Prof. Gilbert Strang
These six brief videos, recorded in 2020, contain ideas and suggestions from Professor Strang about the recommended order of topics in teaching and learning linear algebra. The first topic is called A New Way to Start Linear Algebra. The key point is to start right in with the columns of a matrix A and the multiplication Ax that combines those columns.
That leads to The Column Space of a Matrix and the idea of independent columns and the factorization A = CR that tells so much about A. With good numbers, every student can see dependent columns.
The remaining videos outline very briefly the full course: The Big Picture of Linear Algebra; Orthogonal Vectors; Eigenvalues & Eigenvectors; and Singular Values & Singular Vectors. Singular values have become so important and they come directly from the eigenvalues of A'A.
You can see this new idea developing in the first video lecture of Professor Strang’s 2019 course 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning.