The Four Fundamental Subspaces and Least Squares

This 27-minute lecture explores the four fundamental subspaces in linear algebra and the least squares method, presented by instructor Gilbert Strang as part of MIT OpenCourseWare's "A Vision of Linear Algebra" series. Dive into the column spaces and nullspaces of matrices A and A^T, understanding how these create two perpendicular subspaces in m-dimensional space and two more in n-dimensional space. Learn when a matrix is invertible and what happens when it's not. Discover how to find the vector x in the row space that minimizes the distance between Ax and b in the column space when traditional inverse methods fail. Master the concept of the "pseudoinverse" as an alternative to A^-1 when dealing with non-invertible matrices.