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Unformatted text preview: 18.06 Linear Algebra, Spring 2010 Transcript Lecture 34 OK. Good. The final class in linear algebra at MIT this Fall is to review the whole course. And, you know the best way I know how to review is to take old exams and just think through the problems. So it will be a three-hour exam next Thursday. Nobody will be able to take an exam before Thursday, anybody who needs to take it in some different way after Thursday should see me next Monday. I'll be in my office Monday. OK. May I just read out some problems and, let me bring the board down, and let's start. OK. Here's a question. This is about a 3-by-n matrix. And we're given -- so we're given -- given -- A x equals 1 0 0 has no solution. And we're also given A x equals 0 1 0 has exactly one solution. OK. So you can probably anticipate my first question, what can you tell me about m? It's an m-by-n matrix of rank r, as always, what can you tell me about those three numbers? So what can you tell me about m, the number of rows, n, the number of columns, and r, the rank? OK. See, do you want to tell me first what m is? How many rows in this matrix? Must be three, right? We can't tell what n is, but we can certainly tell that m is three. OK. And, what do these things tell us? Let's take them one at a time. When I discover that some equation has no solution, that there's some right-hand side with no answer, what does that tell me about the rank of the matrix? It's smaller m. Is that right? If there is no solution, that tells me that some rows of the matrix are combinations of other rows. Because if I had a pivot in every row, then I would certainly be able to solve the system. I would have particular solutions and all the good stuff. So any time that there's a system with no solutions, that tells me that r must be below m. What about the fact that if, when there is a solution, there's only one? What does that tell me? Well, normally there would be one solution, and then we could add in anything in the null space. So this is telling me the null space only has the 0 vector in it. There's just one solution, period, so what does that tell me? The null space has only the zero vector in it? What does that tell me about the relation of r to n? So this one solution only, that means the null space of the matrix must be just the zero vector, and what does that tell me about r and n? They're equal. The columns are independent. So I've got, now, r equals n, and r less than m, and now I also know m is three. So those are really the facts I know. n=r and those numbers are smaller than three. Sorry, yes, yes. r is smaller than m, and n, of course, is also. So I guess this summarizes what we can tell. In fact, why not give me a matrix -- because I would often ask for an example of such a matrix -- can you give me a matrix A that's an example? That shows this possibility? Exactly, that there's no solution with that right-hand side, but there's exactly one solution with this right-hand side. Anybody want to suggest a matrix that exactly one solution with this right-hand side....
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