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.

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