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 leftinverses and right
inverses and then something called pseudoinverses.
And pseudoinverses, 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 mbyn
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, twosided, you
know, so the twosided 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.
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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 leftinverse, 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
column rank, I mean the columns are independent, but maybe not the rows. So what
is r equal to in this case? n.
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 Fall '10
 Strang
 Linear Algebra, Algebra, Matrices, Invertible matrix, the deal

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