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.

Intro: A New Way to Start Linear Algebra

Professor Strang describes independent vectors and the column space of a matrix as a good starting point for learning linear algebra. His outline develops the five shorthand descriptions of key chapters of linear algebra.

Part 1: The Column Space of a Matrix

Professor Strang explains why he now starts linear algebra classes by explaining column spaces and A = CR before A = LU. This captures the key idea of a basis for a vector space.

Part 2: The Big Picture of Linear Algebra

Multiplication by A transforms the row space to the column space. Professor Strang then reveals the Big Picture of Linear Algebra where all four fundamental subspaces interact.

Part 3: Orthogonal Vectors

Professor Strang describes in detail orthogonal vectors and matrices and subspaces. He explains Gram-Schmidt orthogonalization, as well as the Least Squares method for line fitting and non-square matrices.

Part 4: Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are a way to look deeper into the matrix. They have applications across all engineering and science disciplines including graphs and networks.

Part 5: Singular Values and Singular Vectors

Data matrices in machine learning are not square, so they require a step beyond eigenvalues: The Singular Value Decomposition (SVD) expresses every matrix by its singular values and vectors.

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