasis is on the methods of creating rigorous and elegant proofs and presenting them clearly in writing. The course starts with the standard linear algebra syllabus and eventually develops the techniques to approach a more advanced topic: abstract root systems in a Euclidean space.
Dept 1818.06CI+spring_2004
MathematicsOpen learning
Linear Partial Differential Equations: Analysis and Numerics
This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples.
Dept 1818.303+fall_2014
PhysicsOpen learning
Mathematical Methods in Nanophotonics
Find out what solid-state physics has brought to Electromagnetism in the last 20 years. This course surveys the physics and mathematics of nanophotonics—electromagnetic waves in media structured on the scale of the wavelength. Topics include computational methods combined with high-level algebraic techniques borrowed from solid-state quantum mechanics: linear algebra and eigensystems, group theory, Bloch’s theorem and conservation laws, perturbation methods, and coupled-mode theories, to understand surprising optical phenomena from band gaps to slow light to nonlinear filters. Note: An earlier version of this course was published on OCW as 18.325 Topics in Applied Mathematics: Mathematical Methods in Nanophotonics, Fall 2005.
Dept 1818.369+spring_2008
MathematicsOpen learning
Mathematics for Materials Scientists and Engineers
This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis.
Dept 33.016+fall_2005
EngineeringOpen learning
Mathematics of Big Data and Machine Learning
This course introduces the Dynamic Distributed Dimensional Data Model (D4M), a breakthrough in computer programming that combines graph theory, linear algebra, and databases to address problems associated with Big Data. Search, social media, ad placement, mapping, tracking, spam filtering, fraud detection, wireless communication, drug discovery, and bioinformatics all attempt to find items of interest in vast quantities of data. This course teaches a signal processing approach to these problems by combining linear algebraic graph algorithms, group theory, and database design. This approach has been implemented in software. The class will begin with a number of practical problems, introduce the appropriate theory, and then apply the theory to these problems. Students will apply these ideas in the final project of their choosing. The course will contain a number of smaller assignments which will prepare the students with appropriate software infrastructure for completing their final projects.
Dept MITRES.LL-005+january-iap_2020
Electrical EngineeringOpen learning
Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
Linear algebra concepts are key 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.
Dept 1818.065+spring_2018
EconomicsOpen learning
Microeconomic Theory I
This half-semester course provides an introduction to microeconomic theory designed to meet the needs of students in an economics Ph.D. program. Some parts of the course are designed to teach material that all graduate students should know. Others are used to introduce methodologies. Students should be comfortable with multivariable calculus, linear algebra, and basic real analysis.
Dept 1414.121+fall_2015
MathematicsOpen learning
Multivariable Calculus Recitation Notes
These lecture notes and exercises (with solutions) cover MIT’s multivariable calculus sequence as taught in Fall 2024. The course 18.02 Multivariable Calculus is a General Institute Requirement (GIR); every MIT student must pass this class in order to graduate. The first third of the course is dedicated to briefly covering some basic linear algebra. The rest of the course covers the traditional multivariable calculus topics including vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2D and 3D space. These notes were created by Evan Chen, a recitation instructor in the Fall 2024 instance of 18.02 Multivariable Calculus . They have not been checked for accuracy by the instructor of that class or by other MIT faculty members. The notes will be updated as needed. Comments, suggestions, and corrections are welcomed.
Dept 18RES.18-016+fall_2024
MathematicsOpen learning
Multivariable Calculus with Theory
This course is a continuation of 18.014 Calculus with Theory . It covers the same material as 18.02 Multivariable Calculus , but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus.
Dept 1818.024+spring_2011
MathematicsOpen learning
Noncommutative Algebra
Noncommutative algebra studies algebraic phenomena involving multiplication for which commutativity law fails, such as product of matrices in linear algebra; such phenomena arise in various disciplines ranging from quantum physics to number theory.
Dept 1818.706+spring_2023
MathematicsOpen learning
Numerical Computation for Mechanical Engineers
This class introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods relevant to mechanical engineering, including approximation (interpolation, least squares and statistical regression), integration, solution of linear and nonlinear equations, ordinary differential equations, and deterministic and probabilistic approaches. Examples are drawn from mechanical engineering disciplines, in particular from robotics, dynamics, and structural analysis.
Dept 22.086+spring_2013
MathematicsOpen learning
Numerical Computation for Mechanical Engineers
This class introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods relevant to mechanical engineering, including approximation (interpolation, least squares and statistical regression), integration, solution of linear and nonlinear equations, ordinary differential equations, and deterministic and probabilistic approaches. Examples are drawn from mechanical engineering disciplines, in particular from robotics, dynamics, and structural analysis. Assignments require MATLAB® programming.
Dept 22.086+fall_2012
EngineeringOpen learning
Numerical Computation for Mechanical Engineers
This class introduces elementary programming concepts including variable types, data structures, and flow control. After an introduction to linear algebra and probability, it covers numerical methods relevant to mechanical engineering, including approximation (interpolation, least squares and statistical regression), integration, solution of linear and nonlinear equations, ordinary differential equations, and deterministic and probabilistic approaches. Examples are drawn from mechanical engineering disciplines, in particular from robotics, dynamics, and structural analysis. Assignments require MATLAB® programming.
Dept 22.086+fall_2014
Data Science, Analytics & Computer TechnologyOpen learning
Numerical Methods Applied to Chemical Engineering
This course focuses on the use of modern computational and mathematical techniques in chemical engineering. Starting from a discussion of linear systems as the basic computational unit in scientific computing, methods for solving sets of nonlinear algebraic equations, ordinary differential equations, and differential-algebraic (DAE) systems are presented. Probability theory and its use in physical modeling is covered, as is the statistical analysis of data and parameter estimation. The finite difference and finite element techniques are presented for converting the partial differential equations obtained from transport phenomena to DAE systems. The use of these techniques will be demonstrated throughout the course in the MATLAB® computing environment.
Dept 1010.34+fall_2005
Data ScienceOpen learning
Numerical Methods Applied to Chemical Engineering
Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: Numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed.
Dept 1010.34+fall_2015
ManagementOpen learning
Optimization Methods in Business Analytics
This course will examine optimization through a business analytics lens. Students will learn the theoretical aspects of linear programming, basic Julia programming, and proficiency with linear and nonlinear solvers. Theoretical components of the course are made approachable and require no formal background in linear algebra or calculus. The primary focus of the course is optimization modeling. As a six-week subject, it covers about half of the material of the MIT OpenCourseWare version, 15.053 Spring 2013. The topics of the 2013 subject were optimization modeling, algorithms, and theory. As part of the Open Learning Library (OLL), this course is free to use. You have the option to sign up and enroll if you want to track your progress, or you can view and use all the materials without enrolling. Resources on OLL allow learners to learn at their own pace while receiving immediate feedback through interactive content and exercises.
Dept 1515.053x+spring_2021
PhysicsOpen learning
Quantum Information Science I
This course is a three-course series that provides an introduction to the theory and practice of quantum computation. The three-course series comprises: 8.370.1x: Foundations of Quantum and Classical computing—quantum mechanics, reversible computation, and quantum measurement 8.370.2x: Simple Quantum Protocols and Algorithms—teleportation and superdense coding, the Deutsch-Jozsa and Simon’s algorithm, Grover’s quantum search algorithm, and Shor’s quantum factoring algorithm 8.370.3x: Foundations of Quantum communication—noise and quantum channels, and quantum key distribution Prior knowledge of quantum mechanics is helpful but not required. It is best if you know some linear algebra. This course was organized as a three-part series on MITx by MIT’s Department of Physics and is now archived on the Open Learning Library, which is free to use. You have the option to sign up and enroll in each module if you want to track your progress, or you can view and use all the materials without enrolling.
Dept 88.370x+spring_2018
MathematicsOpen learning
Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations
This course forms an introduction to a selection of mathematical topics that are not covered in traditional mechanical engineering curricula, such as differential geometry, integral geometry, discrete computational geometry, graph theory, optimization techniques, calculus of variations and linear algebra. The topics covered in any particular year depend on the interest of the students and instructor. Emphasis is on basic ideas and on applications in mechanical engineering. This year, the subject focuses on selected topics from linear algebra and the calculus of variations. It is aimed mainly (but not exclusively) at students aiming to study mechanics (solid mechanics, fluid mechanics, energy methods etc.), and the course introduces some of the mathematical tools used in these subjects. Applications are related primarily (but not exclusively) to the microstructures of crystalline solids.
Dept 22.035+spring_2007
MathematicsOpen learning
Topics in Algebraic Combinatorics
The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.
Dept 1818.318+spring_2006
Business & ManagementOpen learning
Topics in Mathematics with Applications in Finance
This is an updated version of 18.S096 Topics in Mathematics with Applications in Finance from Fall 2013. Please visit the 18.S096 site for more materials and lecture recordings. An investment game is also available as an additional learning resource. The purpose of the class is to expose undergraduate and graduate students to the mathematical concepts and techniques used in the financial industry. The course will consist of a set of mathematics lectures on topics in linear algebra, probability, statistics, stochastic processes, and numerical methods. Mathematics lectures will be mixed with lectures illustrating the corresponding application in the financial industry. MIT mathematicians will teach the mathematics part while industry professionals will give the lectures on applications in finance.
Dept 1818.642+fall_2024
