What did I learn from Gilbert Strang's Linear Algebra course?

As I am trying to work out the new "Mathematics for the Million" (and make the survey course for myself) I watched all of MIT's Gilbert Strang's Spring 2005 Linear Algebra course. If you decide to do this - you should probably start with a newer version - this is just what I hit on youtube here first, and I assume linear algebra 101 can't have changed that much in 12 years. I must confess that I didn't do any of the homework or take the tests. The tests seem quite hard, so don't do them without doing well on the homework first. I would have to put some effort in to pass them, but it would confirm I had learned some mechanics and the principals and ideas about matrices. I would not know why I should care though.

I'm not sure what I have learned. I did NOT learn why this is interesting or necessary - that was either assumed knowledge - or just the practical knowledge of being a pre-requiste to take more interesting classes. To be honest most of the class just seemed to be looking for "nice" patterns in matrices manipulations - where "nice" equals useful or easy calculations.

Things I have learned that seems interesting?

This is a class that enables you to "do" linear algebra - the mechanics are taught, so you the basics by hand, and perhaps understand what is happening in matlab etc.
I think there is a focus on the beautiful connections between the ideas - but because the whole subject is abstracted from the problems, the connections also seem abstract
Is Ax=B really the worlds most done calculation - I guess google and machine language is really pushing that, but this was the 2005 version
I was amazed at how easy least squares regression can be!
I would like to do the small world problem more
When played through an iPhone speaker and heard across the room, my wife Clare says he has an annoying voice, but I think she really just wanted me to go in the other room or do the dishes. His voice sounds fine if you have something that can reproduce some bass.
I understand Eigen Vectors and Eigen values, but have no idea why they are useful from this class - expect perhaps the Hidden Markoff Model example - which is cute.
I am still not convinced that this class tells someone why this is important? It seems you just learn it as a pre-requisite and afterwards find it useful?
The class on compression and linear algebra was OK and got the point about the flexibility of basis across, but there are probably much better compression videos on youtube today.
Best quote from the comments: "In Romania, we learn this thing in the tenth grade..." - but to be fair I think he was talking about the first lecture - unless they do eigen in 10th grade Romania math classes.

Overall, this class has me thinking that a problem solving approach is more interesting for most people to "get it" and then generalize from applied to pure problems to bring in the joy which Professor Strang clearly had teaching this class.

If I was to pick three key ideas to have a chapter on a book about - I am not sure what they would be - perhaps 1. Ax=b and the connection to machine learning, 2. Eigen Vectors and Values, and 3 ...? I guess hte real question is what are the interesting questions to best introduce them?