If the column space is the geometry, the $LU$ decomposition is the algebraic narrative. In many standard texts, Gaussian elimination is presented as a messy, operational necessity—a process of elimination to "solve" a system. In Strang’s notes, elimination becomes construction .
If you’d like, I can help you of his resources by suggesting: Which videos to watch first? Which problem sets are best for exam prep?
The SVD is the climax of linear algebra. Any matrix (A) (even rectangular) can be factored as: [ A = U \Sigma V^T ] lecture notes for linear algebra gilbert strang
The Ultimate Guide to Gilbert Strang’s Linear Algebra Lecture Notes
From these, you get:
factorization splits the matrix into a structured lower triangle (recording the elimination steps) and an upper triangle (the final state of the equations). 3. Vector Spaces and the Big Picture of Linear Algebra
After elimination, the system is upper triangular. Solve from the bottom up. If the column space is the geometry, the
Determinants distill a square matrix into a single scalar value, unlocking the behavior of eigenvalues. Properties of Determinants
To use these resources effectively, you can follow the structure of the MIT course, 18.06. The course progresses logically, building from fundamental ideas to advanced applications. If you’d like, I can help you of
Strang's video lectures are so popular they're hosted in multiple places. They were originally recorded in 1999 (Fall 1999), and a more widely circulated set from Spring 2005 is available on YouTube. They are all fundamentally based on the same core material. Many learners find them easiest to access via YouTube playlists, where they can be watched at their own pace.
) only works for square matrices with enough eigenvectors, . SVD factors an into two orthogonal matrices ( ) and a diagonal matrix of singular values ( Σcap sigma