Lecture 42: Finishing up the Gram-Schmidt process (Nicholson Section 8.1)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 35:56
Description:Reviewed the Gram-Schmidt process. This time I wrote the process in terms of projecting onto subspaces and subtracting off those projections.
10:20 --- Did Gram-Schmidt to a set of six vectors in R^5, they span a subspace S. I’d like a basis for S and the dimension of S. What does it mean if you get zero vectors while applying the Gram-Schmidt process?
20:00 --- I made a mistake here. I should have asked students to accept that Span{v1,v2,v3,v4,v5,v6} = Span{w1,w2,w3,w4,w5,w6}. The v6 is missing on the blackboard. Doh! It’s corrected by a student eventually but still…
22:50 ---- Gave a warning that Gram-Schmidt is super on paper but if you actually implement it on a computer you’ll find that it’s numerically unstable and roundoff error messes stuff up. If you want to do it on a computer you should use the Modified Gram-Schmidt method or something even more sophisticated.
24:50 --- I stated a theorem which will allow us to be confident about Gram-Schmidt not messing up the span at any step. Demonstrated how to use the theorem.
31:10 --- Proved theorem for the special case of three vectors. If you’re curious about the modified Gram-Schmidt method and how it compares to the vanilla Gram-Schmidt method have a look at Solving the Normal Equations by QR and Gram-Schmid or The modified Gram-Schmidt procedure. Here's a nice document: Gram–Schmidt Orthogonalization: 100 Years and More which includes some of the history behind Gram-Schmidt, modified Gram-Schmidt, Least-squares approximation (another way of describing our best approximate solutions) and an interesting sci-fi connection. The notes are from an advanced course so don’t expect to understand all 50 slides.
