WEBVTT
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Okay, good.
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The final class in linear algebra at MIT this fall is to review the whole course.
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And you know the best way I know how to review is to take old exams and just think through
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the problems.
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So it'll be a three hour exam next Thursday.
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Anybody who's got, I mean nobody will be able to take an exam before Thursday.
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Anybody who needs to take it in some different way after Thursday should see me next Monday.
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I'll be in my office Monday.
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Okay.
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Tell, may I just read out some problems and let me bring the board down.
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And let's start.
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Okay.
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Here's a question.
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This is about a three by N matrix.
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And we're given.
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So we're given given a x equals one zero zero has no solution.
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And we're also given a x equals zero one zero has exactly one solution.
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Okay.
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So you can probably anticipate my first question.
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What can you tell me about M?
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It's an M by N matrix of rank R is always.
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What can you tell me about those three numbers?
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So what can you tell me about M?
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The number of rows N, the number of columns and R the rank.
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Okay.
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See, do you want to tell me first what M is?
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How many rows in this matrix?
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Must be three, right?
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We can't tell what M is, but we can certainly tell that M is three.
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Okay.
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And what do we know about?
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So what do these things tell us?
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Let's take them one at a time.
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When this, when I discovered that some equation has no solution, that there's some right hand
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side with no answer, what does that tell me about the rank of the matrix?
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It's smaller than, it's smaller than M.
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Is that right?
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If there's no solution, that tells me that the rank, that some rows of the matrix are
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combinations of other rows.
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Because if I had a pivot in every row, then I would certainly be able to solve the system.
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I would have particular solutions and all the good stuff.
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So if any time that there's a system that has with no solutions, that tells me that R
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must be below M.
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What about the fact that if when there is a solution, there's only one?
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What does that tell me?
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Well normally, there'd be one solution and then we could add in anything in the null space.
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So this is telling me that null space only has the zero vector in it.
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There's just one solution period.
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So what does that tell me?
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The null space has only the zero vector in it.
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That tells me what does that tell me about the relation of R to M.
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So this is one solution only.
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That means the null space of the matrix must be just the zero vector.
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And what does that tell me about R and N?
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They're equal.
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The columns are independent.
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I've got now R equal N and R less than M and now I also know M is three.
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So those are really the facts I know.
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N equals R and those numbers are smaller than three.
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Sorry, yeah.
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R is smaller than M and N, of course, is also.
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So I guess this summarizes what we can tell.
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In fact, why not give me a matrix?
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I would often ask for an example of such a matrix.
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Can you give me a matrix A that's an example that shows this possibility?
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Exactly that.
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There's no solution with that right-hand side, but there's exactly one solution with this
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right-hand side.
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Anybody want to suggest the matrix that does that?
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Let's see.
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What vector do I want in the column space?
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I want zero one zero to be in the column space because I'm able to solve for that.
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So let's put zero one zero in the column space.
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Actually, I could stop right there.
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That would be a matrix with M equals three rows and N and R are both one, rank one, one
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column, and of course, there's no solution to that one.
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So that's perfectly good as it is.
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Or if you kind of have a prejudice against matrices that only have one column, I'll accept
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the second column.
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So what could I include as a second column that would just be a different answer, but
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equally good?
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I could put this vector in the column space two if I wanted.
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That would now be a case with R equal N equal two, but of course, three M is still three
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and this vector is not in the column space.
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So this is just like prompting us to remember all those things, column space, null space,
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all that stuff.
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Now, I probably asked a second question about this type of thing.
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Oh, okay.
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Oh, I even asked.
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Write down an example of a matrix that fits the description.
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Hmm.
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I guess I haven't learned anything in 26 years.
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Okay, cross out all statements that are false about any matrix with these.
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So again, this is the preliminary.
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These are the facts about my matrix.
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This is one example, but of course, by having an example, it will be easy to check some
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of these facts or non-fact.
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Let me write down some facts, some possible facts.
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So this is really true or false.
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The determinant, this is part one, the determinant of A transpose A is the same as the determinant
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of A transpose.
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Is that true or not?
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Second one, A transpose A is invertible.
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Is invertible.
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Good possible fact, A, A transpose is positive definite.
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So you see how on an exam question, I try to connect the different parts of the course.
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So well, I mean, the simplest way would be to try it with that matrix as a good example,
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but maybe we can answer even directly.
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Let me take number two first, because I'm very, very fond of that matrix, A transpose
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A.
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And when is it invertible?
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When is the matrix A transpose A invertible?
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The great thing is that I can tell from the rank of A that I don't have to multiply out
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A transpose A.
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A transpose A is invertible.
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Well, if A has a null space other than the zero vector, then it's no way it's going
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to be invertible.
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But the beauty is, if the null space of A is just the zero vector, so the key fact is
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this is invertible if R equals N, by which I mean independent columns of A in A in the
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matrix A.
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If R equals N, if the matrix A has independent columns, then this combination A transpose
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A is square and still that same null space, only the zero vector independent columns
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all good.
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What's the true false?
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Is this middle one T or F for this in this setup?
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Well we discovered that R was N from that second fact.
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So this is a true.
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That's a true.
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And of course, A transpose A in this example would probably be, what would A transpose A
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B for that matrix?
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Can you multiply A transpose A and see what it looks like for that matrix?
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What shape would it be?
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It'll be 2 by 2.
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And what matrix will it be?
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The identity.
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So, checks out.
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Okay.
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What about A A transpose?
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Well, depending on the shape of A, it could be good or not so good.
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It's all with the metric.
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It's all with square, but what's the size now?
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This is 3 by N and this is N by 3.
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So the result is 3 by 3.
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Is it positive definite?
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I don't think so.
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If I multiply that by A transpose, A A transpose, what would the rank be?
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It would be the same as the rank of A.
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It would be just rank 2.
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And if it's 3 by 3 and it's only rank 2, it's certainly not positive definite.
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So what could I say about A A transpose if I wanted to say something true about it?
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It's true that it is positive semi-definite.
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If I made this semi-definite, it would always be true, always.
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But if I'm looking for positive definite, then I'm looking at the null space of whatever
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is here and in this case, it's got a null space.
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So A A, so we just figure it out here.
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A A transpose for that matrix will be 3 by 3.
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If I multiplied A by A transpose, what would the first row be?
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It's all 0's, right?
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First row of A A transpose could only be all 0's.
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So it's probably a 1 there and a 1 there, something like that.
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But I don't even know if that's right.
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But it's all 0's there, so it's certainly not positive definite.
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Let me not put up anything.
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I'm not, don't check.
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What about this determinant?
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Oh, well, I guess that's a sort of tricky question.
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Is it true or false in this case?
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It's false, apparently.
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Because A transpose A is invertible, we just got a true for this one.
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And we got a false.
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We got a non-invertible one for this one.
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So actually, this one is false.
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Number 1.
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That surprises us, actually, because it's, I mean,
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why was it tricky?
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Because what is true about determinants?
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This would be true if those matrices were square.
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If I have two square matrices, and any other matrix,
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B, could be A transpose, could be somebody else's matrix.
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Then it would be true that the determinant of B A
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would equal the determinant of A B.
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But if the matrices are not square, and it would actually
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be true that it would equal, that this would equal the determinant
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of A times the determinant of A transpose.
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We could even split up those two separate determinants.
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And of course, those would be equal.
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But only when A is square.
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So that's a question that rests on the falseness,
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rests on the fact that the matrix isn't square
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in the first place.
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OK, good.
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Let's see.
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Oh, now, even ask the more.
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Prove that A transpose Y equals C. Oh, God.
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This question goes on and on.
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Now I ask you about A transpose Y equals C.
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So I'm asking you about the matrix A transpose.
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And I want you to prove that it has at least one solution,
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one solution for every C, every right-hand side C.
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And in fact, infinitely many solutions for every C. OK.
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Well, none of this is difficult, but it's been a little while.
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So we just have to think again.
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When I have a system of equations, this matrix A
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transpose is now, instead of being 3 by N, it's N by 3.
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It's N by N, of course.
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To show that a system has at least one solution,
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when does the system always solveable?
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When it has full row rank, when the rows are independent.
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Here we have N rows, and that's the rank.
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So at least one solution, because the number of rows, which
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is N for the transpose, is equal to R the rank.
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This A transpose has independent rows,
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because A had independent columns, right?
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The original A had independent columns.
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When we transpose it, it has independent rows,
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so there's at least one solution.
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But now how do I even know that there are infinitely many solutions?
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Oh, what do I want to know something about the null space?
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What's the dimension of the null space of A transpose?
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So the answer is, it's got to be the dimension
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of the null space of A transpose.
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What's the general fact?
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If A is an N by N matrix of rank R, what's the dimension of A
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transpose?
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Of the null space of A transpose?
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Do you remember that?
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The fourth subspace that's tagging along down in our big picture,
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its dimension was N minus R. And that's bigger than 0.
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M is bigger than R. So there's a lot in that null space.
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So there's always one solution because, and this is speaking about A transpose.
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So for A transpose, the rows of M and N are reversed, of course.
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So keep in mind that this board was about A transpose, so the rows,
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so it's the null space of A transpose,
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and our M minus R free variables.
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OK, that's just some review.
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Can I take another problem that's also sort of supposed the matrix A has three columns?
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V1, V2, V3.
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Those are the columns of the matrix.
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All right.
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Question A solves AX equals V1 minus V2 plus V3.
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Tell me what X is.
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Well, there you're seeing the most.
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The one absolutely essential fact about matrix multiplication, how does it work when we do it a column at a time.
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The very, very first day way back in September, we did multiplication a column at a time.
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So what's X? Just tell me one minus one one.
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Thanks.
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OK. Everybody's got that.
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OK.
249
00:19:25.560 --> 00:19:29.840
Then the next question is, suppose that combination is zero.
250
00:19:29.840 --> 00:19:30.680
Oh, yeah.
251
00:19:30.680 --> 00:19:31.680
OK.
252
00:19:31.680 --> 00:19:44.440
So question B says, but B says, suppose this thing is zero.
253
00:19:44.440 --> 00:19:47.120
Suppose that's zero.
254
00:19:47.120 --> 00:19:51.400
Then the solution is not unique.
255
00:19:51.400 --> 00:19:56.200
So I want true or false and a reason.
256
00:19:56.200 --> 00:20:01.080
Suppose this combination is zero.
257
00:20:01.080 --> 00:20:03.880
V1 minus V2, V3.
258
00:20:03.880 --> 00:20:08.400
Show that, what does that tell me?
259
00:20:08.400 --> 00:20:09.720
So it's a separate question.
260
00:20:09.720 --> 00:20:13.680
Maybe I sort of saved time by writing it that way.
261
00:20:13.680 --> 00:20:16.480
But it's a separate, totally separate question.
262
00:20:16.480 --> 00:20:24.680
If I have a matrix and I know that column one minus column two plus column three is zero,
263
00:20:24.680 --> 00:20:32.640
what does that tell me about whether the solution is unique or not?
264
00:20:32.640 --> 00:20:38.200
Is there more than one solution?
265
00:20:38.200 --> 00:20:42.120
What's uniqueness about?
266
00:20:42.120 --> 00:20:46.560
Uniqueness is about, is there anything in the null space, right?
267
00:20:46.560 --> 00:20:51.400
The solution is unique when there's nobody in the null space except the zero vector.
268
00:20:51.400 --> 00:20:58.400
And if that's zero, then this guy would be in the null space.
269
00:20:58.400 --> 00:21:09.840
So if this were zero, then this X is in the null space of A.
270
00:21:09.840 --> 00:21:19.440
So solutions are never unique.
271
00:21:19.440 --> 00:21:23.160
Because I could always add that to any solution.
272
00:21:23.160 --> 00:21:28.800
And AX wouldn't change.
273
00:21:28.800 --> 00:21:31.520
So it's always that question.
274
00:21:31.520 --> 00:21:33.640
Is there somebody in the null space?
275
00:21:33.640 --> 00:21:35.120
OK.
276
00:21:35.120 --> 00:21:39.320
Oh, now here's a totally different question.
277
00:21:39.320 --> 00:21:45.760
Suppose those three vectors v1, v2, v3 are also normal.
278
00:21:45.760 --> 00:21:50.320
So that's, I mean, this isn't going to happen for orthonormal vectors.
279
00:21:50.320 --> 00:21:50.680
OK.
280
00:21:50.680 --> 00:21:53.640
So part C, forget part v.
281
00:21:53.640 --> 00:22:07.040
C. If v1, v2, v3 are orthonormal, orthonormal.
282
00:22:07.040 --> 00:22:13.520
So that I would usually have called them Q1, Q2, Q3.
283
00:22:13.520 --> 00:22:19.120
Now, what combination, oh, here's a nice question.
284
00:22:19.120 --> 00:22:21.040
If I say so myself.
285
00:22:21.040 --> 00:22:26.800
What combination of v1 and v2 is closest to v3?
286
00:22:26.800 --> 00:22:33.160
What point on the plane of v1 and v2 is the closest point to v3
287
00:22:33.160 --> 00:22:35.160
if these vectors are orthonormal?
288
00:22:35.160 --> 00:22:37.000
So let me, I'll start this in his sense.
289
00:22:37.000 --> 00:22:45.560
Then the combination, something times v1 plus something
290
00:22:45.560 --> 00:22:54.160
times v2 is closest, is the closest combination to v3.
291
00:22:54.160 --> 00:22:55.400
And what's the answer?
292
00:22:55.400 --> 00:23:01.440
What's the closest vector on that plane to v3?
293
00:23:01.440 --> 00:23:09.400
Zeroes, right. We just imagine the x, y, z, x, z, v, v1, v2, v3
294
00:23:09.400 --> 00:23:16.000
could be the standard basis, the x, y, z vectors.
295
00:23:16.000 --> 00:23:23.840
And of course, the point on the x, y plane that's closest to v3
296
00:23:23.840 --> 00:23:25.640
on the z axis is zero.
297
00:23:25.640 --> 00:23:33.200
So if we're orthonormal, then the projection of v3
298
00:23:33.200 --> 00:23:38.680
onto that plane is perpendicular, it hits right at zero.
299
00:23:38.680 --> 00:23:46.200
OK, so that's like a easy question, but still brings it out.
300
00:23:46.200 --> 00:23:49.080
OK, let me see what.
301
00:23:49.080 --> 00:23:58.640
So I write down a, yeah, a Markov matrix.
302
00:23:58.640 --> 00:24:04.200
And I'll ask you for its eigenvalues.
303
00:24:04.200 --> 00:24:06.240
OK, here's a Markov matrix.
304
00:24:06.240 --> 00:24:08.960
This, this.
305
00:24:08.960 --> 00:24:14.400
And tell me its eigenvalues.
306
00:24:14.400 --> 00:24:18.440
So here, I'll call the matrix A,
307
00:24:18.440 --> 00:24:30.800
and I'll call this as 0.2, 0.4, 0.4, 0.2, 0.4, 0.3, 0.3, 0.4.
308
00:24:30.800 --> 00:24:31.320
OK.
309
00:24:35.720 --> 00:24:36.600
Let's see.
310
00:24:36.600 --> 00:24:44.040
It helps out to notice that column 1 plus column 2,
311
00:24:44.040 --> 00:24:49.320
what's interesting about column 1 plus column 2?
312
00:24:49.320 --> 00:24:53.000
It's twice as much as column 3.
313
00:24:53.000 --> 00:24:56.920
So column 1 plus column 2 equals 2 times column 3.
314
00:24:56.920 --> 00:25:00.960
I put that in there, column 1 plus column 2 equals twice
315
00:25:00.960 --> 00:25:02.200
column 3.
316
00:25:02.200 --> 00:25:03.880
That's an observation.
317
00:25:03.880 --> 00:25:09.480
OK, I want to, I tell me the eigenvalues of the matrix.
318
00:25:09.480 --> 00:25:12.360
OK, tell me one eigenvalue.
319
00:25:12.360 --> 00:25:17.680
0, because the matrix is singular.
320
00:25:17.680 --> 00:25:20.200
Tell me another eigenvalue.
321
00:25:20.200 --> 00:25:23.840
1, because it's a Markov matrix.
322
00:25:23.840 --> 00:25:28.000
The columns add to the all 1's vector.
323
00:25:28.000 --> 00:25:34.520
And that will be an eigenvector of a transpose.
324
00:25:34.520 --> 00:25:36.320
And tell me the third eigenvalue.
325
00:25:39.280 --> 00:25:41.800
Let's see, to make the trace come out right,
326
00:25:41.800 --> 00:25:46.120
which is 0.8, we need minus 0.2.
327
00:25:46.120 --> 00:25:47.840
OK.
328
00:25:47.840 --> 00:25:48.880
OK.
329
00:25:48.880 --> 00:25:55.360
And now, suppose I start the Markov process.
330
00:25:55.360 --> 00:25:59.120
Suppose I start with u of 0.
331
00:25:59.120 --> 00:26:04.600
So I'm going to look at the power of a applied to u of 0.
332
00:26:04.600 --> 00:26:07.560
This is u k.
333
00:26:07.560 --> 00:26:10.400
And there's my matrix.
334
00:26:10.400 --> 00:26:12.520
And I'm going to let u of 0 be.
335
00:26:12.520 --> 00:26:17.520
This is going to be 0, 10, 0.
336
00:26:17.520 --> 00:26:23.720
And my question is, what does that approach?
337
00:26:23.720 --> 00:26:27.560
If u of 0 is equal to this, if there is u of 0.
338
00:26:27.560 --> 00:26:30.360
So I write it in, maybe I'll just write in u of 0.
339
00:26:30.360 --> 00:26:40.360
8 of the k, starting with 10 people in state 2.
340
00:26:40.360 --> 00:26:48.320
And every step follows the Markov rule.
341
00:26:48.320 --> 00:26:55.400
What is the solution look like after k steps?
342
00:26:55.400 --> 00:26:57.360
Let me just ask you that.
343
00:26:57.360 --> 00:27:01.200
And then what happens as k goes to infinity?
344
00:27:01.200 --> 00:27:03.920
This is a steady state question, right?
345
00:27:03.920 --> 00:27:05.320
I'm looking for the steady state.
346
00:27:05.320 --> 00:27:08.800
Actually, the question doesn't ask for the k-step answer.
347
00:27:08.800 --> 00:27:11.120
It just jumps right away to infinity.
348
00:27:11.120 --> 00:27:18.080
But how would I express the solution after k steps?
349
00:27:18.080 --> 00:27:24.160
It would be some multiple of the first eigenvalue
350
00:27:24.160 --> 00:27:31.440
times the first eigenvector plus some other multiple
351
00:27:31.440 --> 00:27:34.720
of the second eigenvalue times its eigenvector
352
00:27:34.720 --> 00:27:41.280
and some multiple of the third eigenvalue times its eigenvector.
353
00:27:41.280 --> 00:27:42.280
OK.
354
00:27:42.280 --> 00:27:45.120
Good.
355
00:27:45.120 --> 00:27:55.640
And these eigenvalues are 0, 1, and minus 0.2.
356
00:27:55.640 --> 00:27:58.360
So what happens as k goes to infinity?
357
00:28:01.360 --> 00:28:04.400
The only thing that survives the steady state,
358
00:28:04.400 --> 00:28:09.680
so at u infinity, this is gone.
359
00:28:09.680 --> 00:28:16.280
And this is gone. All that's left is c2x2.
360
00:28:16.280 --> 00:28:21.920
So I better find x2.
361
00:28:21.920 --> 00:28:25.760
I've got to find that eigenvector to complete the answer.
362
00:28:25.760 --> 00:28:29.320
What's the eigenvector that corresponds to lambda equal 1?
363
00:28:29.320 --> 00:28:32.560
That's the key eigenvector in any Markov process
364
00:28:32.560 --> 00:28:35.360
is that eigenvector.
365
00:28:35.360 --> 00:28:37.480
Lambda equal 1 is an eigenvalue.
366
00:28:37.480 --> 00:28:40.560
I need a eigenvector x2.
367
00:28:40.560 --> 00:28:42.720
And then I need to know how much of it
368
00:28:42.720 --> 00:28:47.800
is in the starting vector u0.
369
00:28:47.800 --> 00:28:48.240
OK.
370
00:28:48.240 --> 00:28:50.840
So how do I find that eigenvector?
371
00:28:50.840 --> 00:28:54.400
I guess I subtract 1 from the diagonal.
372
00:28:54.400 --> 00:29:01.080
So I have minus 0.8, minus 0.6.
373
00:29:01.080 --> 00:29:09.360
And the rest, of course, is just still 0.4.4.3.
374
00:29:09.360 --> 00:29:13.040
And hopefully that's a singular matrix.
375
00:29:13.040 --> 00:29:22.960
So I'm looking to solve a minus i equal the a minus i x equals 0.
376
00:29:22.960 --> 00:29:23.400
Let's see.
377
00:29:23.400 --> 00:29:26.120
Can anybody spot the solution here?
378
00:29:26.120 --> 00:29:26.560
I don't know.
379
00:29:26.560 --> 00:29:28.120
I didn't make it easy for myself.
380
00:29:28.120 --> 00:29:33.680
But what do you think there?
381
00:29:33.680 --> 00:29:37.840
Maybe those, and I'm just thinking aloud here.
382
00:29:37.840 --> 00:29:38.800
Let me guess.
383
00:29:38.800 --> 00:29:42.880
Those first two entries might be, oh no.
384
00:29:42.880 --> 00:29:44.600
What do you think?
385
00:29:44.600 --> 00:29:47.760
Anybody see it?
386
00:29:47.760 --> 00:29:52.360
We could use elimination if we were desperate.
387
00:29:52.360 --> 00:29:54.840
Are we that desperate?
388
00:29:54.840 --> 00:29:57.920
Anybody just call out if you see the vector
389
00:29:57.920 --> 00:30:01.400
here that's in that null space?
390
00:30:01.400 --> 00:30:03.560
Is there better be a vector in that null space,
391
00:30:03.560 --> 00:30:04.920
or I'm quitting?
392
00:30:10.840 --> 00:30:12.480
OK.
393
00:30:12.480 --> 00:30:15.680
Well, I guess we could use elimination.
394
00:30:15.680 --> 00:30:18.360
Seems.
395
00:30:18.360 --> 00:30:21.160
I thought maybe somebody might see it from further away.
396
00:30:24.320 --> 00:30:26.360
Do you think, is there a chance of these guys
397
00:30:26.360 --> 00:30:30.280
or could it be that these two are equal?
398
00:30:30.280 --> 00:30:35.080
And this is whatever it takes, like something like 332.
399
00:30:35.080 --> 00:30:37.080
Would that possibly work?
400
00:30:37.080 --> 00:30:38.800
I mean, that's great for this.
401
00:30:38.800 --> 00:30:40.400
No, it's not that great.
402
00:30:40.400 --> 00:30:43.080
334.
403
00:30:43.080 --> 00:30:47.200
Thi24 --> 00:31:51.80
That would be the u infinity.
427
00:31:51.800 --> 00:31:52.400
OK.
428
00:31:52.400 --> 00:31:55.720
So I use there in that process sort of the main fact
429
00:31:55.720 --> 00:32:01.520
about Markov matrices to get a jump on the answer.
430
00:32:01.520 --> 00:32:03.680
OK.
431
00:32:03.680 --> 00:32:04.880
Let's see.
432
00:32:04.880 --> 00:32:10.000
OK, here's some kind of quick short questions.
433
00:32:10.000 --> 00:32:15.560
Maybe I'll move over to this board and leave that for the moment.
434
00:32:15.560 --> 00:32:19.360
I'm looking for 2 by 2 matrices.
435
00:32:19.360 --> 00:32:22.120
And I'll read out the property I want,
436
00:32:22.120 --> 00:32:27.040
and you give me an example or tell me there isn't such a matrix.
437
00:32:27.040 --> 00:32:28.040
All right.
438
00:32:28.040 --> 00:32:28.640
Here we go.
439
00:32:28.640 --> 00:32:32.480
First, so 2 by 2s.
440
00:32:32.480 --> 00:32:40.120
First, I want a projection onto the line
441
00:32:40.120 --> 00:32:45.280
through a equals 4 minus 3.
442
00:32:49.720 --> 00:32:53.080
So it's a 1d, a 1-dimensional projection matrix
443
00:32:53.080 --> 00:32:53.920
I'm looking for.
444
00:32:57.040 --> 00:32:59.800
And what's the formula for it?
445
00:32:59.800 --> 00:33:02.280
What's the formula for the projection matrix
446
00:33:02.280 --> 00:33:05.120
p onto a line through a?
447s is deeper mathematics you're watching now.
404
00:30:47.200 --> 00:30:48.480
334.
405
00:30:48.480 --> 00:30:50.720
That's that.
406
00:30:50.720 --> 00:30:51.480
It works.
407
00:30:51.480 --> 00:30:52.320
Don't mess with it.
408
00:30:52.320 --> 00:30:52.840
It works.
409
00:30:52.840 --> 00:30:56.440
OK, it works.
410
00:30:56.440 --> 00:30:57.360
All right.
411
00:30:57.360 --> 00:30:59.520
And yes, OK.
412
00:30:59.520 --> 00:31:05.000
And so that's x2, 334.
413
00:31:05.000 --> 00:31:15.120
And how much of that vector is in the starting vector?
414
00:31:15.120 --> 00:31:19.680
Well, we could go through a complicated process,
415
00:31:19.680 --> 00:31:22.960
but what's the beauty of Markov's things?
416
00:31:22.960 --> 00:31:27.600
That the total number of the total population,
417
00:31:27.600 --> 00:31:31.160
the sum of these, doesn't change.
418
00:31:31.160 --> 00:31:33.480
That the total number of people, they're moving around,
419
00:31:33.480 --> 00:31:38.400
but they don't get born or dead.
420
00:31:38.400 --> 00:31:40.960
So there's 10 of them at the start.
421
00:31:40.960 --> 00:31:42.120
So there's 10 of them there.
422
00:31:42.120 --> 00:31:44.040
So c2 is actually 1.
423
00:31:44.040 --> 00:31:44.840
Yeah.
424
00:31:44.840 --> 00:31:48.720
So that would be the correct solution.
425
00:31:48.720 --> 00:31:49.240
OK.
426
00:31:49.
00:33:05.120 --> 00:33:08.320
And then we'll just plug in this particular a.
448
00:33:08.320 --> 00:33:11.520
Remember that formula?
449
00:33:11.520 --> 00:33:15.720
There's an a and an a transpose.
450
00:33:15.720 --> 00:33:20.560
And normally we would have an a transpose a inverse in the middle.
451
00:33:20.560 --> 00:33:21.960
But here we just got numbers.
452
00:33:21.960 --> 00:33:24.280
So we just divide by it.
453
00:33:24.280 --> 00:33:28.760
And then plug in a and we've got it.
454
00:33:28.760 --> 00:33:29.360
OK.
455
00:33:29.360 --> 00:33:31.880
So equals.
456
00:33:31.880 --> 00:33:33.960
You can put in the numbers.
457
00:33:33.960 --> 00:33:35.320
Trivial.
458
00:33:35.320 --> 00:33:36.040
OK.
459
00:33:36.040 --> 00:33:37.800
Number 2.
460
00:33:37.800 --> 00:33:41.920
The matrix that has eigenvalue 0.
461
00:33:41.920 --> 00:33:43.560
And so this is a new problem.
462
00:33:43.560 --> 00:33:47.840
The matrix with eigenvalue 0 and 3.
463
00:33:47.840 --> 00:33:51.280
And eigenvectors, well, let me write these down.
464
00:33:51.280 --> 00:34:02.760
eigenvalue 0, eigenvector 1, 2, eigenvalue 3, eigenvector 2, 1.
465
00:34:02.760 --> 00:34:05.600
I'm giving you the eigenvalues and eigenvectors
466
00:34:05.600 --> 00:34:07.720
instead of asking for them.
467
00:34:07.720 --> 00:34:09.200
Now I'm asking for the matrix.
468
00:34:13.720 --> 00:34:15.240
What's the matrix then?
469
00:34:15.240 --> 00:34:15.880
What's a?
470
00:34:20.520 --> 00:34:22.040
What's the here with the formula?
471
00:34:22.040 --> 00:34:23.840
Then we just put in some numbers.
472
00:34:23.840 --> 00:34:25.320
What's the formula here?
473
00:34:25.320 --> 00:34:29.960
Into which we'll just put the given numbers.
474
00:34:29.960 --> 00:34:34.520
It's the s lambda s inverse, right?
475
00:34:34.520 --> 00:34:39.520
So it's s, which is this eigenvector matrix.
476
00:34:39.520 --> 00:34:44.800
It's the lambda, which is the eigenvalue matrix.
477
00:34:44.800 --> 00:34:48.520
It's the s inverse, whatever that turns out to be.
478
00:34:48.520 --> 00:34:50.240
Let me just leave it as inverse.
479
00:34:53.200 --> 00:34:55.760
That has to be it, right?
480
00:34:55.760 --> 00:34:58.320
Because if we went in the other direction,
481
00:34:58.320 --> 00:35:03.920
that matrix s would diagonalize a to produce lambda.
482
00:35:03.920 --> 00:35:06.080
So it's s lambda s inverse.
483
00:35:06.080 --> 00:35:07.320
Good.
484
00:35:07.320 --> 00:35:08.120
OK.
485
00:35:08.120 --> 00:35:09.880
Ready for number three?
486
00:35:09.880 --> 00:35:17.360
A real matrix that cannot be factored into a,
487
00:35:17.360 --> 00:35:23.160
I'm looking for matrix A, that never could equal b transpose b
488
00:35:23.160 --> 00:35:23.960
for any b.
489
00:35:27.760 --> 00:35:31.680
A 2 by 2 matrix that could not be factored in the form b
490
00:35:31.680 --> 00:35:34.400
transpose b.
491
00:35:34.400 --> 00:35:37.320
So all you have to do is think, well, what does b transpose b
492
00:35:37.320 --> 00:35:41.400
look like and then pick something different?
493
00:35:41.400 --> 00:35:42.640
What do you suggest?
494
00:35:46.560 --> 00:35:47.560
Let's see.
495
00:35:47.560 --> 00:35:52.240
What should we take from matrix that could not have this form
496
00:35:52.240 --> 00:35:53.320
b transpose b?
497
00:35:53.320 --> 00:35:55.680
Well, what do we know about b transpose b?
498
00:35:55.680 --> 00:35:58.000
It's always symmetric.
499
00:35:58.000 --> 00:36:00.200
So just give me any non-symmetric matrix
500
00:36:00.200 --> 00:36:02.400
that couldn't possibly have that form.
501
00:36:02.400 --> 00:36:03.160
OK.
502
00:36:03.160 --> 00:36:06.040
And let me ask the fourth part of this question.
503
00:36:06.040 --> 00:36:11.240
A matrix that has orthogonal eigenvectors,
504
00:36:11.240 --> 00:36:14.560
but it's not symmetric.
505
00:36:14.560 --> 00:36:15.760
Tell me a matrix.
506
00:36:15.760 --> 00:36:16.800
How could a matrix?
507
00:36:16.800 --> 00:36:20.120
What matrices have orthogonal eigenvectors?
508
00:36:20.120 --> 00:36:21.760
But they're not symmetric matrices.
509
00:36:24.560 --> 00:36:27.960
What other matrices tell me other families
510
00:36:27.960 --> 00:36:32.280
of matrices that have orthogonal eigenvectors?
511
00:36:32.280 --> 00:36:36.680
We know symmetric matrices do, but others also.
512
00:36:36.680 --> 00:36:45.120
So I'm looking for orthogonal eigenvectors
513
00:36:45.120 --> 00:36:46.840
and what do you suggest?
514
00:36:51.680 --> 00:36:55.560
The matrix could be skew symmetric.
515
00:36:55.560 --> 00:36:58.400
It could be an orthogonal matrix.
516
00:36:58.400 --> 00:37:03.680
So it could be symmetric, but that was too easy.
517
00:37:03.680 --> 00:37:05.240
So I ruled that out.
518
00:37:05.240 --> 00:37:15.200
It could be skew symmetric, like 1 minus 1 like that.
519
00:37:15.200 --> 00:37:19.360
Or it could be an orthogonal matrix.
520
00:37:19.360 --> 00:37:25.080
It could be an orthogonal matrix like cosine sine minus sine
521
00:37:25.080 --> 00:37:26.440
cosine.
522
00:37:26.440 --> 00:37:34.960
All those matrices would have complex orthogonal eigenvectors.
523
00:37:34.960 --> 00:37:37.240
But they would be orthogonal.
524
00:37:37.240 --> 00:37:39.760
And so those examples are fine.
525
00:37:39.760 --> 00:37:40.760
OK.
526
00:37:40.760 --> 00:37:51.480
We can continue a little longer if we would like to with these
527
00:37:51.480 --> 00:37:54.200
from this exam, from these exams.
528
00:37:54.200 --> 00:37:55.200
Least squares.
529
00:37:57.800 --> 00:37:58.160
OK.
530
00:37:58.160 --> 00:38:02.200
Here's a least squares problem in which to make life quick.
531
00:38:02.200 --> 00:38:04.600
I've given the answer.
532
00:38:04.600 --> 00:38:07.280
It's like jeopardy, right?
533
00:38:07.280 --> 00:38:10.480
I just give the answer and you give the question.
534
00:38:10.480 --> 00:38:12.640
OK.
535
00:38:12.640 --> 00:38:13.160
Oops.
536
00:38:16.160 --> 00:38:17.080
Sorry.
537
00:38:17.080 --> 00:38:17.280
Let's see.
538
00:38:17.280 --> 00:38:25.400
And I stay over here for the next question.
539
00:38:25.400 --> 00:38:26.520
OK.
540
00:38:26.520 --> 00:38:27.240
Least squares.
541
00:38:27.240 --> 00:38:29.920
So I'm giving you the problem.
542
00:38:29.920 --> 00:38:38.400
1, 1, 1, 0, 1, 2, 3, d equals 3, 4, 1.
543
00:38:38.400 --> 00:38:40.520
And that's b, of course.
544
00:38:40.520 --> 00:38:44.680
This is a x equal b.
545
00:38:44.680 --> 00:38:47.320
And the least squares solution.
546
00:38:47.320 --> 00:38:50.720
So may I put c hat, d hat, to emphasize
547
00:38:50.720 --> 00:38:54.080
it's not the true solution has lead.
548
00:38:54.080 --> 00:38:56.960
So the least squares solution, maybe I should,
549
00:38:56.960 --> 00:39:02.760
the hats really go here, is 11, 3rd, and minus 1.
550
00:39:02.760 --> 00:39:05.880
Of course, you could have figured that out in no time.
551
00:39:05.880 --> 00:39:09.800
So this theorem, I'll ask you to do it probably.
552
00:39:09.800 --> 00:39:15.520
But suppose we're given the answer, then let's just
553
00:39:15.520 --> 00:39:17.680
remember what happened.
554
00:39:17.680 --> 00:39:19.360
What is the projection?
555
00:39:19.360 --> 00:39:20.480
Good question.
556
00:39:20.480 --> 00:39:24.640
What's the projection p of this vector
557
00:39:24.640 --> 00:39:28.760
onto the column space of that matrix?
558
00:39:28.760 --> 00:39:32.600
What, so I'll write that question down.
559
00:39:32.600 --> 00:39:33.760
What?
560
00:39:33.760 --> 00:39:37.360
What is p, the projection?
561
00:39:37.360 --> 00:39:45.960
The projection of v onto the column space of a is what?
562
00:39:49.240 --> 00:39:54.480
Hopefully, that's what the least squares problem solved.
563
00:39:54.480 --> 00:39:55.320
What is it?
564
00:39:58.840 --> 00:40:01.920
This was the best solution.
565
00:40:01.920 --> 00:40:11.360
It's 11, 3rd times column 1 plus or rather minus 1 times column
566
00:40:11.360 --> 00:40:12.760
2.
567
00:40:12.760 --> 00:40:15.040
That's what the least squares did.
568
00:40:15.040 --> 00:40:18.960
It found the combination of the column that
569
00:40:18.960 --> 00:40:20.880
was as close as possible to b.
570
00:40:20.880 --> 00:40:23.360
That's what the squares was doing.
571
00:40:23.360 --> 00:40:24.960
It found the projection.
572
00:40:24.960 --> 00:40:26.480
OK.
573
00:40:26.480 --> 00:40:29.920
Secondly, draw the straight line problem
574
00:40:29.920 --> 00:40:32.560
that corresponds to this system.
575
00:40:32.560 --> 00:40:36.600
So I guess that the straight line fitting a straight line
576
00:40:36.600 --> 00:40:39.760
problem, we kind of recognize.
577
00:40:39.760 --> 00:40:42.360
We recognize these are the heights.
578
00:40:42.360 --> 00:40:44.440
And these are the points.
579
00:40:44.440 --> 00:40:49.160
So at 0, 1, 2, the heights are 3.
580
00:40:49.160 --> 00:40:55.560
And at t equal to 1, the height is 4, 1, 2, 3, 4.
581
00:40:55.560 --> 00:41:00.800
And at t equal to 2, the height is 1.
582
00:41:00.800 --> 00:41:06.480
So I'm trying to fit the best straight line
583
00:41:06.480 --> 00:41:09.120
through those points.
584
00:41:09.120 --> 00:41:12.320
God.
585
00:41:12.320 --> 00:41:14.280
I could fit a triangle very well.
586
00:41:14.280 --> 00:41:17.080
But where's the best?
587
00:41:17.080 --> 00:41:19.760
I don't even know which way the best straight line goes.
588
00:41:19.760 --> 00:41:22.160
Maybe it goes, oh, I didn't know how it goes,
589
00:41:22.160 --> 00:41:23.040
because there's the answer.
590
00:41:23.040 --> 00:41:24.640
Yes.
591
00:41:24.640 --> 00:41:30.520
It has a height 11, 3rd, and it has slope minus 1.
592
00:41:30.520 --> 00:41:32.640
So it's something like that.
593
00:41:32.640 --> 00:41:33.240
OK.
594
00:41:33.240 --> 00:41:34.600
Great.
595
00:41:34.600 --> 00:41:41.400
Now, finally, and this completes the course,
596
00:41:41.400 --> 00:41:46.080
find a different vector b, not all 0s,
597
00:41:46.080 --> 00:41:50.040
for which the least square solution would be 0.
598
00:41:50.040 --> 00:41:55.560
So I want you to find a different b
599
00:41:55.560 --> 00:41:59.920
so that the least square solution changes to all 0s.
600
00:42:05.280 --> 00:42:09.120
So tell me what I'm really looking for here.
601
00:42:09.120 --> 00:42:12.720
I'm looking for a b, where the best combination
602
00:42:12.720 --> 00:42:16.520
of these two columns is the 0 combination.
603
00:42:16.520 --> 00:42:20.000
So what kind of a vector b am I looking for?
604
00:42:20.000 --> 00:42:21.560
I'm looking for a vector b that's
605
00:42:21.560 --> 00:42:24.040
orthogonal to those columns.
606
00:42:24.040 --> 00:42:25.640
It's orthogonal to those columns.
607
00:42:25.640 --> 00:42:27.760
It's orthogonal to the column space.
608
00:42:27.760 --> 00:42:30.400
The best possible answer is 0.
609
00:42:30.400 --> 00:42:33.720
So a vector b that's orthogonal to those columns,
610
00:42:33.720 --> 00:42:38.800
let's see, maybe one of those minus two of those,
611
00:42:38.800 --> 00:42:43.480
and one of those, that would be orthogonal to those columns.
612
00:42:43.480 --> 00:42:47.520
And the best vector would be 0, 0.
613
00:42:47.520 --> 00:42:48.280
OK.
614
00:42:48.280 --> 00:42:51.440
So that's as many questions as I can do in an hour,
615
00:42:51.440 --> 00:42:53.360
but you get three hours.
616
00:42:53.360 --> 00:42:58.600
And let me just say, as I've said by email,
617
00:42:58.600 --> 00:43:00.920
thanks very much for your patience
618
00:43:00.920 --> 00:43:05.240
as this series of lectures was videotaped.
619
00:43:05.240 --> 00:43:09.360
And thanks for filling out these forms.
620
00:43:09.360 --> 00:43:13.120
Maybe just leave them on the table up there as you go out.
621
00:43:13.120 --> 00:43:16.000
And above all, thanks for taking the course.
622
00:43:16.000 --> 00:43:17.200
Thank you.
623
00:43:17.200 --> 00:43:18.680
Thank you.
624
00:43:18.680 --> 00:43:47.160
Thank you.