Write BIG question to intro ON

[jbunn3]

The previous two modules (AT and GT) had questions regarding understanding linear maps algebraically and geometrically.

What is the big question for ON? We are introducing dot products in order to ultimately define orthogonality, which, after introducing orthogonal complements and projections, we construct "nice" bases for decompositions. However, we also set up the theory required to compute least squares solutions to Ax\approx b. It's worth nothing that in this book, we don't focus too much on Ax=b, rather we usually just talk about the augmented matrix form or the vector equation form, so that will flavor our choice a bit.

So I can start us off with two suggestions:

1. How do we construct nice bases for a vector space? (fits, but feels a bit weak)
2. How do we deal with applications with linear systems with no solution? (feels stronger, but doesn't fit as well as we suggested pushing Least Squares to the appendix)

[fragandi]

I think that one perspective (since the module ends with orthonormality) is the question: how do you represent a vector as a linear combination of basis elements?

In the standard basis, the way that we write the vector is the decomposition

When we have "angles" we can use projections to find the coordinates (ON2) in each direction or we can build an orthonormal basis (ON3) instead

[jbunn3]

OK, so to tie this in with my first suggestion, we are in fact trying to choose a nice basis, but we are doing it with the goal of representing a given vector for a certain purpose. My suggestion (1) doesn't indicate anything about the goal of what we want to do. Your suggestion seems to imply that the basis is already chosen (by which we know how to represent the vector). So I think we need to say something like
4) How do we construct a basis to give a nice representation of a given vector?

I still am not satisfied with this--I'm having trouble thinking of how to succinctly tie these ideas together into a short question.

[jbunn3]

This is where I'm sort of reverting back to thinking that (2) is better as a big picture goal, but it would require making Least Squares the capstone section of the module instead of an applications section. I guess I'm just not convinced that getting an orthonormal basis is interesting enough to be the goal of a module. But I guess this call is for @StevenClontz and @siwelwerd.