Open course ware

PROFESSOR: My watch says five after 2:00, so why don't we get started? This is going to be a strange lecture. I'm expecting some wise guy in the back to say, all your lectures are rather peculiar! But I have a phone call from Europe coming in sometime around 2:15, 2:30. Somebody's coming to visit us next week, and we're going to set up a seminar. So I don't know exactly when it's going to come in, but I have to be in my office. So what I will do is adjourn at quarter after, 20 past the hour. And then so I don't keep you here sitting restively wondering when I'm going to come back, we'll adjourn until 3 o'clock. So you seem to have been taken longer and longer breaks during intermission. So I'll give you a full 3/4 of an hour so you can get it out of your systems. And then, we'll take shorter breaks from here on in. OK this is going to be pretty much our last lecture on symmetry. And we will begin, at least in half of the nex...

Okay. This is lecture five in linear algebra. And, it will complete this chapter of the book. So the last section of this chapter is two point seven that talks about permutations, which finished the previous lecture, and transposes, which also came in the previous lecture. There's a little more to do with those guys, permutations and transposes. But then the heart of the lecture will be the beginning of what you could say is the beginning of linear algebra, the beginning of real linear algebra which is seeing a bigger picture with vector spaces -- not just vectors, but spaces of vectors and sub-spaces of those spaces. So we're a little ahead of the syllabus, which is good, because we're coming to the place where, there's a lot to do. Okay. So, to begin with permutations. Can I just -- so these permutations, those are matrices P and they execute row exchanges. And we may need them. We may have a perfectly good matrix, a perfect matrix A that's invertibl...