Lecture 41: Introduction to the Gram-Schmidt process (Nicholson Section 8.1)

Alternate Video Access via MyMedia | Video Duration: 47:53

Description: Started class with a “why do we care about projections?” word problem --- how much powdered milk, Kraft Dinner, and Gatorade should you take backpacking so as to most-closely satisfy the daily required diet.

11:30 --- Reviewed that there were two different ways of projecting a vector onto a subspace and it’s super-easy if you happen to have an orthogonal basis for the subspace.

14:00 --- what if you have an orthonormal basis for the subspace? This makes the formula for the projection prettier but it makes the vectors in the orthonormal basis look uglier.

19:45 --- Given a basis, how can we create an orthogonal basis from it? I started with a simple example of two vectors in R^2.

25:00 --- what would have happened if I’d done that example but handled the vectors in different order?

29:00 --- Considered S the span of three vectors in R^3. I’d like to find an orthogonal basis for S. But at this point I don’t even have a basis for S! The good news: applying the procedure to a spanning set for S will produce an orthogonal basis for S (and we can then determine the dimension of S). In this particular example, it turns out that the original spanning set was a basis for S.

42:15 ---- I know that S = span of the given vectors because this is how I was given S. I then did a bunch of stuff to the spanning set. How do I know that the final set of vectors spans S? How do I know that I didn’t mess anything up? I raised this question but didn’t answer it.

43:15 --- Did another example where S is the span of three vectors and I sought an orthogonal set of vectors that spans S.