Catalog

Linear Algebra

Mathematics MIT CC BY-NC-SA 4.0 35 lectures

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.

Syllabus

  1. 1 Lecture 1: The geometry of linear equations
  2. 2 Lecture 2: Elimination with matrices
  3. 3 Lecture 3: Multiplication and inverse matrices
  4. 4 Lecture 4: Factorization into A = LU
  5. 5 Lecture 5: Transposes, permutations, spaces R^n
  6. 6 Lecture 6: Column space and nullspace
  7. 7 Lecture 7: Solving Ax = 0: pivot variables, special solutions
  8. 8 Lecture 8: Solving Ax = b: row reduced form R
  9. 9 Lecture 9: Independence, basis, and dimension
  10. 10 Lecture 10: The four fundamental subspaces
  11. 11 Lecture 11: Matrix spaces; rank 1; small world graphs
  12. 12 Lecture 12: Graphs, networks, incidence matrices
  13. 13 Lecture 13: Quiz 1 review
  14. 14 Lecture 14: Orthogonal vectors and subspaces
  15. 15 Lecture 15: Projections onto subspaces
  16. 16 Lecture 16: Projection matrices and least squares
  17. 17 Lecture 17: Orthogonal matrices and Gram-Schmidt
  18. 18 Lecture 18: Properties of determinants
  19. 19 Lecture 19: Determinant formulas and cofactors
  20. 20 Lecture 20: Cramer's rule, inverse matrix, and volume
  21. 21 Lecture 21: Eigenvalues and eigenvectors
  22. 22 Lecture 22: Diagonalization and powers of A
  23. 23 Lecture 23: Differential equations and exp(At)
  24. 24 Lecture 24: Markov matrices; fourier series
  25. 25 Lecture 24b: Quiz 2 review
  26. 26 Lecture 25: Symmetric matrices and positive definiteness
  27. 27 Lecture 26: Complex matrices; fast fourier transform
  28. 28 Lecture 27: Positive definite matrices and minima
  29. 29 Lecture 28: Similar matrices and Jordan form
  30. 30 Lecture 29: Singular value decomposition
  31. 31 Lecture 30: Linear transformations and their matrices
  32. 32 Lecture 31: Change of basis; image compression
  33. 33 Lecture 32: Quiz 3 review
  34. 34 Lecture 33: Left and right inverses; pseudoinverse
  35. 35 Lecture 34: Final course review

Course materials