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Unformatted text preview: 18.06 Linear Algebra, Spring 2010 Transcript Lecture 33 Yes, OK, four, three, two, one, OK, I see you guys are in a happy mood. I don't know if that means 18.06 is ending, or, the quiz was good. Uh, my birthday conference was going on at the time of the quiz, and in the conference, of course, everybody had to say nice things, but I was wondering, what would my 18.06 class be saying, because it was at the exactly the same time. But, what I know from the grades so far, they're basically close to, and maybe slightly above the grades that you got on quiz two. So, very satisfactory. And, then we have a final exam coming up, and today's lecture, as I told you by email, will be a first step in the review, and then on Wednesday I'll do all I can in reviewing the whole course. So my topic today is -- actually, this is a lecture I have never given before in this way, and it will -- well, four subspaces, that's certainly fundamental, and you know that, so I want to speak about left-inverses and right- inverses and then something called pseudo-inverses. And pseudo-inverses, let me say right away, that comes in near the end of chapter seven, and that would not be expected on the final. But you'll see that what I'm talking about is really the basic stuff that, for an m-by-n matrix of rank r, we're going back to the most fundamental picture in linear algebra. Nobody could forget that picture, right? When you're my age, even, you'll remember the row space, and the null space. Orthogonal complements over there, the column space and the null space of A transpose column, orthogonal complements over here. And I want to speak about inverses. OK. And I want to identify the different possibilities. So first of all, when does a matrix have a just a perfect inverse, two-sided, you know, so the two-sided inverse is what we just call inverse, right? And, so that means that there's a matrix that produces the identity, whether we write it on the left or on the right. And just tell me, how are the numbers r, the rank, n the number of columns, m the number of rows, how are those numbers related when we have an invertible matrix? So this is the matrix which was -- chapter two was all about matrices like this, the beginning of the course, what was the relation of th- of r, m, and n, for the nice case? They're all the same, all equal. So this is the case when r=m=n. Square matrix, full rank, period, just -- so I'll use the words full rank. OK, good. Everybody knows that. OK. Then chapter three. We began to deal with matrices that were not of full rank, and they could have any rank, and we learned what the rank was. And then we focused, if you remember on some cases like full column rank. Now, can you remember what was the deal with full column rank? So, now, I think this is the case in which we have a left-inverse, and I'll try to find it. So we have a -- what was the situation there? It's the case of full column rank, and that means -- what does that mean about r? It equals, what's the deal with r, now, if we have full does that mean about r?...
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