WEBVTT
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Okay, this is it.
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This is the second lecture in linear algebra.
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And I've put below my main topics for today.
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I put right there a system of equations that's going to be our example to work with.
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But let me, what are we going to do with it?
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We're going to solve it.
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And the method of solution will not be determinants.
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Determinants are something that will come later.
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The method will use is called elimination.
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And it's the way every software package solves equations.
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And so an elimination, well, if it succeeds, it gets the answer.
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And normally it does succeed.
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If the matrix A that's coming into that system is a good matrix, and I think this one is,
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then elimination will work, will get the answer in an efficient way.
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But why don't we, as long as we're sort of seeing how elimination works, it's always
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good to ask how could it fail.
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So at the same time, we'll see how elimination decides whether the matrix is a good one or
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has problems.
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Then to complete the answer, there's an obvious step of back substitution.
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In fact, the idea of elimination is, you would have thought of it, right?
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I mean, Gauss thought of it before we did, but only because he was born earlier.
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It's a natural idea.
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And died earlier too.
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Okay.
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But then, and you've seen the idea.
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But now, the part that I want to show you is elimination expressed in matrix language,
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and the whole course, all the key ideas get expressed as matrix operations, not as words.
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And one of the operations, of course, that we'll meet is how do we multiply matrices and
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why?
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Okay.
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So there's a system of equations.
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Three equations and three unknowns.
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And there's the matrix, the three by three matrix, so this is the system, AX equal B, you
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could say.
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This is our system to solve, AX equal, and the right hand side is that vector, 2122.
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Okay.
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Now, when I describe elimination, it gets to be a pain to keep writing the equal signs
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and the pluses and so on.
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Is that matrix that totally matters?
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Everything is in that matrix, but behind it is those equations.
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So what does elimination do?
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What's the first step of elimination?
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We accept the first equation, it's okay.
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I'm going to multiply that equation by the right number, the right multiplier, and I'm
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going to subtract it from the second equation with what purpose?
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So what that will decide what the multiplier should be.
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Our purpose is to knock out the X part of equation 2.
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So our purpose is to reduce, to eliminate X.
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So what do I multiply?
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And again, I'll do it with its matrix, because I can do it short.
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What's the multiplier here?
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What do I multiply?
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Equation 1 and subtract.
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Notice I'm saying that word subtract.
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I'd like to stick to that convention, I'll do a subtraction.
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So what, first of all, this is the key number, right, that I'm starting with, and that's
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called the pivot.
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I'll put a box around it and write its name down.
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That's the first pivot, the first pivot.
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Okay, so I'm going to use that sort of like the key number in that equation.
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And now what's the multiplier?
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So I'm going to, my first row won't change, but that's the pivot row, but I'm going to
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use it, and now finally, let me ask you what the multiplier is.
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Yes.
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3, 3 times that first equation will knock out that 3.
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Okay, so what will it leave?
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So the multiplier is 3, 3 times that will leave, will make that 0, that was our purpose.
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3, 2 is away from the 8, will leave a 2, and 3, 1 is away from 1, will leave a minus
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2.
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And this guy then changed.
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Okay.
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Now the next step, so this is forward elimination, that step is completed.
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Oh well, you could say, wait a minute, what about the right hand side?
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Should I carry, the right hand side is like, it's carried along.
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Actually, MATLAB finishes up with the left side before, and then just goes back to do the
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right side.
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Maybe I'll be MATLAB for a moment and do that.
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Okay, so I'm leaving a column for a column of B, the right hand side, but I'll fill it
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in later.
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Okay.
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Now the next step of elimination is what?
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Well strictly speaking, I cleaned up this position that I cleaned up was like the 2, 1
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position, row 2, column 1.
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So I got a 0 in the 2, 1 position.
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Now I'll use 2, 1 as the index of that step.
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The next step should be to finish the column and get a 0 in that position.
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So the next step is really the 3, 1 step, row 3, column 1.
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But of course, I already have 0.
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Okay.
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So the multiplier is 0.
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I take 0 of this equation away from this one, and I'm all set.
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I won't repeat that, but there was a step there which MATLAB would have to look.
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It would look at this number and do that step unless you told it in advance that it was
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0.
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Okay.
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Now what?
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Now we can see the second pivot, which is what?
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The second pivot, see we've eliminated, x is now gone from this equation, right?
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We're down to 2 equations in y and z.
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And so now I just do it again, like everything's recursive.
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This is such a basic algorithm and you've seen it.
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But carry me through one last step.
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So this is still the first pivot.
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Now the second pivot is this guy who has appeared there.
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And what's the multiplier, the appropriate multiplier now?
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And what's my purpose is to clean up this, is the wipe out, the 3, 2 position, right?
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This was the 2, 1 step.
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And now I'm going to take the 3, 2 step.
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So this all stays the same.
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1, 2, 1, 0, 2, minus 1.
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And the pivots are there.
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Now I'm using this pivot, so what's the multiplier?
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2, 2 times this equation, this row, gets subtracted from this row and makes that a 0.
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So it's 0, 0, and what's, is it a 5?
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Yeah, it gets into 5, is that right?
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Because I have a 1 there and I'm subtracting twice this.
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So I think it sits a 5 there.
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It's a 3rd pivot.
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So let me put a box around all 3 pivots.
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Is there a, I think that, oh, did I just invent a negative 1?
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I am sorry that the tape can't correct that as easily as I can.
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OK, is that, thank you very much.
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You get an A in the course now.
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Is that, is that correct?
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Is it correct now?
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OK, so the 3 pivots are there.
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I know right away a lot about this matrix.
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This elimination step from A, this matrix I'm going to call U. U for upper triangular.
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So the whole purpose of elimination was to get from A to U.
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And literally that's the most common calculation in scientific computing.
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And people think of how could I do that faster because it's a major, major thing.
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But we're doing it the straightforward way.
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We found 3 pivots and by the way, I didn't say this.
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Pivots can't be 0.
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I don't accept 0 as a pivot.
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And I didn't get 0.
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So this matrix is great.
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It gave me 3 pivots.
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I didn't have to do anything special.
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I just followed the rule.
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And the pivots are 1, 2, and 5.
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By the way, just because I always anticipate stuff from a later day, if I wanted to know
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the determinant of this matrix, which I never do want to know.
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But I would just multiply the pivots that determinant is 10.
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So even things like the determinant are here.
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OK.
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Now, let me talk about failure for a moment and then come back to success.
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How could this have failed?
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How could, by fail, fail to come up with 3 pivots?
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Tell me, there are a couple of points.
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I would have already been in trouble if this very first number here was 0.
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If it was a 0 there, suppose that had been a 0.
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There were no x's in that equation.
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Does that mean I can't solve the problem?
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Does that mean I quit?
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No.
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What do I do?
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I x switch rows.
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I exchange rows.
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So in case of a 0, I will not say 0 pivot.
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I will never be heard to utter those words, 0 pivot.
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But if there's a 0 in the pivot position, maybe I can say that, I would try to exchange
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for a lower equation and get a proper pivot up there.
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OK.
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Now, for example, this second pivot came out 2.
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Could it have come out 0?
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Actually, if I change that 8 a little bit, I would have got again a little trouble.
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What should I change that 8 to so that I run into trouble a 6?
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If I had had been a 6, then this would have been 0, and I couldn't have used that as a
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pivot, but I could have exchanged again.
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In this case, because when can I get out of trouble?
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I can get out of trouble if there's a non-zero below this troublesome 0.
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And there is here.
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So I would be OK in this case.
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If this was a 6, I would survive by a row exchange.
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Now, of course, it might have happened that I couldn't do the row.
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There was 0 below it, but here there wasn't.
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Now I could also have got in trouble if this number 1 was a little different.
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See, that 1 became a 5, I guess, by the end.
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So can you see what number there would have got me trouble that I really couldn't get
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out of?
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Trouble that I couldn't get out of would mean if the pivot is 0 in the pivot position,
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and I can't, I've got no place to exchange.
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So there must be some number, which if I had had here, it would have meant failure.
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It would have been a negative 4, good.
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If it was a negative 4 here, if it happened to be a negative 4, can I temporarily put it
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up here, if this had been a negative 4, then I would have gone through the same steps.
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This would have been a minus 4.
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It still would have been a minus 4.
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But at the last minute, it would have become 0.
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And there wouldn't have been a third pivot, the matrix would have not been invertible.
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Of course, the inverse of a matrix is coming next week.
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But you've heard these words before.
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So that's how we identify failure.
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There's temporary failure when we can do a row exchange and get out of it.
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Or there's complete failure when we get a 0 and there's nothing below that we can
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use.
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Let's stay with the back to success now.
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In fact, I guess the next topic is back substitution.
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So what's back substitution?
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Well, now I better bring the right hand side in.
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So what would MATLAB do?
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And what should we do?
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Let me bring in the right hand side as an extra column.
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So there comes B.
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So it's 2122.
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This is, I would call this the augmented matrix.
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Augment means you've tacked something on.
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I've tacked on this extra column.
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Because when I'm working with equations, I do the same thing to both sides.
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So at this step, I subtracted two of the first equation away from the second equation so
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that this augmented, I even brought some color chalk, but I don't know if it shows up.
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So this is like the augmented, no, damn, circle the wrong thing.
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OK, here's the, here's B.
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OK, that's the extra column.
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OK, so what happened to that extra column, the right hand side of the equations, when
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I did the first step?
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So that was three of this away from this.
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So it took the two stayed the same, but three two has got taken away from 12, leaving 6,
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and that two stayed the same.
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So this is how it's looking halfway along.
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And let me just carry to the end the two and the six stayed the same.
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But what do I have here?
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Oh gosh.
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Help me out now.
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So now this is still like forward elimination.
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I got to this point, which I think is right.
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And now what did I do with this step?
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I multiplied that pivot by two, or that whole equation by two, and subtracted from that.
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So I think I take two sixes, which is 12 away from the two.
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Do you think minus 10 is my final right hand side, the right hand side that goes with you,
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and let me call that once and forever, the vector C.
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So C is what happens to be, and U is what happens to A.
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Okay.
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There you've seen elimination.
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Clean.
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Okay.
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Oh, what's back substitution?
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So what are my final equations then?
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Can I copy these equations?
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X plus 2y plus z equals 2 is still there.
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And 2y minus 2z equals 6 is there.
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And 5z is minus 10.
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Okay.
245
00:17:37.400 --> 00:17:43.200
Those are the equations that are, that these numbers are telling me about.
246
00:17:43.200 --> 00:17:48.200
Those are the equations Ux equals C.
247
00:17:48.200 --> 00:17:49.200
Okay.
248
00:17:49.200 --> 00:17:51.400
How do I solve them?
249
00:17:51.400 --> 00:17:54.240
What one do I solve for first?
250
00:17:54.240 --> 00:17:55.280
z.
251
00:17:55.280 --> 00:18:01.680
I see immediately that the correct value of z is negative 2.
252
00:18:01.680 --> 00:18:03.880
And what do I do next?
253
00:18:03.880 --> 00:18:05.080
I go back upwards.
254
00:18:05.080 --> 00:18:07.280
I now know z here.
255
00:18:07.280 --> 00:18:13.440
So if z is negative 2, that's 4 there, is that right?
256
00:18:13.440 --> 00:18:16.360
And so 2y plus the 4 is 6.
257
00:18:16.360 --> 00:18:20.040
Maybe y is 1.
258
00:18:20.040 --> 00:18:22.200
This is back substitution.
259
00:18:22.200 --> 00:18:24.840
We're doing it on the fly because it's so easy.
260
00:18:24.840 --> 00:18:31.800
And then x is so x 2y is 2 minus 2.
261
00:18:31.800 --> 00:18:33.040
Maybe x is 2.
262
00:18:39.760 --> 00:18:42.800
So you see what back substitution is.
263
00:18:42.800 --> 00:18:47.360
It's the simple step solving equations in reverse order
264
00:18:47.360 --> 00:18:50.520
because the system is triangular.
265
00:18:50.520 --> 00:18:52.040
Okay.
266
00:18:52.040 --> 00:18:53.240
Good.
267
00:18:53.240 --> 00:18:56.200
So that's elimination and back substitution.
268
00:18:56.200 --> 00:18:58.840
And I kept the right hand side along.
269
00:18:58.840 --> 00:18:59.440
Okay.
270
00:18:59.440 --> 00:19:06.920
Now what do I, so that, that like is first piece of the lecture.
271
00:19:06.920 --> 00:19:10.000
What's the second piece?
272
00:19:10.000 --> 00:19:12.160
Matrices are going to get in.
273
00:19:12.160 --> 00:19:17.400
So I wrote stuff with x, y, and z in there.
274
00:19:17.400 --> 00:19:22.720
Then I really got the right shorthand just
275
00:19:22.720 --> 00:19:25.840
writing the matrix entries.
276
00:19:25.840 --> 00:19:29.240
And now I want to write the operations
277
00:19:29.240 --> 00:19:33.360
that I did in matrices.
278
00:19:33.360 --> 00:19:41.840
I've carried the matrices along, but I haven't said the operation,
279
00:19:41.840 --> 00:19:46.600
those elimination steps, I now want to express as matrices.
280
00:19:46.600 --> 00:19:47.120
Okay.
281
00:19:47.120 --> 00:19:47.720
Here they come.
282
00:19:50.920 --> 00:19:53.160
So now this is elimination matrices.
283
00:19:56.720 --> 00:19:57.720
Okay.
284
00:19:57.720 --> 00:20:03.440
Let me take that first step, which took me from 1, 2, 1,
285
00:20:03.440 --> 00:20:07.640
3, 8, 1, 0, 4, 1.
286
00:20:07.640 --> 00:20:13.560
I want to operate on that.
287
00:20:13.560 --> 00:20:17.640
I want to do elimination on that.
288
00:20:17.640 --> 00:20:19.400
Okay.
289
00:20:19.400 --> 00:20:19.920
Okay.
290
00:20:19.920 --> 00:20:26.600
Now I see I'm remembering a point I want to single out
291
00:20:26.600 --> 00:20:30.200
as, as, as, especially important.
292
00:20:30.200 --> 00:20:40.240
Let me move the board up for that, because when we do matrix
293
00:20:40.240 --> 00:20:43.400
operations, we've got to like be able to see the big picture.
294
00:20:43.400 --> 00:20:43.880
Okay.
295
00:20:43.880 --> 00:20:47.960
Last time I spoke about the big picture of when
296
00:20:47.960 --> 00:20:51.800
I multiply a matrix by a right-hand side.
297
00:20:51.800 --> 00:20:56.120
If I have some matrix there, and I multiply it by 3, 4, 5,
298
00:20:56.120 --> 00:20:57.400
let's say.
299
00:20:57.400 --> 00:21:01.000
So here's a matrix.
300
00:21:01.000 --> 00:21:02.440
What did I say?
301
00:21:02.440 --> 00:21:04.600
Well, I guess I only said it on the video tape.
302
00:21:04.600 --> 00:21:09.960
But do you remember how I look at that matrix multiplication?
303
00:21:09.960 --> 00:21:15.880
The result of multiplying a matrix by some vector
304
00:21:15.880 --> 00:21:23.360
is a combination of the columns of the matrix.
305
00:21:23.360 --> 00:21:26.160
It's 3 times the first column.
306
00:21:26.160 --> 00:21:35.800
It's 3 times column 1 plus 4 times column 2 plus 5 times column 3.
307
00:21:41.200 --> 00:21:41.880
Okay.
308
00:21:41.880 --> 00:21:45.560
I'm going to come back to that multiple times.
309
00:21:45.560 --> 00:21:53.440
What I wanted to do now was to emphasize the parallel thing
310
00:21:53.440 --> 00:21:55.920
with rows.
311
00:21:55.920 --> 00:21:56.640
Why?
312
00:21:56.640 --> 00:21:59.960
Because all our operations here for this,
313
00:21:59.960 --> 00:22:06.160
like this, two weeks of a course, are row operations.
314
00:22:06.160 --> 00:22:10.640
So this isn't what I need for row operations.
315
00:22:10.640 --> 00:22:13.680
Let me do a row operations.
316
00:22:13.680 --> 00:22:20.680
Suppose I have my matrix again.
317
00:22:20.680 --> 00:22:30.280
And suppose I multiply on the left by some, let's say, 1, 2, 7.
318
00:22:30.280 --> 00:22:35.480
Again, I'm just like saying what the result is.
319
00:22:35.480 --> 00:22:39.880
And then we'll say how matrix multiplication works,
320
00:22:39.880 --> 00:22:42.120
and we'll see that it's true.
321
00:22:42.120 --> 00:22:43.200
Okay.
322
00:22:43.200 --> 00:22:50.480
But maybe already I'm bringing up the central idea
323
00:22:50.480 --> 00:22:55.800
of linear algebra is how these matrices work by rows
324
00:22:55.800 --> 00:22:57.400
as well as by columns.
325
00:22:57.400 --> 00:22:59.480
Okay. How does it work by rows?
326
00:22:59.480 --> 00:23:00.480
What?
327
00:23:00.480 --> 00:23:06.440
So this is a row vector.
328
00:23:06.440 --> 00:23:11.080
It's a, I could say, that's a 1 by 3 matrix, a row vector,
329
00:23:11.080 --> 00:23:13.760
multiplying a 3 by 3 matrix.
330
00:23:13.760 --> 00:23:14.280
What?
331
00:23:14.280 --> 00:23:19.040
What's the output?
332
00:23:19.040 --> 00:23:25.200
What's the product of a row times a matrix?
333
00:23:25.200 --> 00:23:28.680
And, okay, it's a row.
334
00:23:28.680 --> 00:23:33.480
A column, sorry, a matrix times a column is a column.
335
00:23:33.480 --> 00:23:43.560
So matrix times the, the matrix times a column is a column.
336
00:23:43.560 --> 00:23:45.880
And we know what column it is.
337
00:23:45.880 --> 00:23:50.200
Over here, I'm doing a row times a matrix, and what is it?
338
00:23:50.200 --> 00:23:51.320
What's the answer?
339
00:23:51.320 --> 00:23:53.640
It's one of that first row.
340
00:23:53.640 --> 00:24:02.160
So it's one times, one times row one plus two times row two
341
00:24:02.160 --> 00:24:06.800
plus seven times row three.
342
00:24:06.800 --> 00:24:10.960
When, as we do matrix multiplication, keep your eye
343
00:24:10.960 --> 00:24:15.520
on what it's doing with whole vectors.
344
00:24:15.520 --> 00:24:20.640
And what it's doing, what it's doing in this case is it's
345
00:24:20.640 --> 00:24:23.560
combining the rows.
346
00:24:23.560 --> 00:24:26.640
We have a combination, a linear combination of the rows.
347
00:24:26.640 --> 00:24:35.040
Okay, I want to use that.
348
00:24:35.040 --> 00:24:40.880
Okay, so my question is, what's the matrix that does this first step?
349
00:24:40.880 --> 00:24:46.600
That takes, subtracts three of equation one from equation two.
350
00:24:46.600 --> 00:24:48.480
That's what I want to do.
351
00:24:48.480 --> 00:24:56.800
So this is going to be a matrix that's going to subtract three times
352
00:24:56.800 --> 00:25:05.720
row one from row two.
353
00:25:05.720 --> 00:25:07.520
And leaves the other rows the same.
354
00:25:07.520 --> 00:25:11.240
Just, I mean, the answer is going to be that.
355
00:25:11.240 --> 00:25:14.920
So whatever matrix this is, and you're
356
00:25:14.920 --> 00:25:18.560
going to like tell me what matrix will do it, it's the matrix
357
00:25:18.560 --> 00:25:23.160
that leaves the first row unchanged, leaves the last row
358
00:25:23.160 --> 00:25:26.480
unchanged, but takes three of these away from this.
359
00:25:26.480 --> 00:25:30.160
So it puts a zero there, a two there, and a minus two.
360
00:25:30.160 --> 00:25:31.560
Good.
361
00:25:31.560 --> 00:25:35.160
What matrix will do it?
362
00:25:35.160 --> 00:25:38.840
It should be a pretty simple matrix, because we're
363
00:25:38.840 --> 00:25:42.480
doing a very simple step.
364
00:25:42.480 --> 00:25:45.960
We're just doing this step that changes row two.
365
00:25:45.960 --> 00:25:48.200
So actually, row one is not changing.
366
00:25:48.200 --> 00:25:50.520
So tell me how the matrix should begin.
367
00:25:53.400 --> 00:26:00.360
One, so the first row of the matrix will be 1,000.
368
00:26:00.360 --> 00:26:05.360
Because that's just the right thing that takes one of that row
369
00:26:05.360 --> 00:26:08.360
and none of the other rows, and that's what we want.
370
00:26:08.360 --> 00:26:11.360
What's the last row of the matrix?
371
00:26:11.360 --> 00:26:13.360
001.
372
00:26:13.360 --> 00:26:18.360
Because that takes one of the third row and none of the other rows.
373
00:26:18.360 --> 00:26:19.360
That's great.
374
00:26:19.360 --> 00:26:20.360
OK.
375
00:26:20.360 --> 00:26:24.360
Now, suppose I didn't want to do anything at all.
376
00:26:24.360 --> 00:26:29.360
Suppose I had a case here when I was doing this.
377
00:26:29.360 --> 00:26:34.160
A case here when I already had a zero and didn't have
378
00:26:34.160 --> 00:26:38.360
to do anything, what matrix does nothing,
379
00:26:38.360 --> 00:26:42.760
like just leaves you where you were?
380
00:26:42.760 --> 00:26:52.360
If I put in 0,1, 0, that would be, that's the matrix.
381
00:26:52.360 --> 00:26:54.560
What's the name of that matrix?
382
00:26:54.560 --> 00:26:57.360
The identity matrix, right.
383
00:26:57.360 --> 00:26:59.360
So it does absolutely nothing.
384
00:26:59.360 --> 00:27:01.360
It just multiplies everything and leaves it where it is.
385
00:27:01.360 --> 00:27:05.360
It's like a 1, like the number 1, for matrices.
386
00:27:05.360 --> 00:27:10.360
But that's not what we want, because we want to change this row 2.
387
00:27:10.360 --> 00:27:13.360
So what's the correct?
388
00:27:13.360 --> 00:27:18.360
What should I put in here now to come out to do it right?
389
00:27:18.360 --> 00:27:20.360
What do I want?
390
00:27:20.360 --> 00:27:21.360
What am I am asked?
391
00:27:21.360 --> 00:27:26.360
Or I want to subtract my, I want three of row 1 to get subtracted off.
392
00:27:26.360 --> 00:27:30.360
So what's that row, what's the right matrix,
393
00:27:30.360 --> 00:27:34.360
the finish that matrix for me?
394
00:27:34.360 --> 00:27:38.360
Negative 3 goes here.
395
00:27:38.360 --> 00:27:43.360
And what goes here, that 1, and what goes here, the 0?
396
00:27:43.360 --> 00:27:45.360
That's the good matrix.
397
00:27:45.360 --> 00:27:50.360
That's the matrix that takes minus 3 of row 1 plus the row 2
398
00:27:50.360 --> 00:27:52.360
and gives a new row 2.
399
00:27:52.360 --> 00:27:59.360
Should we just like check some particular entry?
400
00:27:59.360 --> 00:28:05.360
How do I check a particular entry of a matrix, in matrix multiplication?
401
00:28:05.360 --> 00:28:13.360
Like, suppose I wanted to check the entry here that's in row 2, column 3.
402
00:28:13.360 --> 00:28:18.360
So where does the entry in row 2, column 3 come from?
403
00:28:18.360 --> 00:28:28.360
I would look at row 2 of this guy and column 3 of this one to get that number.
404
00:28:28.360 --> 00:28:32.360
That number comes from the second row and the third column.
405
00:28:32.360 --> 00:28:41.360
And I just take this dot product minus 3, I'm multiplying minus 3 plus 1 and 0 gives the minus 2.
406
00:28:41.360 --> 00:28:44.360
Yeah, it works.
407
00:28:44.360 --> 00:28:49.360
So we've got various ways to multiply matrices now.
408
00:28:49.360 --> 00:28:52.360
We're sort of like informally.
409
00:28:52.360 --> 00:29:00.360
We've got by columns, we've got, well, we will have by columns, by rows, by each entry at a time.
410
00:29:00.360 --> 00:29:06.360
But it's good to see that matrix multiplication when one of the matrices is so simple.
411
00:29:06.360 --> 00:29:14.360
So this guy is our elementary matrix. Let's call it E for elementary or elimination.
412
00:29:14.360 --> 00:29:24.360
And let me put the indexes 2,1 because it's the matrix that we needed to fix the 2,1 position.
413
00:29:24.360 --> 00:29:30.360
It's the matrix that we needed to get this 2,1 position to be 0.
414
00:29:30.360 --> 00:29:32.360
Okay. Good enough.
415
00:29:32.360 --> 00:29:37.360
So what do I do next? I need another matrix, right?
416
00:29:37.360 --> 00:29:41.360
I need to, there's another step here.
417
00:29:41.360 --> 00:29:49.360
And I want to express the whole elimination process in matrix language.
418
00:29:49.360 --> 00:29:56.360
So tell me what's now, so next step, step 2, which was what?
419
00:29:56.360 --> 00:30:04.360
What was the actual step that we did? I think I subtracted, do you remember?
420
00:30:04.360 --> 00:30:17.360
I had a 2 in the pivot and a 4 below it, so I subtracted 2 times row 2 from row 3.
421
00:30:17.360 --> 00:30:26.360
Tell me the matrix that'll do that. And tell me it's Nate.
422
00:30:26.360 --> 00:30:33.360
Okay. It's going to be E for elementary or elimination matrix.
423
00:30:33.360 --> 00:30:41.360
And what's the index number that I used to tell me what E32, right?
424
00:30:41.360 --> 00:30:50.360
Because it's fixing this 3,2 position. And what is the matrix now?
425
00:30:50.360 --> 00:30:56.360
Okay. You remember? So E32 is supposed to multiply my guy that I have.
426
00:30:56.360 --> 00:31:02.360
And it's supposed to produce the right result, which was it leaves,
427
00:31:02.360 --> 00:31:07.360
supposed to leave the first row. It's supposed to leave the second row.
428
00:31:07.360 --> 00:31:13.360
And it's supposed to straighten out the third row to this.
429
00:31:13.360 --> 00:31:18.360
And what's the matrix that does that?
430
00:31:18.360 --> 00:31:23.360
1,00, right? Because we don't change the first row.
431
00:31:23.360 --> 00:31:30.360
And the next row, we don't change either. And the last row is the one we do change.
432
00:31:30.360 --> 00:31:38.360
And what do I do? Let's see. I've subtracted 2 times. So what's this row?
433
00:31:38.360 --> 00:31:44.360
What's this here? 0, right? Because the first row is not involved.
434
00:31:44.360 --> 00:31:47.360
It's just in the 3,2 position, isn't it?
435
00:31:47.360 --> 00:31:54.360
The key number is this minus the multiplier that goes sitting there in that 3,2 position.
436
00:31:54.360 --> 00:32:09.360
So is it a minus 2 to subtract 2? And then this is 1 so that the overall effect is to take minus 2 of this row plus 1 of that.
437
00:32:09.360 --> 00:32:16.360
Okay. So I've now given you the pieces.
438
00:32:16.360 --> 00:32:24.360
The elimination matrices, the elementary matrices, that take each step.
439
00:32:24.360 --> 00:32:35.360
So now what? Now the next point in the lecture is to put those steps together into a matrix that does it all.
440
00:32:35.360 --> 00:32:42.360
And see how it all happens. So now I'm going to express the whole, everything we did today,
441
00:32:42.360 --> 00:32:53.360
so far on A, was to start with A. We multiplied it by E2 1.
442
00:32:53.360 --> 00:33:01.360
That was the first step. And then we multiplied that result by E3 2.
443
00:33:01.360 --> 00:33:13.360
And that led us to this thing. And what was that matrix? U.
444
00:33:13.360 --> 00:33:21.360
You see why I like matrix notation? Because there in like little space,
445
00:33:21.360 --> 00:33:28.360
a few bits when it's compressed on the web is everything, is this whole lecture.
446
00:33:28.360 --> 00:33:38.360
Okay. Now, there are important facts about matrix multiplication.
447
00:33:38.360 --> 00:33:45.360
And we're close to maybe the most important. And that is this.
448
00:33:45.360 --> 00:33:51.360
Suppose I ask you this question. Suppose I start with a matrix A,
449
00:33:51.360 --> 00:34:00.360
and I want to end with a matrix U, and I want to say what matrix does the whole job?
450
00:34:00.360 --> 00:34:09.360
What matrix takes me from A to U? Using the letters I've got.
451
00:34:09.360 --> 00:34:18.360
So, and the answer is simple. I'm not asking this is, but it's highly important.
452
00:34:18.360 --> 00:34:24.360
How could I, how would I create the matrix that does the whole job at once,
453
00:34:24.360 --> 00:34:32.360
that does all of elimination in one shot? It would be, I would just put these together, right?
454
00:34:32.360 --> 00:34:37.360
In other words, this is the thing I'm struggling to say.
455
00:34:37.360 --> 00:34:44.360
I can move those parentheses. If I keep the matrices in order,
456
00:34:44.360 --> 00:34:52.360
I can't mess around with the order of the matrices, but I can change the order that I do the multiplications.
457
00:34:52.360 --> 00:35:02.360
I can multiply these two first. In other words, you see what those parentheses are doing?
458
00:35:02.360 --> 00:35:10.360
It's saying do, multiply the E's first, and that gives you the matrix that does everything at once.
459
00:35:10.360 --> 00:35:17.360
Okay. So this fact that this is automatically the same as this.
460
00:35:17.360 --> 00:35:23.360
For every matrix multiplication, which I'm conscious of,
461
00:35:23.360 --> 00:35:29.360
still not telling you in every detail, but like you're seeing how it works,
462
00:35:29.360 --> 00:35:36.360
and this is highly important, and maybe tell me the long word that describes this law from matrices,
463
00:35:36.360 --> 00:35:43.360
that you can move the parentheses. It's the called the associative law.
464
00:35:43.360 --> 00:35:53.360
I think you can now forget that. But don't forget the law. I mean, like forget the word associative, I don't know.
465
00:35:53.360 --> 00:36:03.360
But don't forget the law. Because actually, we'll see so many steps in linear algebra,
466
00:36:03.360 --> 00:36:10.360
so many proofs, even, of main facts, come from just moving the parentheses.
467
00:36:10.360 --> 00:36:22.360
And it's not that easy to prove that this is correct. You have to go into the gory details of matrix multiplication,
468
00:36:22.360 --> 00:36:30.360
do it both ways, and see that you come out the same. Maybe I'll leave the author to do that.
469
00:36:30.360 --> 00:36:37.360
Okay. So there we go. That's how I...
470
00:36:37.360 --> 00:36:47.360
So there's a single matrix, I could call it E. Now, oh, why were we talking about these matrices?
471
00:36:47.360 --> 00:36:52.360
Tell me one other... There's another type of elementary matrix,
472
00:36:52.360 --> 00:36:57.360
and we already said why we might need it. We didn't need it in this case.
473
00:36:57.360 --> 00:37:04.360
But it's the matrix that exchanges two rows. It's a called a permutation matrix.
474
00:37:04.360 --> 00:37:12.360
Can you just like tell me what that would be? So I'm just like this is a slight digression.
475
00:37:12.360 --> 00:37:19.360
And we'll... Yeah, so let me get some... Let me figure out where I'm going to put a permutation matrix.
476
00:37:19.360 --> 00:37:28.360
You'll see I'm always squeezing stuff in. So permutation, or in fact,
477
00:37:28.360 --> 00:37:41.360
this one that like exchange rows... So like exchange rows one and two, just to make life easy.
478
00:37:41.360 --> 00:37:47.360
So if I had my matrix... No, let me just do two by two. ABCD.
479
00:37:47.360 --> 00:37:58.360
Suppose I want to find the matrix that exchanges those rows.
480
00:37:58.360 --> 00:38:07.360
What is it? So the matrix that exchanges those rows, the row I want is CD, and it's there.
481
00:38:07.360 --> 00:38:14.360
So I better take one of it. And the row I want here is up top, so I'll take one of that.
482
00:38:14.360 --> 00:38:25.360
So actually, I've just the easy... This is my matrix that I'll call P for permutation.
483
00:38:25.360 --> 00:38:33.360
It's the matrix... Actually, the easy way to find it is just do the thing to the identity matrix.
484
00:38:33.360 --> 00:38:42.360
Exchange rows... Exchange the rows of the identity matrix, and then that's the matrix that'll do row exchanges for you.
485
00:38:42.360 --> 00:38:52.360
Suppose I wanted to exchange columns instead. Collons have hardly got into today's lecture, but they certainly are going to be around.
486
00:38:52.360 --> 00:39:04.360
How could I... If I started with this matrix ABCD, then I wouldn't... I'm not even going to write this down. I'm just going to ask you.
487
00:39:04.360 --> 00:39:16.360
Because in elimination, we're doing rows. But suppose we wanted to exchange the columns of a matrix.
488
00:39:16.360 --> 00:39:25.360
How would I do that? What matrix multiplication would do that job? Actually, why not? I'll write it down.
489
00:39:25.360 --> 00:39:48.360
So this is like... I'll write it under here and then hide it again. Suppose I had my matrix ABCD, and I want to get to AC over here and VD here.
490
00:39:48.360 --> 00:40:03.360
What matrix does that job? Can I multiply... Can I cook up some matrix that produces that answer?
491
00:40:03.360 --> 00:40:15.360
And you can see from where I put my hand, I was really asking, can I put a matrix here on the left that will exchange columns?
492
00:40:15.360 --> 00:40:26.360
And the answer is... No. If I'm just bringing out again this point that when I multiply on the left, I'm doing row operations.
493
00:40:26.360 --> 00:40:34.360
So if I want to do a column operation, where do I put that permutation matrix on the right?
494
00:40:34.360 --> 00:40:46.360
If I put it here, where I just barely left room for it, so I'll exchange the two columns of the identity. Then it comes out right.
495
00:40:46.360 --> 00:40:57.360
Because now I'm multiplying a column at a time. This is the first column and says take none of that column, one of this one, and then you got it.
496
00:40:57.360 --> 00:41:11.360
Over here, take one of this one, none of this one, and you've got AC. So, in short, to do column operations, the matrix multiplies on the right, to do row operations, it multiplies on the left.
497
00:41:11.360 --> 00:41:16.360
Okay. Okay. And it's row operations that we're really doing.
498
00:41:16.360 --> 00:41:36.360
Okay. And of course, I mentioned in passing, but I better say it very clearly, that you can't exchange the orders of matrices.
499
00:41:36.360 --> 00:41:51.360
And that's just the point I was making again here. A times B is not the same as B times A. You have to keep these matrices in their gals given order here.
500
00:41:51.360 --> 00:42:08.360
But you can move the parentheses. So, in other words, the commutative law, which would allow you to take it into the other order, is false.
501
00:42:08.360 --> 00:42:18.360
So, we have to keep it in that order. Okay. So, what next?
502
00:42:18.360 --> 00:42:29.360
I could do this multiplication. I could do E32. So, let me come back to see what that was.
503
00:42:29.360 --> 00:42:46.360
Here was E21. And here is E32. And if I multiply those matrices together, E32 and then E21,
504
00:42:46.360 --> 00:43:01.360
that's a single matrix that does elimination. I don't want to do it that. If I do that multiplication,
505
00:43:01.360 --> 00:43:12.360
there's a better way to do this. And so, in this last few minutes of today's lecture, can I anticipate that better way?
506
00:43:12.360 --> 00:43:24.360
The better way is to think not how do I get from A to U, but how do I get from U back to A?
507
00:43:24.360 --> 00:43:33.360
So, reversing steps is going to come in. Inverse. I'll use the word inverse here. Okay.
508
00:43:33.360 --> 00:43:47.360
So, let me make the first step at what's the inverse matrix? All the matrices you've seen on this board have inverses.
509
00:43:47.360 --> 00:43:57.360
I didn't write any bad matrices down. We spoke about possible failure, and for a moment we put in a matrix that would fail.
510
00:43:57.360 --> 00:44:09.360
But right now, all these matrices are good. They're all invertible. And let's take the inverse. Well, let me say first what does the inverse mean and find it.
511
00:44:09.360 --> 00:44:21.360
Okay. So, we're getting a little leg up on inverses. Okay. So, this is the final moments of today.
512
00:44:21.360 --> 00:44:39.360
Sorry. He's still there. Okay. Inverses. Okay. And I'm just going to take one example, and then we're done.
513
00:44:39.360 --> 00:45:00.360
The example I'll take will be that E. So, my matrix is 1 0 0 minus 3 1 0 0 0 1. And I want to find the matrix that undoes that step.
514
00:45:00.360 --> 00:45:15.360
So, what was that step? The step was subtract 3 times row 1 from row 2. So, what matrix will get me back?
515
00:45:15.360 --> 00:45:30.360
What matrix will bring back, you know, if I started with a 2 12 2 and it changed to 2 6 2 because of this guy, I want to get back to the 2 12 2.
516
00:45:30.360 --> 00:45:43.360
I want to find the matrix which undoes elimination. The matrix which multiplies this to give the identity.
517
00:45:43.360 --> 00:45:50.360
And you can tell me what I should do in words first and then we'll write down the matrix that does it.
518
00:45:50.360 --> 00:45:59.360
If this step subtracted 3 times row 1 from row 2, what's the inverse step?
519
00:45:59.360 --> 00:46:10.360
I add 3 times row 1 to row 2, right? I added back. That's what I subtracted the way I add back.
520
00:46:10.360 --> 00:46:27.360
So, the inverse matrix in this case is I now want to add 3 times row 1 to row 2. So, I won't change row 1, I won't change row 3, and I'll add 3 times row 1 to row 2.
521
00:46:27.360 --> 00:46:41.360
That's a case where the inverse is clear. It's clear in words what to do because what this did was simple to express.
522
00:46:41.360 --> 00:46:50.360
It just changed row 2 by subtracting 3 of row 1. So, to invert it, I go that way.
523
00:46:50.360 --> 00:46:58.360
If we do that calculation 3 times this row plus 1 times this row comes out the right row of the identity.
524
00:46:58.360 --> 00:47:12.360
So, if this matrix was E and this matrix is I for identity, then what's the notation for this guy?
525
00:47:12.360 --> 00:47:25.360
E to the minus 1, E inverse. Okay, let's stop there for today. That's a little jump on what's coming on Monday.
526
00:47:25.360 --> 00:47:52.360
So, say a Monday.
527
00:49:25.360 --> 00:49:46.360
So, I'll add 3 times row 1 to the minus 1, and I'll add 3 times row 1 to the minus 1.
528
00:49:46.360 --> 00:50:01.360
So, I'll add 3 times row 1 to the minus 1.