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If you would have been driving through the hills of Tennessee about 40 miles south of Nashville on I-65 in the 1990s, you would have come across a very curious exit called the Saturn Parkway. And if you would've taken that exit and gone a little farther, you would've come across something called the Donald Ephlin Parkway. That was the entrance to the Saturn Corporation, one of the boldest, most innovative, new experiments in labor management relations and organizations of the past 30 years. So let's take a look at what Saturn was all about and, unfortunately, why it failed. Well, Saturn started in the 1980s when a number of innovative labor and management leaders said, we've got to adjust the way in which we work. General Motors came to Donald Ephlin, who was the vice president of the auto workers union, and said, we can't build a small car in the United States competitively. We've got to outsource that work. Don, in his innovative way, said, well,...

GILBERT STRANG: I would like you to see the big picture of linear algebra. We're not doing, in this set of videos, a full course on linear algebra. That's already on OpenCourseWare 1806. And now I'm concentrating on differential equations, but you got to see linear algebra this way. And this way means subspaces. And there are four of them in the big picture. And we-- previous video described the column space and the null space. Now, we've got two more, making four. And let me look at this matrix-- it's for subspaces-- and put them into the big picture. So the first space I'll look at is the row space. Now, the row space has these rows-- has the vector 1, 2, 3 and the vector 4, 5, 6, two vectors there, and all their combinations. That's the key idea in linear algebra, linear combinations. So 1, 2, 3 is a vector in three dimensional space. 4, 5, 6 is another one. Now, if I take all their combinations, do you visualize that if I have two vectors, ...

DR. PETER DOURMASHKIN: I would like to now analyze the motion of a system of particles that has both translational and rotational motion. So I'm going to consider a pulley, and the pulley has radius R. And there is a string wrapped around THE pulley and a block of object 1 that's on a plane, and another block of object 2. And as object 2 falls down, the pulley rotates and object 1 moves to the right. And there's a coefficient of friction between block 1 and the surface. Now, in order to analyze this problem, I'm going to apply, for the pulley, our torque equals I cm alpha, and for each of the blocks, I'll apply F1 equals m1 a1 and F2 equals m2 a2. But the important thing to realize is that these three quantities, the acceleration of block 1, the acceleration of block 2, and the angular acceleration of the pulley, are constrained because this string is not slipping around the pulley. And so let's begin to analyze this type of problem. So we'll s...

PROFESSOR: All right, I would like to then get back to a discussion of some of the basic relations that we have been discussing. We didn't get terribly far, but I'd like to start with the Cartesian coordinate system that we set up. Rather than using x, y, and z, I'm labeling the axes x1, x2, and x3. And we'll see that the subscripts play a very useful role in the formalism we're about to develop. Now, the first thing we might want to specify in this coordinate is the orientation of a vector and its components. So let's suppose that this is some vector P. And what I will do to define its orientation is to use the three angles that the vector makes, or the direction makes with respect to x1, x2, x3. And we could define these angles as theta1, that's the angle between the direction and x1, theta2, the angle between our direction or our vector and x2, and finally, not surprisingly, I'll call this one theta3. So the three components of the vecto...

I actually would like to take this occasion to praise and thank the TAs. We've been teaching this course for about a dozen years and we've had some really excellent groups of TAs, but this year's crop is really off scale, really outstanding. And we're all very grateful. [APPLAUSE] You know their names. I won't go through them all. But the fact of the matter is these TAs teach this course because they're told to teach the course, and at the same time they're doing all their thesis research, so they end spending about 168 hours a week on various kinds of work. So it's not a natural thing for them to spend an enormous amount of time, as they have been this year, really just off scale extraordinarily good. Here they are, Winston, Susan, Michelle, Sara, Divia, Jim, Sydney, Yasmine and Cha. So thank you all. At the same time, I'd also like to thank Claudette Gardel who runs this thing. This is a l...

I thought I would start by maybe the most uh outstanding example uh in this direction of linear programming so linear programming it's an whole area it's linear algebra with inequalities that's what it really is about and um the whole area just sprang into existence after World War II or in the in the developments of World War II uh I guess the name the most famous name in linear programming is uh daning who created this uh Simplex method pretty unusual for a method to um survive fundamentally unchanged for such a long time that the that the first method suggested to solve linear programs should turn out to have such a a long life and still still very alive uh with competition now from uh a different uh a different class of methods but the I hope you'll feel that this subject is kind of fun uh I'm picking linear programming well I'm just creating an example here uh to show what Duality is about um and then the second lecture this afternoon please s...