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GILBERT STRANG: OK well, here we're at the beginning. And that I think it's worth thinking about what we know. Calculus. Differential equations is the big application of calculus, so it's kind of interesting to see what part of calculus, what information and what ideas from calculus, actually get used in differential equations. And I'm going to show you what I see, and it's not everything by any means, it's some basic ideas, but not all the details you learned. So I'm not saying forget all those, but just focus on what matters. OK. So the calculus you need is my topic. And the first thing is, you really do need to know basic derivatives. The derivative of x to the n, the derivative of sine and cosine. Above all, the derivative of e to the x, which is e to the x. The derivative of e to the x is e to the x. That's the wonderful equation that is solved by e to the x. Dy dt equals y. We'll have to do more with that. And then the inverse fun...

GILBERT STRANG: OK, this is the video about two neat functions-- the step function and its derivative the delta function. So if I can just introduce you to those functions and show you that they're very natural inputs to a differential equation. They happen all the time in real life. And so we need to understand how to compute these formulas and compute with them. OK, so the first one is the step function and it's-- I'll call it h after its inventor who was an engineer named Heaviside, started with an H. And the step function, let me write the formula. h of t is 0 for t negative and 1 for t greater or equal to 0. OK. So, that's the step function. It just has two values and it has a jump. You could say jump function also. Jump function, step function. All right. And notice I've also graphed the shifted step function. What happens to any function including this one if I change from t, which jumps at 0, to h of t minus t? If I put in t minus some fixed nu...

TINA SRISVASTAVA: OK, so when we were talking about radar, we had a great reminder about how the origins of that came right here at MIT. And in fact, physically right here, where the Stata building is located. Well, when we're talking about instrument flying, there is also a very strong connection here to MIT. So does anyone know the story of Jimmy Dolittle and his first blind flight? Yes, do you want to share? AUDIENCE: [INAUDIBLE] TINA SRISVASTAVA: Yes, you're right. He has a lot of good stories. AUDIENCE: Takeoff and landing [? just on ?] flight instruments. TINA SRISVASTAVA: Takeoff and landing just on flight instruments, yes. Good, that's exactly right. AUDIENCE: He also got a doctorate here. TINA SRISVASTAVA: He also got his doctorate here, that's right. You have a good understanding of him. AUDIENCE: [INAUDIBLE] TINA SRISVASTAVA: So Jimmy Dolittle, before he came to MIT, was flying in the military. And he was doing a lot of things. He's known fo...

GILBERT STRANG: OK, thanks. Here's a second video that involves the matrix exponential. But it has a new idea in it, a basic new idea. And that idea is two matrices being called "similar." So that word "similar" has a specific meaning, that a matrix A, is similar to another matrix B, if B comes from A this way. Notice this way. It means there's some matrix M-- could be any invertible matrix. So that I take A, multiply on the right by M and on the left by M inverse. That'd probably give me a new matrix. Call it B. That matrix is called "similar" to B. I'll show you examples of matrices that are similar. But first is to get this definition in mind. So in general, a lot of matrices are similar to-- if I have a certain matrix A, I can take any M, and I'll get a similar matrix B. So there are lots of similar matrices. And the point is all those similar matrices have the same eigenvalues. So there's a little family of matric...

PROFESSOR: OK, well let's move on then, and just talk about the amino acids. Amino acids side chains. And you won't have to memorize these structures. We will give you a chart if you have a problem. On the other hand, you need to get very familiar with them, so they're old friends even if you can't quite remember how many methylene are in a chain, or something like that. And you will find that they fall into certain categories. And I'm just going to try and give you examples of the major categories. There are negatively charged side chains. An example would be amino acids known as aspartate, or Asp, in which the side chain which corresponds to the R1 or to the R2 over there, has a methylene group, and then a carboxyl group. But at pH sevenish, which is the pH that you find inside a cell, that carboxyl group would be deprotonated so it would have a negative charge. The other negatively charged amino acid is glutamate, which also, as you'll see has a...

GILBERT STRANG: OK, so I was speaking about eigenvalues and eigenvectors for a square matrix. And then I said for data for many other applications, the matrices are not square. We need something that replaces eigenvalues and eigenvectors. And what they are-- and it's perfect-- is singular values and singular vectors. So may I explain singular values and singular vectors? This slide shows a lot of them. The point is that there will be-- now I don't say eigenvectors-- two-- different left singular vectors. They will go into this matrix u. Right singular vectors will go into v. It was the other case that was so special. When the matrix was symmetric, then the left equals left eigenvector. They're the same as the right one. That's sort of sensible. But a general matrix and certainly a rectangular matrix-- well, we don't call them eigenvectors, because that would be confusing-- we call them singular vectors. And then, inbetween are not eigenvalues, but sing...

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,...