The Four Fundamental Subspaces and Least Squares

Description

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.

Slides Used in this Video: The Four Fundamental Subspaces and Least Squares (PDF)

Instructor: Gilbert Strang