Linear Algebra

Massachusetts Institute of Technology via MIT OpenCourseWare

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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 {{% resource_link "4cdcc772-348a-4e48-9678-0a6cc9328fde" "_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 {{% resource_link "4cdcc772-348a-4e48-9678-0a6cc9328fde" "_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.

Syllabus

An Interview with Gilbert Strang on Teaching Linear Algebra.
Course Introduction | MIT 18.06SC Linear Algebra.
1. The Geometry of Linear Equations.
Geometry of Linear Algebra.
Rec 1 | MIT 18.085 Computational Science and Engineering I, Fall 2008.
An Overview of Key Ideas.
2. Elimination with Matrices..
Elimination with Matrices.
3. Multiplication and Inverse Matrices.
Inverse Matrices.
4. Factorization into A = LU.
LU Decomposition.
5. Transposes, Permutations, Spaces R^n.
Subspaces of Three Dimensional Space.
6. Column Space and Nullspace.
Vector Subspaces.
7. Solving Ax = 0: Pivot Variables, Special Solutions.
Solving Ax=0.
8. Solving Ax = b: Row Reduced Form R.
Solving Ax=b.
9. Independence, Basis, and Dimension.
Basis and Dimension.
10. The Four Fundamental Subspaces.
Computing the Four Fundamental Subspaces.
11. Matrix Spaces; Rank 1; Small World Graphs.
Matrix Spaces.
12. Graphs, Networks, Incidence Matrices.
Graphs and Networks.
13. Quiz 1 Review.
Exam #1 Problem Solving.
14. Orthogonal Vectors and Subspaces.
Orthogonal Vectors and Subspaces.
15. Projections onto Subspaces.
Projection into Subspaces.
16. Projection Matrices and Least Squares.
Least Squares Approximation.
17. Orthogonal Matrices and Gram-Schmidt.
Gram-Schmidt Orthogonalization.
18. Properties of Determinants.
Properties of Determinants.
19. Determinant Formulas and Cofactors.
Determinants.
20. Cramer's Rule, Inverse Matrix, and Volume.
Determinants and Volume.
21. Eigenvalues and Eigenvectors.
Eigenvalues and Eigenvectors.
22. Diagonalization and Powers of A.
Powers of a Matrix.
23. Differential Equations and exp(At).
Differential Equations and exp (At).
24. Markov Matrices; Fourier Series.
Markov Matrices.
24b. Quiz 2 Review.
Exam #2 Problem Solving.
25. Symmetric Matrices and Positive Definiteness.
Symmetric Matrices and Positive Definiteness.
26. Complex Matrices; Fast Fourier Transform.
Complex Matrices.
27. Positive Definite Matrices and Minima.
Positive Definite Matrices and Minima.
28. Similar Matrices and Jordan Form.
Similar Matrices.
29. Singular Value Decomposition.
Computing the Singular Value Decomposition.
30. Linear Transformations and Their Matrices.
Linear Transformations.
31. Change of Basis; Image Compression.
Change of Basis.
33. Left and Right Inverses; Pseudoinverse.
Pseudoinverses.
32. Quiz 3 Review.
Exam #3 Problem Solving.
34. Final Course Review.
Final Exam Problem Solving.

Taught by

Prof. Gilbert Strang

Reviews

4.9 rating, based on 8 Class Central reviews

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Muhammad Amir Naseer

Linear algebra course was a game-changer. Initially, I was struggling with vectors and matrices, but the professor's way of explaining concepts made it so easy to grasp. The course covered everything from basics to advanced topics like eigenvectors and SVD. Practice problems and assignments helped solidify my understanding. The best part was learning how to apply linear algebra to real-world problems like data science and machine learning. Highly recommend this course to anyone looking to strengthen their math foundation. Overall, I'd give it 5/5 stars.
Gabriel Putra Nandra Christian

The Linear Algebra course from MIT OpenCourseWare, taught by Professor Gilbert Strang, is widely regarded as one of the best foundational resources in mathematics education. Strang’s teaching style focuses on deep conceptual understanding rather tha…

The Linear Algebra course from MIT OpenCourseWare, taught by Professor Gilbert Strang, is widely regarded as one of the best foundational resources in mathematics education. Strang’s teaching style focuses on deep conceptual understanding rather than rote computation. He explains the "why" behind key ideas like vector spaces, matrix transformations, and eigenvalues in a clear, engaging way that makes abstract topics feel intuitive. The course balances theory and real-world applications, helping learners see how linear algebra is used in areas like engineering, computer science, and physics.

Although the video quality reflects its early 2000s production, the content remains timeless. The course includes full lecture videos, problem sets, and exams, all freely available, making it an exceptional resource for self-learners. While it may move quickly and require strong motivation, it's ideal for students seeking a deeper, more meaningful understanding of linear algebra. Prof. Strang’s enthusiasm and clarity make this course a classic that continues to inspire learners worldwide.
Joel Antony

Reviewers note that the course successfully connects core linear algebra concepts (vectors, matrices, systems of equations) to real-world data science and machine learning problems, making the abstract material more engaging. The inclusion of practical, short coding exercises is a major plus for those seeking applied skills. The video lectures and accompanying materials are considered high-quality and effective for self-study. The main limitation mentioned in general reviews of similar rigorous courses is the significant time commitment required, but the content is considered essential for a strong foundation in related technical fields.
Arjun Pavan Mandekar

i learnt a lot by this course, the couse was really helpful for me. i am very happy because i got to khow and explore this educational platform and i watched all the lecture which were amazing and easy to understand.
Anonymous

Professor Gilbert find the way to explain and navigate us thought through the rationale behind those rule and the core concept. This completely change my thought about mathmetics. Thank you so much
Priyadharshini S

Good teaching, and easy to learning linear algebra course teaches well and I learning well. Our problem solving method is very easy to understand.
Rangu Ajay

Nice experience and I learnt many things on the course which I took and nive explained 👌.
And Avery line to line they I have been explained nice thank for explain in understanding way
KIRAN KUMAR

This course has been incredibly beneficial. I've gained extensive knowledge in linear algebra, thanks to the clear and thorough explanations of the concepts.

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