WEBVTT
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Text as soon as the
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stairs start.
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Fixed
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XXXVI
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I'm going to go to the other side.
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He's going to go to the other side.
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He's going to get at the start.
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He's going to get at the start.
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And the family is going to raise his hand
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and take over what I know on.
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Just a minute, though.
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Let him settle.
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Okay, Johnny.
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Okay, give me this thing a little bit when you want me to start.
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Okay, this is living around for about a minute, like so forth.
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And the first thing I have to do is something that was all on the list for the last time,
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but here it is now, once the inverse of a product.
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If I multiply two matrices together and I know there is versus how I get the inverse of a times a.
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So I know what inverse of b for a single matrix a and for a matrix b,
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what matrix do I multiply by the x, the identity if I have a.
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Okay, that will be simple, but so basic.
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Then I'm going to use that to I will have a product of matrix b.
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And the product that will be these of the matrix b.
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And the next result of today's vector is the base formula for the matrix.
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So the next result of today's vector is this.
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It is a great way to look at Gaussian elimination.
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We know that we get from a to u by elimination.
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We know that this step, but now we get the right way to look at it, a, e, for L, u.
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So that's the high point for today.
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Can I say that e is the first step first? So suppose a is invertible.
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And of course it's going to be a big question.
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When is the matrix invertible?
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But let's say a is invertible and b is invertible.
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Then what matrix gives me the inverse of a b?
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So that's the direct question.
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What's the inverse of a b?
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Do I multiply the most separate inverses?
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Yes.
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I multiply the two matrixes a and inverse and b and inverse.
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But what over here do I multiply?
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In reverse order.
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And you see why?
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So the right thing to put here is b and inverse a.
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That's the inverse of matrix.
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And we can just take the a b times that matrix gives the identity.
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OK.
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So why?
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Once again it's the fact that I can move for everything around.
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I can actually, I can just erase them all and do the multiplications any way I want to.
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So what's the right multiplications to do first?
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b times the inverse.
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This product here is the identity.
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Then a times the identity is the identity.
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And then finally a times the inverse gives the identity.
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So again the dumb example in the book is just why do you do that?
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Inverse things in reverse order.
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It's just like taking it, take off your shoes, take off your socks.
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Then the good way to invert that product is to back on first then.
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Sorry.
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OK.
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Sorry, that's OK.
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But sorry.
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And of course on the other side we should really check on the other side.
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I have b inverse and inverse.
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That does multiply a b.
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And this time the C guy just gives the identity, sweep down, make it the identity, we're in sync.
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OK.
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So there's the everything.
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Good.
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Well, we're at it.
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Let me do a transpose.
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So the next lecture has got a lot to involve transpose.
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So how do I, if I transpose the matrix, I'm talking about square and vertical matrices right now.
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If I transpose one, what's it in reverse?
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Well, the nice one is, let's take it.
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Let me start from a a and root equal the identity.
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And let me transpose all stuff.
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That will bring a transpose into the picture.
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So if I transpose the identity matrix, what do I have?
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The identity, right?
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If I exchange, roll in times, the identity is a symmetric matrix that doesn't know the difference.
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If I transpose these guys, that's right.
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Then again, it turns out that I have to reverse the order.
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I can transpose them separately, but when I multiply those transpose that come in the opposite order.
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So it's a inverse transpose times a transpose, giving the identity.
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So that's this equation.
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It just comes directly from that one.
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But this equation tells me what I wanted to know.
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And then, what is the inverse of this guy, a transverse?
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What's the inverse of the result?
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And this is quite a result.
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Here it is.
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This is the inverse of a transverse.
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The inverse of a transverse.
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Well, a transverse.
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I just, so I'll point a big circle around that, because that's the answer we could take for.
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If we want to know the inverse of a transverse, and you know the inverse of a, then you just transpose that.
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So in other words, to put it in another way, transpose the inverse thing you can do with either order for a single order.
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Okay.
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Well, these are like basic facts that we can now use.
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So now I put it to you.
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I put it to you by thinking we're really completely the subject of elimination.
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Actually, I, let's think about elimination here.
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It's the right way to understand what the matrix has got.
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This A for L U is the most basic fact of elimination of a matrix.
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I always worry that you will think that the court is all elimination.
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It's just roll-off race.
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And please don't.
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It will be beyond that.
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So it's the right, it's the right, the algorithm is in first.
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Okay.
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So now I'm coming near the end of it, but I want to get it into these at four.
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So my piece at four is matrix four.
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I have a matrix A.
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Let's suppose it's a good matrix.
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I can do elimination.
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No low exchanges.
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So no low exchanges for now.
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Pivots all find nothing zero in the pivot position.
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I guess in the very end, which is you.
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So I guess from A to U.
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And I want to know what's the connection?
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How is A related to you?
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And this is going to tell me that there's a matrix L that's connected.
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Okay.
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Can I do it for two by two?
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Okay. Two by two.
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Eliminating.
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Okay.
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So I'll do an underview.
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So that's my matrix A, B.
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Let's just, let's see for example.
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Say two and an eight.
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So we know that the first evidence is two.
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And the multiplier is going to be up four.
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And then that means one here.
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And what number do I not want to put there?
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Four.
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I don't want to pour there because in that case,
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the second pivot would not, we wouldn't have a second pivot.
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The matrix would be singular general.
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Okay.
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So let me put another number here like that.
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Okay.
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Okay.
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Now I want to operate on that with my elementary matrix.
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So what the elementary matrix and I'll strictly be speaking at E21
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because it's the guy that's going to produce a zero in that position.
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And it's going to produce you in one shot because it's the two by two matrix.
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So two one.
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And I'm going to take four of those away from those.
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So produce that zero and leave a three there and that's you.
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And what's the matrix-positives?
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Quick review then.
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What's the elimination of elementary matrix E21?
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If one's the real thing and negative four, one.
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Right.
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Good.
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Okay.
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So that's the fact that you see the difference between this and one two before.
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I'm shooting for A on one side and the other matrix are things on the other side of the place.
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Okay.
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So I can do that right away.
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Now here's going to be my A equal to LV and you won't have any trouble telling me what.
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So A is still two one eight seven.
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L is what you're going to tell me and you are still two one zero three.
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Okay.
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So what's L in this case?
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Well, first, so how is L related to this E?
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It's re-experts because I want to multiply it through by an inverse of it,
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which will put the identity here and the inverse will show up there and I'll call it L.
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So what is the inverse of this?
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Remember those elimination matrices are easy to invert.
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The inverse matrix for this one is 104.
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It has a plus sign because it adds that to what this remote.
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Okay.
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If we did the number right, this should be correct.
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Okay.
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And of course it is.
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That says the first row is right.
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Four times the first row plus the second row is 8.
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Good.
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Okay.
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That's the two I choose.
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This shows the form that we're heading for.
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It shows what the L has been for.
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Why the letter L?
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If you chose for upper triangular, then of course L stands for lower triangular.
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And actually it has ones on the side where this thing has a cubic on the side.
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Oh, sometimes we may want to separate out the pivot.
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So can I just mention that sometimes we could also write this as we could have this 1041.
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It shows how I would divide out this matrix up here.
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Two, three.
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There's a diagonal matrix.
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And whatever is left is here, what's left?
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If I divide this first row by two to pull out the two, then I have one and one half.
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And if I divide the second row by three to pull out the three, then I don't want.
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So that's the, if this is L, U, this is maybe called L, B for pivot, U.
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And now it's a little more balanced because we have ones on the diagonal here and here.
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And the diagonal matrix in the middle.
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So both of those, that might have to reduce either one.
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I'll basically stay with L U.
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Okay.
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Now I actually think about bigger than two by two.
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Right now this is just like the easy exercise.
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And the title of two, that this one was the minus sign and this one was the plus sign.
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I mean that's the only difference.
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But, which three by three, there's a more significant difference.
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That means, let me show you how that works.
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That means, let me move up to a three by three.
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Let's say some matrix A.
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Okay.
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It's the magic in its three by three.
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I won't write numbers down for now.
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So what's the first elimination step that I do?
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The first matrix I multiply it by what letter will I use for that?
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It's a big E to 1.
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The first step will be to get a new row in that two ones with this.
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And then the next step will be to get a zero in the three ones with this.
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And the final step will be to get a zero in the three two with this.
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That's what a elimination is and it's a new.
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And again, no, we'll row it.
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We will...
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I'm taking the nice piece now, the typical case two.
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When I don't have to do any row exchange, all I do is be until the next one is there.
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Okay.
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Now, suppose I want that stuff over on the right hand side.
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And as I really do, that's like my point here.
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I can multiply these together to get a matrix E, but I want it over on the right.
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I want it inverse over there.
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So what's the right expression now?
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If I write A and U, what's over there?
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Okay.
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So I've got the inverse of this.
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I've got three matrices in the row now.
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And if there are invoices that are going to show up, because each one is easy to invert.
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Question is, what about the whole bunch?
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How easy is it to invert the whole bunch?
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That's what we know how to do.
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We know how to invert.
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We should take some separate invoices, but they go in the opposite order.
242
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So what goes here?
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E, three two, inverse.
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Right.
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I'll multiply from the left by E, three two, inverse.
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I'll pop it up next to U.
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And then we'll come E, three one.
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And then this will be, I mean, this will be the only guy left standing.
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And that's gone when I do an E, two one.
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So there is L. That's L, U.
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L is this product of inverse.
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Now you fill it and ask why does this guy separate inverse?
253
00:17:49.960 --> 00:17:51.960
And let me explain why.
254
00:17:51.960 --> 00:17:57.960
Let me explain why is this product nicer than this one?
255
00:17:57.960 --> 00:18:02.960
This product turns out to be better than this one.
256
00:18:02.960 --> 00:18:06.960
Let me take a typical case here.
257
00:18:06.960 --> 00:18:14.960
So let me, I have to do three by three for you to see the infertile.
258
00:18:14.960 --> 00:18:19.960
Two by two, there was just one E, no problem.
259
00:18:19.960 --> 00:18:21.960
But let me go up to this case.
260
00:18:21.960 --> 00:18:26.960
Suppose my matrix is E, two one.
261
00:18:26.960 --> 00:18:33.960
Suppose E, two one has a minus two in there.
262
00:18:33.960 --> 00:18:41.960
Suppose that, and now suppose, oh, I don't even suppose E, three one is the identity.
263
00:18:41.960 --> 00:18:46.960
So I'm just going to, I'm going to make the point with just a couple of things.
264
00:18:46.960 --> 00:18:50.960
Okay, now this guy will have two sub-saying.
265
00:18:50.960 --> 00:18:53.960
Now let's suppose my spot.
266
00:18:53.960 --> 00:18:54.960
One.
267
00:18:54.960 --> 00:18:59.960
Okay, there's the test.
268
00:18:59.960 --> 00:19:03.960
That's a typical case in which we didn't need an E3 one.
269
00:19:03.960 --> 00:19:06.960
Maybe we already have a zero on that three one.
270
00:19:06.960 --> 00:19:09.960
Okay.
271
00:19:09.960 --> 00:19:16.960
Let me see if that, is that going to be enough to show my point?
272
00:19:16.960 --> 00:19:21.960
Let me do that multiplication.
273
00:19:21.960 --> 00:19:26.960
Okay, so if I do that multiplication, this is like a subtractor for multiple identity matrices.
274
00:19:26.960 --> 00:19:31.960
Tell me what's above the diagonal when I do this multiplication.
275
00:19:31.960 --> 00:19:33.960
Oh, there.
276
00:19:33.960 --> 00:19:40.960
When I do this multiplication, I'm going to get one on the diagonal and zero's above.
277
00:19:40.960 --> 00:19:42.960
Because what does that say?
278
00:19:42.960 --> 00:19:46.960
That says that I'm subtracting rows from lower rows.
279
00:19:46.960 --> 00:19:52.960
So nothing is moving upwards, but it did last time in Gauss-Storting.
280
00:19:52.960 --> 00:19:59.960
Okay, now so really what I have to do is check this minus 2, 1, 0.
281
00:19:59.960 --> 00:20:03.960
Now this is, so what's that number?
282
00:20:03.960 --> 00:20:07.960
This is the number that I really have in mind.
283
00:20:07.960 --> 00:20:11.960
That number is 10.
284
00:20:11.960 --> 00:20:21.960
And this one is what those here row three against column two looks like the minus five.
285
00:20:21.960 --> 00:20:26.960
What?
286
00:20:26.960 --> 00:20:29.960
If that's 10, how did that 10 get in there?
287
00:20:29.960 --> 00:20:31.960
I don't like that 10.
288
00:20:31.960 --> 00:20:38.960
I mean, of course I don't want to erase it, so it's right, but I don't want it in there.
289
00:20:38.960 --> 00:20:46.960
It's because the 10 got in there because I subtracted two of row one from row two.
290
00:20:46.960 --> 00:20:51.960
And then I subtracted five of that new row two from row three.
291
00:20:51.960 --> 00:20:57.960
So doing it in that order, how did row one affect row three?
292
00:20:57.960 --> 00:21:04.960
When they did, the two of it got removed from row two and then five of those got removed from row three.
293
00:21:04.960 --> 00:21:10.960
So all together 10 of row one got thrown into row three.
294
00:21:10.960 --> 00:21:13.960
Now my point is in the reverse direction.
295
00:21:13.960 --> 00:21:20.960
And now I'll do below it, I'll do the end.
296
00:21:20.960 --> 00:21:28.960
Okay, and of course opposite order, reverse order.
297
00:21:28.960 --> 00:21:34.960
Okay, so now this is going to, this is the, this is the E.
298
00:21:34.960 --> 00:21:40.960
It goes on the left, the left of A.
299
00:21:40.960 --> 00:21:45.960
Now I'm going to do the inverses in the opposite order.
300
00:21:45.960 --> 00:21:50.960
So the opposite order means I took this inverse first.
301
00:21:50.960 --> 00:21:52.960
And what is this inverse?
302
00:21:52.960 --> 00:21:54.960
What's the inverse of E21?
303
00:21:54.960 --> 00:21:57.960
Same thing with a plus sign, right?
304
00:21:57.960 --> 00:22:05.960
For the individual matrix, instead of taking away two, I add back two of row one to row two.
305
00:22:05.960 --> 00:22:13.960
No problem, and now in reverse order, I want to invert that.
306
00:22:13.960 --> 00:22:18.960
Just, just, right, I'm doing this, just, this.
307
00:22:18.960 --> 00:22:28.960
So now the inverse is again the same thing, but add in the box.
308
00:22:28.960 --> 00:22:37.960
And now do that multiplication, and I'll get a happy rhythm.
309
00:22:37.960 --> 00:22:40.960
I hope.
310
00:22:40.960 --> 00:22:42.960
Did I do it right so far?
311
00:22:42.960 --> 00:22:45.960
Yes, okay, let me do the multiplication.
312
00:22:45.960 --> 00:22:46.960
I believe it's something else.
313
00:22:46.960 --> 00:22:49.960
So row one of the answer is 100.
314
00:22:49.960 --> 00:22:53.960
Oh, I know that all this is going to be right.
315
00:22:53.960 --> 00:23:00.960
Then I have 2, 1, 0. So I get 2, 1, 0 there, right?
316
00:23:00.960 --> 00:23:02.960
And what's the third row?
317
00:23:02.960 --> 00:23:07.960
What's the third row in this product?
318
00:23:07.960 --> 00:23:09.960
Let's read it out to me.
319
00:23:09.960 --> 00:23:11.960
The third row.
320
00:23:11.960 --> 00:23:15.960
0, 5, 1.
321
00:23:15.960 --> 00:23:22.960
Because that, the one way to say it, this is, say, take one of the last row, and there it is.
322
00:23:22.960 --> 00:23:28.960
And this is my matrix L, and it's the one that goes to the left.
323
00:23:28.960 --> 00:23:31.960
It goes on the left of U.
324
00:23:31.960 --> 00:23:37.960
It goes into, this is, what do I do here?
325
00:23:37.960 --> 00:23:44.960
So maybe, maybe, rather than say, left of A, left of U, let me write down again what I mean.
326
00:23:44.960 --> 00:23:51.960
E A is U, whereas A is L U.
327
00:23:51.960 --> 00:23:54.960
Okay.
328
00:23:54.960 --> 00:23:58.960
Let me make the point now in words.
329
00:23:58.960 --> 00:24:03.960
The order of the matrix is found for L is the right order.
330
00:24:03.960 --> 00:24:13.960
The 2 and the 5 don't show the inner sphere to produce this fence.
331
00:24:13.960 --> 00:24:19.960
In the right order, the multiplier just fit in the matrix L.
332
00:24:19.960 --> 00:24:22.960
That's the point.
333
00:24:22.960 --> 00:24:28.960
So if I want to know L, I have no work to do.
334
00:24:28.960 --> 00:24:33.960
I just keep a record of what those multipliers work.
335
00:24:33.960 --> 00:24:36.960
And that gives me L.
336
00:24:36.960 --> 00:24:48.960
So let me say, this is the A equal L U.
337
00:24:48.960 --> 00:25:01.960
So if no work changes, the multipliers,
338
00:25:01.960 --> 00:25:09.960
the numbers that we multiply, those five, and subtract it when we did an elimination set,
339
00:25:09.960 --> 00:25:20.960
the multipliers go directly into L.
340
00:25:20.960 --> 00:25:23.960
Okay.
341
00:25:23.960 --> 00:25:33.960
So L is, this is the way to look at elimination.
342
00:25:33.960 --> 00:25:40.960
That you go through the elimination steps.
343
00:25:40.960 --> 00:25:49.960
And actually, if you do it right, you can throw away A as you create L will do.
344
00:25:49.960 --> 00:25:54.960
If you think about it, those steps of elimination,
345
00:25:54.960 --> 00:26:05.960
as you, when you've finished with row 2 of A, you've created a new row 2 of U.
346
00:26:05.960 --> 00:26:07.960
What you have to say.
347
00:26:07.960 --> 00:26:11.960
And you've created the multipliers that you use.
348
00:26:11.960 --> 00:26:13.960
What you have to say.
349
00:26:13.960 --> 00:26:16.960
And then you can forget A.
350
00:26:16.960 --> 00:26:19.960
Because you've all there in L and U.
351
00:26:19.960 --> 00:26:37.960
So that's like, this moment is maybe the new insight in elimination that comes from doing it in matrix form.
352
00:26:37.960 --> 00:26:42.960
So it was a product of E.
353
00:26:42.960 --> 00:26:49.960
We can't see what that product of E is. The matrix E is not particularly attractive.
354
00:26:49.960 --> 00:26:55.960
What's great is when we put them on the other side, they're inverses in the opposite order,
355
00:26:55.960 --> 00:26:58.960
there the L comes out to the problem.
356
00:26:58.960 --> 00:26:59.960
Okay.
357
00:26:59.960 --> 00:27:06.960
Now, now that's, so today is a sort of like practical day.
358
00:27:06.960 --> 00:27:13.960
Can we take together how expensive is elimination?
359
00:27:13.960 --> 00:27:17.960
How much, how many operations do we do?
360
00:27:17.960 --> 00:27:21.960
So what? So this is not a kind of new topic.
361
00:27:21.960 --> 00:27:27.960
If I didn't list as on the program, but here it comes.
362
00:27:27.960 --> 00:27:41.960
How many operations on an N by N matrix?
363
00:27:41.960 --> 00:27:48.960
Hey.
364
00:27:48.960 --> 00:27:51.960
I mean, it's a very practical question.
365
00:27:51.960 --> 00:28:01.960
Can we solve systems of order 1000 in a second or a minute or a week?
366
00:28:01.960 --> 00:28:09.960
Can we solve systems of order of illness in a second or an hour or a week?
367
00:28:09.960 --> 00:28:18.960
I mean, what if it's N by N, we often want to take N bigger.
368
00:28:18.960 --> 00:28:21.960
I mean, we put in more information.
369
00:28:21.960 --> 00:28:26.960
We make this is the whole thing is more accurate for the bigger matrix.
370
00:28:26.960 --> 00:28:31.960
But it's more expensive too and the question is how much more expensive?
371
00:28:31.960 --> 00:28:35.960
If I go, if I don't make the system order 100.
372
00:28:35.960 --> 00:28:37.960
Let's say 100 by 100.
373
00:28:37.960 --> 00:28:40.960
Let me take N to be 100.
374
00:28:40.960 --> 00:28:43.960
Say any problem.
375
00:28:43.960 --> 00:28:48.960
How many steps do I have to do?
376
00:28:48.960 --> 00:28:51.960
How many operations are we actually doing that we deserve?
377
00:28:51.960 --> 00:28:53.960
Or not we?
378
00:28:53.960 --> 00:28:56.960
And let's suppose there are many zeros.
379
00:28:56.960 --> 00:28:58.960
Because that core fits up.
380
00:28:58.960 --> 00:29:01.960
The matrix has got a lot of zeros and good places.
381
00:29:01.960 --> 00:29:03.960
We don't have to do those operations.
382
00:29:03.960 --> 00:29:06.960
And it will be much faster.
383
00:29:06.960 --> 00:29:15.960
So we just think for a moment about the first step.
384
00:29:15.960 --> 00:29:20.960
So here's our matrix A, 100 by 100.
385
00:29:20.960 --> 00:29:28.960
And the first step will be, that's called its got 0 down here.
386
00:29:28.960 --> 00:29:33.960
So it's down to 99 by 99.
387
00:29:33.960 --> 00:29:40.960
That's really like the first stage of elimination.
388
00:29:40.960 --> 00:29:44.960
To get from this 100 by 100 non-zero matrix.
389
00:29:44.960 --> 00:29:48.960
So this stage where the first pivot is sitting up here.
390
00:29:48.960 --> 00:29:50.960
And the first row is okay.
391
00:29:50.960 --> 00:29:53.960
And the first column is okay.
392
00:29:53.960 --> 00:29:58.960
So essentially, what's the how many steps does that take?
393
00:29:58.960 --> 00:30:03.960
It's not, I'm trying to get an idea.
394
00:30:03.960 --> 00:30:05.960
If the answer for force will turn in.
395
00:30:05.960 --> 00:30:08.960
Is the total number of steps in elimination.
396
00:30:08.960 --> 00:30:11.960
The total number is the proportions of the N.
397
00:30:11.960 --> 00:30:17.960
In which case if I double N from 100 to 200, does it take me twice as long?
398
00:30:17.960 --> 00:30:19.960
Does it square?
399
00:30:19.960 --> 00:30:21.960
So it would take me four times as long?
400
00:30:21.960 --> 00:30:22.960
Does it skew?
401
00:30:22.960 --> 00:30:24.960
So it would take me eight times as long?
402
00:30:24.960 --> 00:30:31.960
Or is it N is factorial? So it would take me a hundred times as long?
403
00:30:31.960 --> 00:30:38.960
I think, you know, from a practical point of view, we have to have some idea of the cost here.
404
00:30:38.960 --> 00:30:40.960
So these are the questions.
405
00:30:40.960 --> 00:30:42.960
So let me ask those questions again.
406
00:30:42.960 --> 00:30:43.960
Is it proportional?
407
00:30:43.960 --> 00:30:45.960
Is it going like N?
408
00:30:45.960 --> 00:30:46.960
Like N squared?
409
00:30:46.960 --> 00:30:47.960
Like N cubed?
410
00:30:47.960 --> 00:30:49.960
Like some higher power of N?
411
00:30:49.960 --> 00:30:53.960
Like N factorial?
412
00:30:53.960 --> 00:31:01.960
Where every step out multiplies by 100 and then by 101 and then by 102, which is it?
413
00:31:01.960 --> 00:31:09.960
Okay, so that's the only way I'm known to answer that is to think through what we actually have to do.
414
00:31:09.960 --> 00:31:16.960
Okay, so how much, what was the cost here?
415
00:31:16.960 --> 00:31:24.960
Well, let's see, what do I mean by an operation?
416
00:31:24.960 --> 00:31:27.960
Well, an addition or, yeah, no big deal.
417
00:31:27.960 --> 00:31:35.960
I guess I mean an addition or subtraction or an multiplication or division.
418
00:31:35.960 --> 00:31:41.960
And actually, what operation am I doing all the time?
419
00:31:41.960 --> 00:31:55.960
When I multiply row one by a multiplier L and I subtract from row six, what's happening there individually?
420
00:31:55.960 --> 00:31:57.960
What's going on?
421
00:31:57.960 --> 00:32:02.960
If I multiply, I do a multiplication by L and then it subtracts.
422
00:32:02.960 --> 00:32:09.960
So I guess, operation, can I count that for the moment as like one operation?
423
00:32:09.960 --> 00:32:12.960
You may want to count them separately.
424
00:32:12.960 --> 00:32:20.960
The typical operation is multiply plus a subtract.
425
00:32:20.960 --> 00:32:28.960
So if I count those together, my answer's going to come out half as many as this.
426
00:32:28.960 --> 00:32:32.960
I mean, if I count them separately, I have a certain number of multiplies, certain number of subtracts.
427
00:32:32.960 --> 00:32:34.960
That's really what I want to do.
428
00:32:34.960 --> 00:32:37.960
Okay, how many of I got to?
429
00:32:37.960 --> 00:32:43.960
So I think, let's see.
430
00:32:43.960 --> 00:32:54.960
It's about, well, how many of I got to get from here to here?
431
00:32:54.960 --> 00:33:00.960
Well, maybe one way to look at it is all these numbers have a good change.
432
00:33:00.960 --> 00:33:06.960
The first row didn't get changed, but all the other rows got changed at this step.
433
00:33:06.960 --> 00:33:20.960
Well, I guess maybe, shall I say, it's caused about, this caused about, about 100 squared.
434
00:33:20.960 --> 00:33:26.960
I mean, if I change the first row, that would have been exactly 100 squared.
435
00:33:26.960 --> 00:33:29.960
Because, because that's how many numbers are here.
436
00:33:29.960 --> 00:33:38.960
100 squared numbers is the total count of the entries, and all of us in significant first row got changed.
437
00:33:38.960 --> 00:33:42.960
So I would say about 100 squared.
438
00:33:42.960 --> 00:33:48.960
Okay, now what about the next step?
439
00:33:48.960 --> 00:33:51.960
So now the first row is fine.
440
00:33:51.960 --> 00:33:58.960
The second row is fine, and I'm changing these zeros are all fine.
441
00:33:58.960 --> 00:34:01.960
So what's the second step?
442
00:34:01.960 --> 00:34:03.960
And then you're with me.
443
00:34:03.960 --> 00:34:05.960
Roughly what's the cost?
444
00:34:05.960 --> 00:34:14.960
If it's first step, cause 100 squared about operations, then this one, which is really working on this guy,
445
00:34:14.960 --> 00:34:21.960
to produce this, caused about what?
446
00:34:21.960 --> 00:34:25.960
How many operations to figure?
447
00:34:25.960 --> 00:34:30.960
About 99 squared, or 99 times 98.
448
00:34:30.960 --> 00:34:31.960
But less, right?
449
00:34:31.960 --> 00:34:36.960
Less, cause our problem's getting smaller, about 99 squared.
450
00:34:36.960 --> 00:34:41.960
And then I go down and down, the next one will be 98 squared, the next 97 squared,
451
00:34:41.960 --> 00:34:48.960
and find the answer down around 1 squared, or where, this is like, just little numbers.
452
00:34:48.960 --> 00:34:50.960
The baking numbers are here.
453
00:34:50.960 --> 00:34:58.960
So the number of operations is about n squared plus, that was n, right?
454
00:34:58.960 --> 00:35:00.960
n was 100.
455
00:35:00.960 --> 00:35:07.960
It n squared for the first step, then n minus 1 squared, then n minus 2 squared,
456
00:35:07.960 --> 00:35:17.960
finally down to 3 squared and 2 squared and even 1 squared.
457
00:35:17.960 --> 00:35:22.960
No way I should have written that, squeeze that in.
458
00:35:22.960 --> 00:35:36.960
Let me try it, so the count is n squared plus, n minus 1 squared plus, all the way down to 1 squared.
459
00:35:36.960 --> 00:35:39.960
That's a pretty decent count.
460
00:35:39.960 --> 00:35:51.960
Admittedly, we didn't catch every single tiny operation, but we got the right, the right meeting term here.
461
00:35:51.960 --> 00:35:56.960
And what does that up to?
462
00:35:56.960 --> 00:36:04.960
Okay, so now we're coming to the punch of this question, it's operation time.
463
00:36:04.960 --> 00:36:14.960
So this is the operation on the left side, on the matrix A, to finally get to you.
464
00:36:14.960 --> 00:36:23.960
And anybody, so which of these quantities is the right fault bar for that count?
465
00:36:23.960 --> 00:36:33.960
If I add 100 squared to 99 squared and 98 squared, 97 squared, all the way down to 2 squared and 1 squared,
466
00:36:33.960 --> 00:36:37.960
what am I, what have I got about?
467
00:36:37.960 --> 00:36:42.960
Okay, so it's one of these, so count what's identified first.
468
00:36:42.960 --> 00:36:46.960
Is it n, certainly not?
469
00:36:46.960 --> 00:36:50.960
Is it n factorial?
470
00:36:50.960 --> 00:36:56.960
No, if it was n factorial, we would, with determinants that is n factorial,
471
00:36:56.960 --> 00:37:07.960
put it up, bad mark against determinants because that's, okay, so what is it?
472
00:37:07.960 --> 00:37:12.960
n, well, this is the answer.
473
00:37:12.960 --> 00:37:15.960
It's this order, n cubed.
474
00:37:15.960 --> 00:37:21.960
It's like I have n term, right?
475
00:37:21.960 --> 00:37:28.960
I can answer in this sum and the biggest one is n squared, so the, the worst it could be would be n cubed.
476
00:37:28.960 --> 00:37:39.960
But it's not as bad as n cubed times, it's about 1 third of n cubed.
477
00:37:39.960 --> 00:37:48.960
That's the, that's the magic operation.
478
00:37:48.960 --> 00:37:57.960
Somehow that 1 third takes the count of the fact that the, the numbers are getting smaller.
479
00:37:57.960 --> 00:38:02.960
If they weren't getting smaller, we would have n terms times n squared would be exactly n cubed.
480
00:38:02.960 --> 00:38:05.960
But our numbers are getting smaller.
481
00:38:05.960 --> 00:38:10.960
Actually, do you remember where, where is 1 third in this?
482
00:38:10.960 --> 00:38:14.960
I'll even allow a mention of calculus.
483
00:38:14.960 --> 00:38:22.960
Calculus can be mentioned, integrations can be mentioned now in the next minute and not again for weeks.
484
00:38:22.960 --> 00:38:28.960
It's not that I don't like 18 or 1, but, if you know, it's better.
485
00:38:28.960 --> 00:38:37.960
Okay, so, so what, what's the calculus formula that that looks like?
486
00:38:37.960 --> 00:38:45.960
It looks like a thumb is like, if we were in calculus instead of summing stuff, we would integrate.
487
00:38:45.960 --> 00:38:53.960
So I would integrate x squared and I would get 1 third x cubed.
488
00:38:53.960 --> 00:39:04.960
So, so, if that was like a integral from 1 to n of x squared vx, it's the answer would be 1 third n cubed.
489
00:39:04.960 --> 00:39:09.960
And it's correct for the thumb also, because that's like the whole point of calculus.
490
00:39:09.960 --> 00:39:15.960
The whole point of calculus is, oh, I don't want to tell you the whole, I mean, you know the whole point of calculus.
491
00:39:15.960 --> 00:39:22.960
Calculus is like sum except it's continuous.
492
00:39:22.960 --> 00:39:27.960
Okay, and we're, and algorithm is free.
493
00:39:27.960 --> 00:39:29.960
Okay, so the answer is 1 third n cubed.
494
00:39:29.960 --> 00:39:34.960
Now, let me, let me say one more thing about operation.
495
00:39:34.960 --> 00:39:36.960
What about the right hand sub?
496
00:39:36.960 --> 00:39:39.960
This was what it cost on the left sub.
497
00:39:39.960 --> 00:39:44.960
Now, this is on A.
498
00:39:44.960 --> 00:39:47.960
But this is A that we're working with.
499
00:39:47.960 --> 00:39:55.960
But what's the cost on the extra column vector b that we're hanging around here?
500
00:39:55.960 --> 00:40:02.960
So b, b, it cost a lot less obviously, because it's just one total.
501
00:40:02.960 --> 00:40:09.960
We carry it through elimination, and then actually we do back substitutes.
502
00:40:09.960 --> 00:40:13.960
Let me just tell you the answer there. It's n squared.
503
00:40:13.960 --> 00:40:18.960
So the cost for every right hand side is n squared.
504
00:40:18.960 --> 00:40:32.960
So let me, I'll just fit that in here for the right, for the cost of b of b turns out to be n squared.
505
00:40:32.960 --> 00:40:43.960
So you see, if we have, as we often have, a matrix A and several right hand side,
506
00:40:43.960 --> 00:40:52.960
then we pay the price on A, the higher price on A, to get it split up into L and U,
507
00:40:52.960 --> 00:40:54.960
to do elimination on A.
508
00:40:54.960 --> 00:40:58.960
But then we can process every right hand side at the low top.
509
00:40:58.960 --> 00:41:08.960
Okay, so that, that, we really have discussed the most fundamental algorithm of,
510
00:41:08.960 --> 00:41:14.960
it says for a system of that way. Okay.
511
00:41:14.960 --> 00:41:23.960
So I'm ready to allow rowing stitches.
512
00:41:23.960 --> 00:41:32.960
I'm ready to allow now what happens to this whole today's lecture if there are rowing stitches.
513
00:41:32.960 --> 00:41:36.960
When would there be rowing stitches?
514
00:41:36.960 --> 00:41:42.960
There are rowing, we need to do rowing changes that the zero shows up in the pivot position.
515
00:41:42.960 --> 00:41:50.960
So I'm moving then into the final section of this chapter, which is about transposes.
516
00:41:50.960 --> 00:41:53.960
Well, we've already seen some transposes.
517
00:41:53.960 --> 00:42:00.960
And so what's the title, the title of this section is,
518
00:42:00.960 --> 00:42:09.960
transposes and permutations. Okay.
519
00:42:09.960 --> 00:42:15.960
So can I say now where does the permutation come in?
520
00:42:15.960 --> 00:42:18.960
Let me talk a little about permutations.
521
00:42:18.960 --> 00:42:29.960
So that'll be up here permutation.
522
00:42:29.960 --> 00:42:36.960
So these are the matrices that I need to do rowing changes.
523
00:42:36.960 --> 00:42:39.960
And I may have to do two rowing changes.
524
00:42:39.960 --> 00:42:46.960
Can you invent some matrix where I would have to do two rowing changes?
525
00:42:46.960 --> 00:42:49.960
And then would come out fine.
526
00:42:49.960 --> 00:42:53.960
Yeah, let's put the heck of it.
527
00:42:53.960 --> 00:42:59.960
So I'll put it here. Let me do three by three.
528
00:42:59.960 --> 00:43:05.960
Actually, why don't I just plain the list all the three by three permutation matrices?
529
00:43:05.960 --> 00:43:08.960
There a nice little group of them.
530
00:43:08.960 --> 00:43:15.960
What are all the matrices that exchange no rows at all?
531
00:43:15.960 --> 00:43:21.960
Well, I'll include the identity.
532
00:43:21.960 --> 00:43:25.960
So that's a permutation matrix that doesn't do it.
533
00:43:25.960 --> 00:43:29.960
Now what's the permutation matrix that exchanges?
534
00:43:29.960 --> 00:43:32.960
What is P12?
535
00:43:32.960 --> 00:43:43.960
The permutation matrix that exchanges rows 1 and 2 would be 01010,
536
00:43:43.960 --> 00:43:47.960
I just exchange those rows of the identity.
537
00:43:47.960 --> 00:43:51.960
And I've got it.
538
00:43:51.960 --> 00:43:57.960
Let me not clutter this up.
539
00:43:57.960 --> 00:44:02.960
Okay, give me a complete list of all the rowing changes matrices.
540
00:44:02.960 --> 00:44:03.960
So what are they?
541
00:44:03.960 --> 00:44:09.960
They're all the ways I can take the identity matrix and rearrange this row.
542
00:44:09.960 --> 00:44:12.960
How many will it be?
543
00:44:12.960 --> 00:44:16.960
How many three by three permutations matrices?
544
00:44:16.960 --> 00:44:19.960
So we keep going and get the answer.
545
00:44:19.960 --> 00:44:23.960
So tell me some more.
546
00:44:23.960 --> 00:44:28.960
0, okay. What are you going to do now?
547
00:44:28.960 --> 00:44:32.960
Switch row 1 and 3.
548
00:44:32.960 --> 00:44:36.960
Okay. 1 and 3.
549
00:44:36.960 --> 00:44:39.960
Leaving 2 up.
550
00:44:39.960 --> 00:44:43.960
Okay. Now what else?
551
00:44:43.960 --> 00:44:47.960
Which, what do we do? The next easy one is switch to the next three.
552
00:44:47.960 --> 00:44:50.960
Okay. So these 1, 0, 0 alone.
553
00:44:50.960 --> 00:44:56.960
And I'll switch up, move number 3 up and number 2 down.
554
00:44:56.960 --> 00:45:00.960
Okay. That was one of the just exchange single or single
555
00:45:00.960 --> 00:45:02.960
a pair of rows.
556
00:45:02.960 --> 00:45:06.960
This guy, this guy, this guy, exchanges a pair of rows.
557
00:45:06.960 --> 00:45:12.960
But now there are more possibilities for what's left.
558
00:45:12.960 --> 00:45:15.960
There is another one here.
559
00:45:15.960 --> 00:45:18.960
What's left?
560
00:45:18.960 --> 00:45:19.960
It's going to move.
561
00:45:19.960 --> 00:45:21.960
It's going to change all rows.
562
00:45:21.960 --> 00:45:24.960
Right? Where should we put them?
563
00:45:24.960 --> 00:45:26.960
So give me a first row.
564
00:45:26.960 --> 00:45:28.960
0, 1, 0.
565
00:45:28.960 --> 00:45:37.960
Okay. Now a second row, say 0, 0, 1, and the third guy, 1, 0, 0.
566
00:45:37.960 --> 00:45:40.960
So that is like a cycle.
567
00:45:40.960 --> 00:45:46.960
Next put row 2 moves up to row 1, row 3 moves up to row 2,
568
00:45:46.960 --> 00:45:49.960
and row 1 moves down to row 3.
569
00:45:49.960 --> 00:45:52.960
And there is one more.
570
00:45:52.960 --> 00:45:55.960
Which is, let's see.
571
00:45:55.960 --> 00:45:59.960
What's left?
572
00:45:59.960 --> 00:46:03.960
Is it 0, 1? Okay.
573
00:46:03.960 --> 00:46:05.960
1, 0, 0, 0, 0.
574
00:46:05.960 --> 00:46:07.960
Okay.
575
00:46:07.960 --> 00:46:13.960
Great.
576
00:46:13.960 --> 00:46:14.960
Six.
577
00:46:14.960 --> 00:46:17.960
Six to the other.
578
00:46:17.960 --> 00:46:19.960
Six p.
579
00:46:19.960 --> 00:46:21.960
Out.
580
00:46:21.960 --> 00:46:24.960
And they're pretty nice.
581
00:46:24.960 --> 00:46:31.960
Because what happens if I like multiply two of them together?
582
00:46:31.960 --> 00:46:34.960
If I multiply two of these matrices together,
583
00:46:34.960 --> 00:46:37.960
how can you tell me about the answer?
584
00:46:37.960 --> 00:46:40.960
It's all a little bit.
585
00:46:40.960 --> 00:46:44.960
If I do some more exchanges and then I do some more row exchanges,
586
00:46:44.960 --> 00:46:46.960
I know together I've done row exchanges.
587
00:46:46.960 --> 00:46:50.960
So if I multiply, but I don't know.
588
00:46:50.960 --> 00:46:56.960
And if I invert, then I'm just doing more exchanges to get back again.
589
00:46:56.960 --> 00:46:58.960
So the inverse is a row there.
590
00:46:58.960 --> 00:47:01.960
It's a little family of matrices.
591
00:47:01.960 --> 00:47:07.960
That's like they've got their own.
592
00:47:07.960 --> 00:47:10.960
If I multiply, I'm still inside this group.
593
00:47:10.960 --> 00:47:12.960
If I invert, I'm inside this group.
594
00:47:12.960 --> 00:47:14.960
Actually, the group is the right thing for this.
595
00:47:14.960 --> 00:47:17.960
It's a group of six matrices.
596
00:47:17.960 --> 00:47:20.960
And what about the inverse?
597
00:47:20.960 --> 00:47:23.960
What's the inverse of this guy, for example?
598
00:47:23.960 --> 00:47:25.960
What's the inverse?
599
00:47:25.960 --> 00:47:30.960
If I change rows 1 and 2, what's the inverse matrix?
600
00:47:30.960 --> 00:47:33.960
Is this something bad?
601
00:47:33.960 --> 00:47:41.960
The inverse of that matrix is, if I change rows 1 and 2,
602
00:47:41.960 --> 00:47:45.960
then what I should do is that it's just to get back to where it's started is.
603
00:47:45.960 --> 00:47:47.960
The same thing.
604
00:47:47.960 --> 00:47:50.960
So this thing is its own inverse.
605
00:47:50.960 --> 00:47:52.960
That's probably its own inverse.
606
00:47:52.960 --> 00:47:54.960
This is probably not.
607
00:47:54.960 --> 00:47:56.960
Actually, I think these are inverses of each other.
608
00:47:56.960 --> 00:47:57.960
Oh, yeah.
609
00:47:57.960 --> 00:48:02.960
Actually, the inverse is this transpose.
610
00:48:02.960 --> 00:48:08.960
There's a curious fact about permutation matrices.
611
00:48:08.960 --> 00:48:12.960
That the inverses are the transpose.
612
00:48:12.960 --> 00:48:15.960
And final moment, how many are there?
613
00:48:15.960 --> 00:48:18.960
If I have how many 4x4 permutation?
614
00:48:18.960 --> 00:48:26.960
So let me take 4x4, how many AP?
615
00:48:26.960 --> 00:48:29.960
Well, OK.
616
00:48:29.960 --> 00:48:33.960
Make sure you get 24.
617
00:48:33.960 --> 00:48:34.960
24p.
618
00:48:34.960 --> 00:48:35.960
OK.
619
00:48:35.960 --> 00:48:40.960
So we've got these permutation matrices.
620
00:48:40.960 --> 00:48:45.960
And in the next lecture, we use them.
621
00:48:45.960 --> 00:48:51.960
So the next lecture finishes chapter 2 and moves to chapter 3.
622
00:48:51.960 --> 00:48:59.960
Thank you.
623
00:49:21.960 --> 00:49:28.960
Thank you.
624
00:49:51.960 --> 00:49:58.960
Thank you.
625
00:49:58.960 --> 00:50:05.960
Thank you.
626
00:50:05.960 --> 00:50:13.960
Thank you.