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# Linear Algebra (Fall 2011)

### Overview

Course Features
• Video lectures
• Captions/transcript
• Lecture notes
• Assignments: problem sets with solutions
• Exams and solutions
• Recitation videos
• Resource Index
Educator Features
• Instructor insights
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. 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.

### 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

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