The Best Approximation Theorem

In this video, we present, with proof, the Best Approximation Theorem of vectors, that is, the shortest distance between a vector and a subspace is the orthogonal projection. We also explain how to find the distance between a vector and an arbitrary affine set.

Linear Algebra Done Openly is an open source linear algebra textbook developed by Dr. Andrew Misseldine. It is supported and improved by student contributions done by Math 2700 students at Southern Utah University, since Fall 2018. The textbook can be found at:

This lecture covers section 4.5 - Orthogonal Projections from the book. This is part 4 of 4 videos from this section.