Linear Algebra: We verify the Spectral Theorem for the 3x3 real symmetric matrix A = [ 0 1 1 / 1 0 1 / 1 1 0 ]. That is, we show that the eigenvalues of A are real and that there exists an orthonormal basis of eigenvectors. In other words, we can put A in real diagonal form using an orthogonal matrix P. (Eigenvalues and eigenvectors for this A are found in the video "Eigenvalues and Eigenvectors.")

1. Row Reduction for a System of Two Linear Equations
2. Solving a 2x2 SLE Using a Matrix Inverse
3. Solving a SLE in 3 Variables with Row Operations 1
4. Solving a SLE in 3 Variables with Row Operations 2
5. Consistency of a System of Linear Equations
6. Inverse of 3 x 3 Matrix Using Row Operations 1
7. Inverse of 3x3 Matrix Using Row Operations 2
8. Inverse of 4x4 Matrix Using Row Operations
9. Example of Determinant Using Row Echelon Form
10. Inverse of 3 x 3 Matrix Using Adjugate Formula
11. Inverse of 4x4 Matrix Using Adjugate Formula
12. Cramer's Rule for Three Linear Equations
13. Determinant of a 4 x 4 Matrix Using Cofactors
14. Determinant of a 4 x 4 Matrix Using Row Operations
15. Examples of Linear Maps
16. Example of Linear Combination
17. Example of Linear Combination (Visual)
18. Evaluating Linear Transformations Using a Basis
19. Linear Transformations on R^2
20. Example of Checking for Basis Property
21. Example of Basis for a Null Space
22. Example of Basis for a Span
23. Example of Linear Independence Using Determinant
24. Example of Kernel and Range of Linear Transformation
25. Linear Transformations: One-One
26. Linear Transformations: Onto
27. Example of Change of Basis
28. Eigenvalues and Eigenvectors
29. Example of Eigenvector: Markov Chain
30. Example of Diagonalizing a 2 x 2 Matrix
31. Example of Power Formula for a Matrix
32. The Fibonacci Numbers Using Linear Algebra (HD Version)
33. Vector Length in R^n
34. The Standard Inner Product on R^n
35. Example of Fourier's Trick
36. Example of Orthogonal Complement
37. Orthogonal Transformations 1: 2x2 Case
38. Orthogonal Transformations 2: 3x3 Case
39. Example of Gram-Schmidt Orthogonalization
40. QR-Decomposition for a 2x2 Matrix
41. Beyond Eigenspaces: Real Invariant Planes
42. Beyond Eigenspaces 2: Complex Form
43. Spectral Theorem for Real Matrices: General 2x2 Case
44. Spectral Theorem for Real Matrices: General nxn Case
45. Example of Spectral Theorem (3x3 Symmetric Matrix)
46. Example of Spectral Decomposition
47. Example of Diagonalizing a Symmetric Matrix (Spectral Theorem)

This course contains 47 short video lectures by Dr. Bob on basic and advanced concepts from Linear Algebra. He walks you through basic ideas such as how to solve systems of linear equations using row echelon form, row reduction, Gaussian-Jordan elimination, and solving systems of 2 or more equations using determinants, Cramer's rule, and more.

He also looks over concepts of vector spaces such as span, linear maps, linear combinations, linear transformations, basis of a vector, null space, changes of basis, as well as finding eigenvalues and eigenvectors.

Finally, he finishes the course covering some advanced concepts involving eigenvectors, including the diagonalization of the matrix, the power formula for a matrix, solving Fibonacci numbers using linear algebra, inner product on R^n, orthogonal transformations, Gram-Schmidt orthogonalization, QR-decomposition, the spectral theorem, and much more.