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GILBERT STRANG: OK, ready for part three of this vision of linear algebra. So the key word in part three is orthogonal, which again means perpendicular. So we have perpendicular vectors. We can imagine those. We have something called orthogonal matrices. That's when-- I've got one here. An orthogonal matrix is when we have these columns. I'm always going to use the letter Q for an orthogonal matrix. And I look at its columns, and every column is perpendicular to every other column. So I don't just have two perpendicular vectors going like this. I have n of them because I'm in n dimensions. And you just imagine xyz axes or xyzw axes, go up to 4D for relativity, go up to 8D for string theory, 8 dimensions. We just have vectors. After all, it's just this row of numbers or a column of numbers. And we can decide when things are perpendicular by that test. Like say the test for Q1 to be perpendicular to Qn is that row times that column. When I say times,...

PROFESSOR: OK, ready for more? I defined a problem for you. Now let's address it. Said that if we have one coordinate system, and if we have some vector, q, that's defined as a second-rank tensor, aij times some other vector p sub j-- and let me digress in passing. I am very careful to say "a second-rank tensor" and not a "second-order tensor," because higher-order order terms means negligible and non-important. And when I say "second-order tensor," I don't mean to say it's not important and negligible. It's very important, so I say "rank," which has some sort of dignity to it. So I don't like the term "order," because it has another meaning. OK, so here is a tensor that relates a vector pj to give us the components of a vector qi. If we change coordinate system, the components of p, representing exactly one in the same vector, wink on and off and take different values. The values for q take on diff...

okayy ready for lecture 7 uh and the topics then would be the first time we're seeing differential equations uh let me copy that equation up here and I hope you might guess that it's completely analogous to our Matrix equation so this is going to be a transpose minus d by DX this you remember we had the diagonal matrix C well now we've got with with a discrete set of uh spring constants now we have an elastic constant all along the bar so I'm thinking instead of before we had a line of Springs with masses and now so this is for a transpose C a u equal F that was the discrete one The Continuous one will be if we want to so you can tell part of the point here is to get the analogies because I think that's sort of a part something that mathematics can contribute to see the same pattern appearing in different places and here we have uh the pattern appearing in the a discrete problem and in a continuous problem so I'm thinking here of an elastic bar you...

We're about ready to get started. A couple things. First of all, this is for those of you who are working on GN&C for your projects. I have spoken with Phil Hattis from Draper Lab, who is going to be giving the lecture on GN&C, but he is not giving the lecture until the third of November. Now, that is kind of late. And so I asked him if he would be willing to meet with all of you who are working on GN&C just so you could talk with him and ask questions. If you want him to give you a short version of his presentation that he's going to give in November, he can do that. But he is our in-house resident expert. And I figured it's better if you get a chance to talk to him early. He has some time free tomorrow and he has some time next week. He is out of town through today. I know we have two teams working on GN&C. If you guys want to get together and find out if there is some time when you could all meet with Phil, I will give you his contact inform...