The Four Fundamental Subspaces and Least Squares

A Vision of Linear Algebra
Instructor: Gilbert Strang
View the complete course: https://ocw.mit.edu/2020-vision
YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61iQEFiWLE21EJCxwmWvvek

The four subspaces are the column spaces and the nullspaces of A and A^T: Two perpendicular subspaces in m-dimensional space and two more in n-dimensional space. A is invertible when m=n and both nullspaces contain only the zero vector. When A is NOT invertible, we look for the vector x in the row space that makes || Ax - b || AS SMALL AS POSSIBLE in the column space. This video finds that winning vector! Instead of x = A^-1 b (that inverse doesn't exist) we introduce the  "pseudoinverse" of A.

License: Creative Commons BY-NC-SA
More information at https://ocw.mit.edu/terms
More courses at https://ocw.mit.edu

We encourage constructive comments and discussion on OCW’s YouTube and other social media channels. Personal attacks, hate speech, trolling, and inappropriate comments are not allowed and may be removed. More details at https://ocw.mit.edu/comments.

Post from MIT OpenCourseWare on March 12, 2025

View Original Post