In this video lecture, Prof. Gilbert Strang lectures on Symmetric Matrices and Positive Definiteness.

1. The Geometry of Linear Equations
2. Elimination with Matrices
3. Multiplication and Inverse Matrices
4. Factorization into A = LU
5. Transposes, Permutations, Spaces Rⁿ
6. Column Space and Nullspace
7. Solving Ax = 0: Pivot Variables, Special Solutions
8. Solving Ax = b: Row Reduced Form R
9. Independence, Basis, and Dimension
10. The Four Fundamental Subspaces
11. Matrix Spaces; Rank 1; Small World Graphs
12. Graphs, Networks, Incidence Matrices
13. Quiz 1 Review
14. Orthogonal Vectors and Subspaces
15. Projections onto Subspaces
16. Projection Matrices and Least Squares
17. Orthogonal Matrices and Gram-Schmidt
18. Properties of Determinants
19. Determinant Formulas and Cofactors
20. Cramer's Rule, Inverse Matrix, and Volume
21. Eigenvalues and Eigenvectors
22. Diagonalization and Powers of A
23. Differential Equations and exp(At)
24. Markov Matrices; Fourier Series
25. Quiz 2 Review
26. Symmetric Matrices and Positive Definiteness
27. Complex Matrices; Fast Fourier Transform
28. Positive Definite Matrices and Minima
29. Similar Matrices and Jordan Form
30. Singular Value Decomposition
31. Linear Transformations and Their Matrices
32. Change of Basis; Image Compression
33. Quiz 3 Review
34. Left and Right Inverses; Pseudoinverse
35. Final Course Review

18.06 Linear Algebra 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.