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Okay.
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This is it.
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The second lecture in linear
algebra, and I've put below my
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main topics for today.
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I put right there a system of
equations that's going to be our
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example to work with.
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But 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 we'll use is called
elimination.
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And it's the way every software
package solves equations.
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And elimination,
well, if it succeeds,
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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
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matrix, and I think this one is,
then elimination will work.
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We'll get the answer in an
efficient way.
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But why don't we,
as long as we're sort of seeing
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how elimination works -- it's
always good to ask how could it
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fail?
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So at the same time,
we'll see how elimination
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decides whether the matrix is a
good one or has problems.
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Then to complete the answer,
there's an obvious step of back
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substitution.
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In fact, the idea of
elimination is -- you would have
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thought of it,
right?
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I mean Gauss thought of it
before we did,
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but only because he was born
earlier.
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It's a natural idea...
and died earlier,
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too.
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Okay, and you've seen the idea.
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But now, the part that I want
to show you is elimination
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expressed in matrix language,
because the whole course --
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all the key ideas get expressed
as matrix operations,
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not as words.
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And one of the operations,
of course, that we'll meet is
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how do we multiply matrices and
why?
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Okay, 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
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this is the system Ax = b.
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This is our system to solve,
Ax equal -- and the right-hand
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side is that vector 2,
12, 2.
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Okay.
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Now, when I describe
elimination -- it gets to be a
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pain to keep writing the equal
signs and the pluses and so on.
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It's that matrix that totally
matters.
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Everything is in that matrix.
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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,
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the right multiplier and I'm
going to subtract it from the
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second equation.
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With what purpose?
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So that will decide what the
multiplier should be.
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Our purpose is to knock out the
x part of equation two.
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So our purpose is to eliminate
x.
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So what do I multiply -- and
again, I'll do it with this
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matrix, because I can do it
short.
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What's the multiplier here?
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What do I multiply -- equation
one 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.
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I'll do a subtraction.
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First of all this is the key
number that I'm starting with.
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And that's 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.
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The first pivot.
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Okay.
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So I'm going to use -- that's
sort of like the key number in
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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,
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that's the pivot row.
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But I'm going to use it -- and
now, finally,
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let me ask you what the
multiplier is.
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Yes?
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3. --> 00:04:50
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3 times that first equation
will knock out that 3.
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Okay.
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So what will it leave?
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So the multiplier is 3.
3 times that will make that 0.
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That was our purpose.
3 2s away from the 8 will leave
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a 2 and three 1s away from 1
will leave a minus 2.
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And this guy didn't change.
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Okay.
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Now the next step -- this is
forward elimination and that
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step's completed.
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Oh, well, you could say wait a
minute, what about the right
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hand side?
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Shall I carry -- the right-hand
side gets carried along.
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Actually MatLab finishes up
with the left side before -- and
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then just goes back to do the
right side.
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Maybe I'll be MatLab for a
moment and do that.
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Okay.
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I'm leaving a room for a column
of b, the right-hand side.
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But I'll fill it 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...
this position that I cleaned up
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was like the 2,
1 position, row 2,
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column 1.
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So I got a 0 in the 2,
1 position.
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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
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that position.
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So the next step is really the
3,1 step, row three,
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column one.
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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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So I won't repeat that,
but there was a step there
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which, MatLab would have to look
-- it would look at this number
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and, do that step,
unless you told it in advance
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that it was 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
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gone from this equation,
right?
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We're down to two equations in
y and z.
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And so now I just do it again.
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Like, everything's very cursive
at this -- this is like --
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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?
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Is it to 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,
1 2 1, 0 2 -1 and the pivots
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are there.
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Now I'm using this pivot,
so what's the multiplier?
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2 times this equation,
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this row, gets subtracted from
this row and makes that a 0.
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So it's 0, 0 and is it a 5?
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Yeah, I guess it's a 5,
is that right?
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Because I have a one there and
I'm subtracting twice of twice
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this, so I think it's a 5 there.
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There's the third pivot.
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So let me put a box around all
three pivots.
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Is there a -- oh,
did I just invent a negative
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one?
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I'm sorry that the tape can't,
correct that as easily as I
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can.
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Okay.
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Thank you very much.
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You get an A in the course now.
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Is that correct?
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Is it correct now?
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Okay.
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So the three pivots are there
-- I know right away a lot about
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this matrix.
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This elimination step from A --
this matrix I'm going to call U.
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U for upper triangular.
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So the whole purpose of
elimination was to get from A to
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U.
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And, literally,
that's the most common
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calculation in scientific
computing.
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And people think of how could I
do that faster?
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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 three pivots,
and by the way,
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I didn't say this,
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 three pivots,
I didn't have to do anything
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special, I just followed the
rules and, and the pivots are 1,
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2 and 5.
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By the way, just because I
always anticipate stuff from a
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later day, if I wanted to know
the determinant of this matrix
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--
which I never do want to know,
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but I would just multiply the
pivots.
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The determinant is 10.
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So even things like the
determinant are here.
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Okay.
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Now -- oh, let me talk about
failure for a moment,
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and then --
and then come back to success.
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How could this have failed?
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How could -- by fail,
I mean to come up with three
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pivots.
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I mean, there are a couple of
points.
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I would have already been in
trouble if this very first
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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 Xs in that
equation -- first 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 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,
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I would try to exchange for a
lower equation and get a proper
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pivot up there.
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Okay.
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Now, for example,
this second pivot came out two.
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Could it have come out 0?
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What -- actually,
if I change that 8 a little
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bit, I would have got a little
trouble.
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What should I change that 8 to
so that I run into trouble?
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A 6.
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If that had been a 6,
then this would have been 0 and
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I couldn't have used that as the
pivot.
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But I could have exchanged
again.
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In this case.
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In this case,
because when can I get out of
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trouble?
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I can get out of trouble if
there's a non-0 below this
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troublesome 0.
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And there is here.
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So I would be okay in this
case.
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If this was a 6,
I would survive by a row
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exchange.
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Now -- of course,
it might have happened that I
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couldn't do the row,
that -- that there was 0s below
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it, but here there wasn't.
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Now, I could also have got in
trouble if this number 1 was a
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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
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00:13:05 --> 00:13:09
that I really couldn't get out
of?
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Trouble that I couldn't get out
of would mean if 0 is in the
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00:13:16 --> 00:13:22
pivot position and I've got no
place to exchange.
236
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So there must be some number
which if I had had here it would
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have meant failure.
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Negative 4, good.
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If it was a negative 4 here --
if it happened to be a negative
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4, I'll temporarily put it up
here.
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If this had been a negative 4
z, then I would have gone
242
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through the same steps.
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This would have been a minus 4,
it still would have been a
244
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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.
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The matrix would have not been
invertible.
248
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Well, of course,
the inverse of a matrix is
249
00:14:06 --> 00:14:09
coming next week,
but, you've heard these words
250
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before.
251
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So, that's how we identify
failure.
252
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There's temporary failure when
we can do a row exchange --
253
00:14:17 --> 00:14:21.26
and get out of it,
or there's complete failure
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00:14:21.26 --> 00:14:26
when we get a 0 and -- and
there's nothing below that we
255
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can use.
256
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Okay.
257
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Let's stay with -- back to
success now.
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In fact, I guess the next topic
is back substitution.
259
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So what's back substitution?
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Well, now I'd better bring the
right-hand side in.
261
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So what would MatLab do 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.
264
00:14:56 --> 2.
So it's 2, 12,
265
2. --> 00:14:58
266
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I would call this the augmented
matrix.
267
00:15:02 --> 00:15:06
"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
270
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to both sides.
271
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So, at this step,
I subtracted 2 of the first
272
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equation away from the second
equation so that this augmented
273
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-- I even brought some colored
chalk, but I don't know if it
274
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shows up.
275
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So this is like the augmented
-- no!
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Damn, circled the wrong thing.
277
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Okay.
278
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Here is b.
279
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Okay, that's the extra column.
280
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Okay.
281
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So what happened to that extra
column, the right-hand side of
282
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the equations,
when I did the first step?
283
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So that was 3 of this away from
this, so it took -- the 2 stayed
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the same, but three 2s got taken
away from 12,
285
00:15:56 --> 00:15:59
leaving 6, and that 2 stayed
the same.
286
00:15:59 --> 00:16:04
So this is how it's looking
halfway along.
287
00:16:04 --> 00:16:09
And let me just carry to the
end.
288
00:16:09 --> 00:16:16
The 2 and the 6 stay the same,
but -- what do I have here?
289
00:16:16 --> 00:16:17.86
Oh, gosh.
290
00:16:17.86 --> 00:16:19
Help me out,
now.
291
00:16:19 --> 00:16:25
What -- so now I'm --
This is still like forward
292
00:16:25 --> 00:16:26
elimination.
293
00:16:26 --> 00:16:29
I got to this point,
which I think is right,
294
00:16:29 --> 00:16:32.04
and now what did I do at this
step?
295
00:16:32.04 --> 00:16:36
I multiplied that pivot by 2 or
that whole equation by 2 and
296
00:16:36 --> 00:16:39
subtracted from that,
so I think I take two 6s,
297
00:16:39 --> 00:16:42.07
which is 12,
away from the 2.
298
00:16:42.07 --> 00:16:47
Do you think minus 10 is my
final right-hand side -- the
299
00:16:47 --> 00:16:54.63
right-hand side that goes with
U, and let me call that once and
300
00:16:54.63 --> 00:16:56
forever the vector c.
301
00:16:56 --> 00:17:02
So c is what happens to b,
and U is what happens to A.
302
00:17:02 --> 00:17:03
Okay.
303
00:17:03 --> 00:17:06
There you've seen elimination
clean.
304
00:17:06 --> 00:17:07
Okay.
305
00:17:07 --> 00:17:11
Oh, what's back substitution?
306
00:17:11 --> 00:17:16
So what are my final equations,
then?
307
00:17:16 --> 00:17:23
Can I copy these equations?
x+2y+z=2 is still there and
308
00:17:23 --> 00:17:27
2y-2z=6 is there,
and 5z=-10.
309
00:17:27 --> 00:17:28
Okay.
310
00:17:28 --> 00:17:36
Those are the equations that
these numbers are telling me
311
00:17:36 --> 00:17:38
about.
312
00:17:38 --> 00:17:42
Those are the equations U x
equals c.
313
00:17:42 --> 00:17:45
Okay, how do I solve them?
314
00:17:45 --> 00:17:49
What one do I solve for first?
z.
315
00:17:49 --> 00:17:56
I see immediately that the
correct value of z is negative
316
00:17:56 --> 2.
317
2. --> 00:17:56
318
00:17:56 --> 00:17:59
And what do I do next?
319
00:17:59 --> 00:18:02
I go back upwards.
320
00:18:02 --> 00:18:04
I now know z here.
321
00:18:04 --> 00:18:08
So, if z is negative 2,
that's 4 there,
322
00:18:08 --> 00:18:10
is that right?
323
00:18:10 --> 00:18:14.63
And so 2 y plus a 4 is 6,
maybe y is 1.
324
00:18:14.63 --> 00:18:18
Going -- this is back
substitution.
325
00:18:18 --> 00:18:23.62
We're doing it on the fly
because it's so easy.
326
00:18:23.62 --> 00:18:29
And then x is --
so x -- 2y is 2 minus 2,
327
00:18:29 --> 00:18:30
maybe x is 2?
328
00:18:30 --> 00:18:34
So you see what back
substitution is.
329
00:18:34 --> 00:18:40
It's the simple step solving
the equations in reverse order
330
00:18:40 --> 00:18:43
because the system is
triangular.
331
00:18:43 --> 00:18:43
Okay.
332
00:18:43 --> 00:18:44
Good.
333
00:18:44 --> 00:18:48
So that's elimination and back
substitution,
334
00:18:48 --> 00:18:53
and I kept the right-hand side
along.
335
00:18:53 --> 00:19:00
Okay, now what do I -- that,
like, is first piece of the
336
00:19:00 --> 00:19:00
lecture.
337
00:19:00 --> 00:19:03.57
What's the second piece?
338
00:19:03.57 --> 00:19:06.8
Matrices are going to get in.
339
00:19:06.8 --> 00:19:11
So I wrote stuff with x,
y-s and z-s in there,
340
00:19:11 --> 00:19:17
then I really,
got the right shorthand,
341
00:19:17 --> 00:19:23
just writing the matrix
entries, and now I want to write
342
00:19:23 --> 00:19:28
the operations that I did in
matrices, right?
343
00:19:28 --> 00:19:34
I've carried the matrices
along, but I haven't said the
344
00:19:34 --> 00:19:41
operation those elimination
steps, I now want to express as
345
00:19:41 --> 00:19:43.67
matrices.
346
00:19:43.67 --> 00:19:44
Okay.
347
00:19:44 --> 00:19:46
Here they come.
348
00:19:46 --> 00:19:52
So now this is elimination
matrices.
349
00:19:52 --> 00:19:53
Okay.
350
00:19:53 --> 00:20:02
Let me take that first step,
which took me from 1 2 1 3 8 1
351
00:20:02 --> 00:20:03
0 4 1.
352
00:20:03 --> 00:20:14
I want to operate on that -- I
want to do elimination on that.
353
00:20:14 --> 00:20:14
Okay.
354
00:20:14 --> 00:20:20
Okay, now I'm remembering a
point I want to single out as
355
00:20:20 --> 00:20:23
especially important.
356
00:20:23 --> 00:20:26
Let me move the board up for
that.
357
00:20:26 --> 00:20:32
Because when we do matrix
operations, we've got to,
358
00:20:32 --> 00:20:37
like, be able to see the big
picture.
359
00:20:37 --> 00:20:37
Okay.
360
00:20:37 --> 00:20:43
Last time, I spoke about the
big picture of -- when I
361
00:20:43 --> 00:20:47
multiply a matrix by a
right-hand side.
362
00:20:47 --> 00:20:52
If I have some matrix there and
I multiply it by 3 4 5,
363
00:20:52 --> 00:20:57
let's say -- so here's a matrix
--
364
00:20:57 --> 00:21:01
what did I say -- well,
I guess I only said it on the
365
00:21:01 --> 00:21:06
videotape, but -- do you
remember how I look at that
366
00:21:06 --> 00:21:08
matrix multiplication?
367
00:21:08 --> 00:21:13
The result of multiplying a
matrix by some vector is a
368
00:21:13 --> 00:21:17
combination of the columns of
the matrix.
369
00:21:17 --> 00:21:19
It's 3 times the first column.
370
00:21:19 --> 00:21:25
It's 3 times column one plus 4
times column two plus 5 times
371
00:21:25 --> 00:21:27
column three.
372
00:21:27 --> 00:21:28
Okay.
373
00:21:28 --> 00:21:34
I'm going to come back to that
multiple times.
374
00:21:34 --> 00:21:43
What I wanted to do now was to
emphasize the parallel thing
375
00:21:43 --> 00:21:44
with rows.
376
00:21:44 --> 00:21:45
Why?
377
00:21:45 --> 00:21:54
Because all our operations here
for this two weeks of the course
378
00:21:54 --> 00:21:58
are row operations.
379
00:21:58 --> 00:22:04
So this isn't what I need for
row operations.
380
00:22:04 --> 00:22:08
Let me do a row operation.
381
00:22:08 --> 00:22:16
Suppose I have my matrix again
and suppose I multiply on the
382
00:22:16 --> 00:22:20
left by some -- let's say 1 2 7.
383
00:22:20 --> 00:22:27
Again, I'm just,
like, saying what the result
384
00:22:27 --> 00:22:28.78
is.
385
00:22:28.78 --> 00:22:35
And then we'll say how matrix
multiplication works and we'll
386
00:22:35 --> 00:22:37
see that it's true.
387
00:22:37 --> 00:22:37
Okay.
388
00:22:37 --> 00:22:44
But maybe already I'm making --
I'm sort of bringing up -- the
389
00:22:44 --> 00:22:50
central idea of linear algebra
is how these matrices work by
390
00:22:50 --> 00:22:54
rows as well as by columns.
391
00:22:54 --> 00:22:54
Okay.
392
00:22:54 --> 00:22:57
How does it work by rows?
393
00:22:57 --> 00:23:01
What -- so that's a row vector.
394
00:23:01 --> 00:23:06
I could say that's a one by
three matrix,
395
00:23:06 --> 00:23:12
a row vector multiplying a
three by three matrix.
396
00:23:12 --> 00:23:14.57
What's the output?
397
00:23:14.57 --> 00:23:20
What's the product of a row
times a matrix?
398
00:23:20 --> 00:23:23
And -- okay,
it's a row.
399
00:23:23 --> 00:23:26
A row -- a column -- I'm sorry.
400
00:23:26 --> 00:23:30
A matrix times a column is a
column.
401
00:23:30 --> 00:23:33
So matrix times a -- yeah.
402
00:23:33 --> 00:23:36
Matrix times a column is a
column.
403
00:23:36 --> 00:23:39
And we know what column it is.
404
00:23:39 --> 00:23:45
Over here, I'm doing a row
times a matrix.
405
00:23:45 --> 00:23:47
And what's the answer?
406
00:23:47 --> 00:23:54
It's one of that first row,
so it's 1 times -- 1 times row
407
00:23:54 --> 00:24:00
one, plus 2 times row two plus 7
times row three.
408
00:24:00 --> 00:24:04
When -- as we do matrix
multiplication,
409
00:24:04 --> 00:24:11
keep your eye on what it's
doing with whole vectors.
410
00:24:11 --> 00:24:17
And what it's doing -- what
it's doing in this case is it's
411
00:24:17 --> 00:24:19.38
combining the rows.
412
00:24:19.38 --> 00:24:24
And we have a combination,
a linear combination of the
413
00:24:24 --> 00:24:25
rows.
414
00:24:25 --> 00:24:27
Okay, I want to use that.
415
00:24:27 --> 00:24:33
Okay, so my question is what's
the matrix that does this first
416
00:24:33 --> 00:24:39
step, that takes -- subtracts 3
of equation one from equation
417
00:24:39 --> 00:24:40
two?
418
00:24:40 --> 00:24:44
That's what I want to do.
419
00:24:44 --> 00:24:52
So this is going to be a matrix
that's going to subtract 3 times
420
00:24:52 --> 00:24:59
row one from row two,
and leaves the other rows the
421
00:24:59 --> 00:24:59
same.
422
00:24:59 --> 00:25:07
Just in -- I mean,
the answer is going to be that.
423
00:25:07 --> 00:25:10
So whatever matrix this is --
and you're going to,
424
00:25:10 --> 00:25:15
like, tell me what matrix will
do it, it's the matrix that
425
00:25:15 --> 00:25:19
leaves the first row unchanged,
leaves the last row unchanged,
426
00:25:19 --> 00:25:23
but takes 3 of these away from
this so it puts a 0 there,
427
00:25:23 --> 00:25:25
a 2 there and a minus 2.
428
00:25:25 --> 00:25:26
Good.
429
00:25:26 --> 00:25:28.63
What matrix will do it?
430
00:25:28.63 --> 00:25:29
It's these.
431
00:25:29 --> 00:25:34.97
It should be a pretty simple
matrix, because we're doing a
432
00:25:34.97 --> 00:25:36
very simple step.
433
00:25:36 --> 00:25:40
We're just doing this step that
changes row two.
434
00:25:40 --> 00:25:44
So actually,
row one is not changing.
435
00:25:44 --> 00:25:48
So tell me how the matrix
should begin.
436
00:25:48 --> 00:25:54
One -- the first row of the
matrix will be 1 0 0,
437
00:25:54 --> 00:26:00
because that's just the right
thing that takes one of that row
438
00:26:00 --> 00:26:06
and none of the other rows,
and that's what we want.
439
00:26:06 --> 00:26:11
What's the last row of the
matrix?
440
00:26:11 --> 00:26:15
0 0 1, because that takes one
of the third row and none of the
441
00:26:15 --> 00:26:17
other rows, that's great.
442
00:26:17 --> 00:26:18
Okay.
443
00:26:18 --> 00:26:22
Now, suppose I didn't want to
do anything at all.
444
00:26:22 --> 00:26:26
Suppose my row -- well,
I guess maybe I had a case here
445
00:26:26 --> 00:26:31
when I already had a 0 and,
didn't have to do anything.
446
00:26:31 --> 00:26:36.94
What matrix does nothing,
like, just leaves you where you
447
00:26:36.94 --> 00:26:37
were?
448
00:26:37 --> 00:26:43
If I put in -- if I put in 0 1
0, that would be -- that would
449
00:26:43 --> 00:26:48
be -- that's the matrix --
what's the name of that matrix?
450
00:26:48 --> 00:26:52
The identity matrix,
right.
451
00:26:52 --> 00:26:54
So it does absolutely nothing.
452
00:26:54 --> 00:26:58
It just multiplies everything
and leaves it where it is.
453
00:26:58 --> 00:27:01
It's like a one,
like the number one,
454
00:27:01 --> 00:27:02.1
for matrices.
455
00:27:02.1 --> 00:27:06
But that's not what we want,
because we want to change this
456
00:27:06 --> 00:27:09
row to -- so what's the correct
--
457
00:27:09 --> 00:27:14
what should I put in here now
to do it right?
458
00:27:14 --> 00:27:17
I want to get -- what do I
want?
459
00:27:17 --> 00:27:23
What I -- I'm after -- I want 3
of row one to get subtracted
460
00:27:23 --> 00:27:23
off.
461
00:27:23 --> 00:27:29
So what's the right matrix,
finish that matrix for me.
462
00:27:29 --> 00:27:31
Negative 3 goes here?
463
00:27:31 --> 00:27:33
And what goes here?
464
00:27:33 --> 00:27:34
That 1.
465
00:27:34 --> 00:27:35
And what goes here?
466
00:27:35 --> 00:27:36
The 0.
467
00:27:36 --> 00:27:38
That's the good matrix.
468
00:27:38 --> 00:27:43
That's the matrix that takes
minus 3 of row one plus the row
469
00:27:43 --> 00:27:46
two and gives the new row 2.
470
00:27:46 --> 00:27:50
Should we just,
like, check some particular
471
00:27:50 --> 00:27:50
entry?
472
00:27:50 --> 00:27:55
How do I check a particular
entry of a matrix in matrix
473
00:27:55 --> 00:27:58
multiplication?
474
00:27:58 --> 00:28:05
Like, suppose I wanted to check
the entry here that's in row
475
00:28:05 --> 00:28:07
two, column three.
476
00:28:07 --> 00:28:14
So where does the entry in row
two, column three come from?
477
00:28:14 --> 00:28:21
I would look at row two of this
guy and column three of this one
478
00:28:21 --> 00:28:24
to get that number.
479
00:28:24 --> 00:28:29
That number comes from the
second row and the third column
480
00:28:29 --> 00:28:35
and I just take this dot product
minus 3 -- I'm multiplying --
481
00:28:35 --> 00:28:38
minus 3 plus 1 and 0 gives the
minus 2.
482
00:28:38 --> 00:28:39
Yeah.
483
00:28:39 --> 00:28:39
It works.
484
00:28:39 --> 00:28:44
So we got various ways to
multiply matrices now.
485
00:28:44 --> 00:28:47
We're sort of,
like -- informally.
486
00:28:47 --> 00:28:50
We've got by columns,
we've got -- well,
487
00:28:50 --> 00:28:55
we will have by columns,
by rows, by each entry at a
488
00:28:55 --> 00:28:55
time.
489
00:28:55 --> 00:28:59
But it's good to see that
matrix multiplication when one
490
00:28:59 --> 00:29:02
of the matrices is so simple.
491
00:29:02 --> 00:29:06
So this guy is our elementary
matrix.
492
00:29:06 --> 00:29:10
Let's call it E for elementary
or elimination.
493
00:29:10 --> 00:29:16.61
And let me put the indexes 2 1,
because it's the matrix that we
494
00:29:16.61 --> 00:29:19.59
needed to fix the 2 1 position.
495
00:29:19.59 --> 00:29:25
It's the matrix that we needed
to get this 2 1 position to be
496
00:29:25 --> 0.
497
0. --> 00:29:25
498
00:29:25 --> 00:29:26.03
Okay.
499
00:29:26.03 --> 00:29:28
Good enough.
500
00:29:28 --> 00:29:30
So what do I do next?
501
00:29:30 --> 00:29:33
I need another matrix,
right?
502
00:29:33 --> 00:29:37
I need to -- there's another
step here.
503
00:29:37 --> 00:29:44
And I want to express the whole
elimination process in matrix
504
00:29:44 --> 00:29:45
language.
505
00:29:45 --> 00:29:49
So tell me what -- so next
step, step two,
506
00:29:49 --> 00:29:52
which was what?
507
00:29:52 --> 00:29:59
Subtract -- what was -- what
was the actual step that we did?
508
00:29:59 --> 00:30:03
I think I subtracted -- do you
remember?
509
00:30:03 --> 00:30:10
I had a 2 in the pivot and a 4
below it, so I subtracted two
510
00:30:10 --> 00:30:14.52
times -- times row two from row
three.
511
00:30:14.52 --> 00:30:16.19
From row three.
512
00:30:16.19 --> 00:30:21
Tell me the matrix that will do
that.
513
00:30:21 --> 00:30:23.49
And tell me its name.
514
00:30:23.49 --> 00:30:28
Okay, it's going to be E,
for elementary or elimination
515
00:30:28 --> 00:30:34
matrix and what's the index
number that I used to tell me
516
00:30:34 --> 00:30:36
what E -- 3, 2,
right?
517
00:30:36 --> 00:30:40
Because it's fixing this 3 2
position.
518
00:30:40 --> 00:30:43.69
And what's the matrix,
now?
519
00:30:43.69 --> 00:30:48
Okay, you remember -- so E 3 2
is supposed to multiply my guy
520
00:30:48 --> 00:30:52
that I have and it's supposed to
produce the right result,
521
00:30:52 --> 00:30:57
which was -- it leaves -- it's
supposed to leave the first row,
522
00:30:57 --> 00:31:02
it's supposed to leave the
second row and it's supposed to
523
00:31:02 --> 00:31:05
straighten out that third row to
this.
524
00:31:05 --> 00:31:09
And what's the matrix that does
that?
525
00:31:09 --> 00:31:11
1 0 0, right?
526
00:31:11 --> 00:31:17
Because we don't change the
first row and the next row we
527
00:31:17 --> 00:31:23
don't change either,
and the last row is the one we
528
00:31:23 --> 00:31:24
do change.
529
00:31:24 --> 00:31:26
And what do I do?
530
00:31:26 --> 00:31:33
Let's see, I subtract two times
-- so what's this row?
531
00:31:33 --> 00:31:37
What's this here?
0, right, because the first
532
00:31:37 --> 00:31:39
row's not involved.
533
00:31:39 --> 00:31:42
It's just in the 3 2 position,
isn't it?
534
00:31:42 --> 00:31:47
This the key number is this
minus the multiplier that goes
535
00:31:47 --> 00:31:51
-- sitting there in that 3 2
position.
536
00:31:51 --> 00:31:58
Is it a minus 2 to subtract 2
and then this is a 1 so that --
537
00:31:58 --> 00:32:05
the overall effect is to take
minus 2 of this row plus 1 of
538
00:32:05 --> 00:32:05
that.
539
00:32:05 --> 00:32:06
Okay.
540
00:32:06 --> 00:32:11
So, I've now given you the
pieces, the elimination
541
00:32:11 --> 00:32:19
matrices, the elementary
matrices that take each step.
542
00:32:19 --> 00:32:20
So now what?
543
00:32:20 --> 00:32:25.91
Now the next point in the
lecture is to put those steps
544
00:32:25.91 --> 00:32:31
together into a matrix that does
it all and see how it all
545
00:32:31 --> 00:32:32
happens.
546
00:32:32 --> 00:32:39
So now I'm going to express the
whole -- everything we did today
547
00:32:39 --> 00:32:45
so far on A was to start with A,
we multiplied it by E 2 1,
548
00:32:45 --> 00:32:52
that was the first step --
and then we multiplied that
549
00:32:52 --> 00:33:00
result by E 3 2 and that led us
to this thing and what was that
550
00:33:00 --> 00:33:01
matrix?
551
00:33:01 --> 00:33:01.8
U.
552
00:33:01.8 --> 00:33:08.57
You see why I like matrix
notation, because there in,
553
00:33:08.57 --> 00:33:16
like, little space -- a few
bits when its compressed on the
554
00:33:16 --> 00:33:23
web -- is everything -- is this
whole lecture.
555
00:33:23 --> 00:33:23
Okay.
556
00:33:23 --> 00:33:28
Now there -- there are
important facts about matrix
557
00:33:28 --> 00:33:30
multiplication.
558
00:33:30 --> 00:33:34
And we're close to maybe the
most important.
559
00:33:34 --> 00:33:36
And that is this.
560
00:33:36 --> 00:33:39
Suppose I ask you this
question.
561
00:33:39 --> 00:33:45
Suppose I start with a matrix A
and I want to end with a matrix
562
00:33:45 --> 00:33:52
U and I want to say what matrix
does the whole job?
563
00:33:52 --> 00:33:59.12
What matrix takes me from A to
U, using the letters I've got?
564
00:33:59.12 --> 00:34:01
And the answer is simple.
565
00:34:01 --> 00:34:07
I'm not asking this as -- but
it's highly important.
566
00:34:07 --> 00:34:14
How would I create the matrix
that does the whole job at once,
567
00:34:14 --> 00:34:20
that does all of elimination in
one shot?
568
00:34:20 --> 00:34:24
It would be -- I would just put
these together,
569
00:34:24 --> 00:34:25
right?
570
00:34:25 --> 00:34:28
In other words,
this is the thing I'm
571
00:34:28 --> 00:34:30
struggling to say.
572
00:34:30 --> 00:34:32
I can move those parentheses.
573
00:34:32 --> 00:34:38
If I keep the matrices in order
-- I can't mess around with the
574
00:34:38 --> 00:34:44
order of the matrices,
but I can change the order that
575
00:34:44 --> 00:34:46
I do the multiplications.
576
00:34:46 --> 00:34:51
I can multiply these two first
-- in other words,
577
00:34:51 --> 00:34:55.64
you see what those parentheses
are doing?
578
00:34:55.64 --> 00:35:01
It's saying -- multiply the Es
first and that gives you the
579
00:35:01 --> 00:35:05
matrix that does everything at
once.
580
00:35:05 --> 00:35:06
Okay.
581
00:35:06 --> 00:35:09
So this fact,
that this is automatically the
582
00:35:09 --> 00:35:14
same as this -- for every matrix
multiplication,
583
00:35:14 --> 00:35:19
which I'm conscious of still
not telling you in every detail,
584
00:35:19 --> 00:35:24.42
but, like, you're seeing how it
works -- and this is highly
585
00:35:24.42 --> 00:35:28
important --
and maybe tell me the long word
586
00:35:28 --> 00:35:33
that describes this law for
matrices, that you can move the
587
00:35:33 --> 00:35:34
parentheses?
588
00:35:34 --> 00:35:37
It's called the associative
law.
589
00:35:37 --> 00:35:39
I think you can now forget
that.
590
00:35:39 --> 00:35:41
But don't forget the law.
591
00:35:41 --> 00:35:46
I mean, like,
forget the word associative.
592
00:35:46 --> 00:35:47
I don't know.
593
00:35:47 --> 00:35:50
But don't forget the law.
594
00:35:50 --> 00:35:56
Because actually,
we'll see so many steps in
595
00:35:56 --> 00:35:59
linear algebra,
so many proofs,
596
00:35:59 --> 00:36:07
even, of main fact come from
just moving the parentheses.
597
00:36:07 --> 00:36:12
And it's not that easy to prove
that this is correct,
598
00:36:12 --> 00:36:16
you have to go into the gory
details of matrix
599
00:36:16 --> 00:36:20
multiplication,
do it both ways and see that
600
00:36:20 --> 00:36:22
you come out the same.
601
00:36:22 --> 00:36:25
Maybe I'll leave the author to
do that.
602
00:36:25 --> 00:36:26
Okay.
603
00:36:26 --> 00:36:28
So there we go.
604
00:36:28 --> 00:36:34
So there's a single matrix,
I could call it E -- while
605
00:36:34 --> 00:36:40
we're talking about these
matrices, tell me one other --
606
00:36:40 --> 00:36:45
there's another type of
elementary matrix,
607
00:36:45 --> 00:36:50
and we already said why we
might need it.
608
00:36:50 --> 00:36:53
We didn't need it in this case.
609
00:36:53 --> 00:36:56
But it's the matrix that
exchanges two rows.
610
00:36:56 --> 00:36:59
It's called a permutation
matrix.
611
00:36:59 --> 00:37:02.55
Can you just,
like, tell me what that would
612
00:37:02.55 --> 00:37:02
be?
613
00:37:02 --> 00:37:06
So I'm just -- like,
this is a slight digression and
614
00:37:06 --> 00:37:11
we'll --
yes, so let me get some -- let
615
00:37:11 --> 00:37:17
me figure out where I'm going to
put a permutation matrix.
616
00:37:17 --> 00:37:21
You'll see I'm always squeezing
stuff in.
617
00:37:21 --> 00:37:23.44
So permutation.
618
00:37:23.44 --> 00:37:29
Or, in fact this one you'll,
like, exchange rows -- shall I
619
00:37:29 --> 00:37:35
exchange rows one and two,
just to make life easy?
620
00:37:35 --> 00:37:42
So if I had my matrix -- no,
let -- let me just do two by
621
00:37:42 --> 00:37:44
two.
|a b; c d|.
622
00:37:44 --> 00:37:50.04
Suppose I want to find the
matrix that exchanges those
623
00:37:50.04 --> 00:37:50.61
rows.
624
00:37:50.61 --> 00:37:51
What is it?
625
00:37:51 --> 00:37:58
So the matrix that exchanges
those rows -- the row I want is
626
00:37:58 --> 00:38:02
c d and it's there.
627
00:38:02 --> 00:38:04
So I better take one of it.
628
00:38:04 --> 00:38:09
And the row I want here is up
top, so I'll take one of that.
629
00:38:09 --> 00:38:12.43
So actually,
I'm just -- the easy way --
630
00:38:12.43 --> 00:38:16
this is my matrix that I'll call
P, for permutation.
631
00:38:16 --> 00:38:21
It's the matrix -- actually,
the easy way to find it is just
632
00:38:21 --> 00:38:25
do the thing to the identity
matrix.
633
00:38:25 --> 00:38:29
Exchange the rows of the
identity matrix and then that's
634
00:38:29 --> 00:38:33
the matrix that will do row
exchanges for you.
635
00:38:33 --> 00:38:36.99
Suppose I wanted to exchange
columns instead.
636
00:38:36.99 --> 00:38:40
Columns have hardly got into
today's lecture,
637
00:38:40 --> 00:38:44.61
but they certainly are going to
be around.
638
00:38:44.61 --> 00:38:49
How could I -- if I started
with this matrix |a b;
639
00:38:49 --> 00:38:55
c d| then I wouldn't -- I'm not
even going to write this down,
640
00:38:55 --> 00:39:00
I'm just going to ask you,
because in elimination,
641
00:39:00 --> 00:39:03
we're doing rows.
642
00:39:03 --> 00:39:07
But suppose we wanted to
exchange the columns of a
643
00:39:07 --> 00:39:08
matrix.
644
00:39:08 --> 00:39:10
How would I do that?
645
00:39:10 --> 00:39:14
What matrix multiplication
would do that job?
646
00:39:14 --> 00:39:16.23
Actually, why not?
647
00:39:16.23 --> 00:39:18
I'll write it down.
648
00:39:18 --> 00:39:23
So this is -- I'll write it
under here and then hide it
649
00:39:23 --> 00:39:24.71
again.
650
00:39:24.71 --> 00:39:25
Okay.
651
00:39:25 --> 00:39:32
Suppose I had my matrix |a b;
c d| and I want to get to a c
652
00:39:32 --> 00:39:35
over here and b d here.
653
00:39:35 --> 00:39:39
What matrix does that job?
654
00:39:39 --> 00:39:46
Can I multiply -- can I cook up
some matrix that produces that
655
00:39:46 --> 00:39:47
answer?
656
00:39:47 --> 00:39:55
You can see from where I put my
hand I was really asking can I
657
00:39:55 --> 00:40:04
put a matrix here on the left
that will exchange columns?
658
00:40:04 --> 00:40:06
And the answer is no.
659
00:40:06 --> 00:40:12
I'm just bringing out again
this point that when I multiply
660
00:40:12 --> 00:40:16
on the left, I'm doing row
operations.
661
00:40:16 --> 00:40:23
So if I want to do a column
operation, where do I put that
662
00:40:23 --> 00:40:26
permutation matrix?
663
00:40:26 --> 00:40:27
On the right.
664
00:40:27 --> 00:40:30
If I put it here,
where I just barely left room
665
00:40:30 --> 00:40:35
for it -- so I'll exchange the
two columns of the identity.
666
00:40:35 --> 00:40:39
Then it comes out right,
because now I'm multiplying a
667
00:40:39 --> 00:40:41
column at a time.
668
00:40:41 --> 00:40:45
This is the first column and
says take one --
669
00:40:45 --> 00:40:50
take none of that column,
one of this one and then you
670
00:40:50 --> 00:40:50
got it.
671
00:40:50 --> 00:40:55
Over here, take one of this
one, none of this one and you've
672
00:40:55 --> 00:40:56
got a c.
673
00:40:56 --> 00:40:58
So, in short,
to do column operations,
674
00:40:58 --> 00:41:02
the matrix multiplies on the
right.
675
00:41:02 --> 00:41:08
To do row operations,
it multiplies on the left.
676
00:41:08 --> 00:41:14
Okay, okay, and it's row
operations that we're really
677
00:41:14 --> 00:41:14
doing.
678
00:41:14 --> 00:41:15
Okay.
679
00:41:15 --> 00:41:19.99
And of course,
I mentioned in passing,
680
00:41:19.99 --> 00:41:26
but I better say it very
clearly that you can't exchange
681
00:41:26 --> 00:41:30
the orders of matrices.
682
00:41:30 --> 00:41:35
And that's just the point I was
making again here.
683
00:41:35 --> 00:41:39.75
A times B is not the same as B
times A.
684
00:41:39.75 --> 00:41:46
You have to keep these matrices
in their Gauss given order here,
685
00:41:46 --> 00:41:47
right?
686
00:41:47 --> 00:41:51
But you can move the
parentheses, so that,
687
00:41:51 --> 00:41:56
in other words,
the commutative law,
688
00:41:56 --> 00:42:03
which would allow you to take
it in the other order is false.
689
00:42:03 --> 00:42:08
So we have to keep it in that
order.
690
00:42:08 --> 00:42:08
Okay.
691
00:42:08 --> 00:42:10
So what next?
692
00:42:10 --> 00:42:14
I could do this multiplication.
693
00:42:14 --> 00:42:15
I could do E 32.
694
00:42:15 --> 00:42:22
So let me come back to see what
that was.
695
00:42:22 --> 00:42:24
Here was E 2 1.
696
00:42:24 --> 00:42:26
And here is E 3 2.
697
00:42:26 --> 00:42:35
And if I multiply those two
matrices together -- E 3 2 and
698
00:42:35 --> 00:42:43
then E 2 1, I'll get a single
matrix that does elimination.
699
00:42:43 --> 00:42:52
I don't want to do it that --
if I do that multiplication --
700
00:42:52 --> 00:42:57
there -- there's a better way
to do this.
701
00:42:57 --> 00:43:03
And so in this last few minutes
of today's lecture,
702
00:43:03 --> 00:43:06
can I anticipate that better
way?
703
00:43:06 --> 00:43:13
The better way is to think not
how do I get from A to U,
704
00:43:13 --> 00:43:16
but how do I get from U back to
A?
705
00:43:16 --> 00:43:22
So reversing steps is going to
come in.
706
00:43:22 --> 00:43:26
Inverse -- I'll use the word
inverse here.
707
00:43:26 --> 00:43:26.96
Okay.
708
00:43:26.96 --> 00:43:32
So let me make the first step
at what's the inverse matrix?
709
00:43:32 --> 00:43:37.55
All the matrices you've seen on
this board have inverses.
710
00:43:37.55 --> 00:43:40
I didn't write any bad matrices
down.
711
00:43:40 --> 00:43:45
We spoke about possible
failure, and for a moment,
712
00:43:45 --> 00:43:49
we put in a matrix that would
fail.
713
00:43:49 --> 00:43:53
But right now,
all these matrices are good,
714
00:43:53 --> 00:43:55
they're all invertible.
715
00:43:55 --> 00:44:00
And let's take the inverse --
well, let me say first what does
716
00:44:00 --> 00:44:03
the inverse mean and find it?
717
00:44:03 --> 00:44:03
Okay.
718
00:44:03 --> 00:44:07
So we're getting a little leg
up on inverses.
719
00:44:07 --> 00:44:12
Okay, so this is the final
moments of today.
720
00:44:12 --> 00:44:15
Sorry, he's still there.
721
00:44:15 --> 00:44:16
Okay.
722
00:44:16 --> 00:44:18
Inverses.
723
00:44:18 --> 00:44:27
Okay, and I'm just going to
take one example and then we're
724
00:44:27 --> 00:44:28
done.
725
00:44:28 --> 00:44:33
The example I'll take will be
that E.
726
00:44:33 --> 00:44:39
So my matrix is 1 0 0 minus 3 1
0 0 0 1.
727
00:44:39 --> 00:44:49
And I want to find the matrix
that undoes that step.
728
00:44:49 --> 00:44:52
So what was that step?
729
00:44:52 --> 00:44:58
The step was subtract 3 times
row one from row two.
730
00:44:58 --> 00:45:02
So what matrix will get me
back?
731
00:45:02 --> 00:45:09.73
What matrix will bring back --
you know, if I started with a 2
732
00:45:09.73 --> 00:45:16
12 2 and I changed it to a 2 6 2
because of this guy,
733
00:45:16 --> 2.
I want to get back to the 2 12
734
2. --> 00:45:21
735
00:45:21 --> 00:45:27
I want to find the matrix which
-- which undoes elimination,
736
00:45:27 --> 00:45:32
the matrix which multiplies
this to give the identity.
737
00:45:32 --> 00:45:37
And you can tell me what I
should do in words first,
738
00:45:37 --> 00:45:43
and then we'll write down the
matrix that does it.
739
00:45:43 --> 00:45:49
If this step subtracted 3 times
row 1 from row 2,
740
00:45:49 --> 00:45:52
what's the inverse step?
741
00:45:52 --> 00:45:56
I add 3 times row one to row
two, right?
742
00:45:56 --> 00:45:58
I add it back.
743
00:45:58 --> 00:46:03
The -- what I subtracted away,
I add back.
744
00:46:03 --> 00:46:08.94
So the inverse matrix in this
case is --
745
00:46:08.94 --> 00:46:13
I now want to add 3 times row
one to row two,
746
00:46:13 --> 00:46:20.17
so I won't change row one,
I won't change row three and
747
00:46:20.17 --> 00:46:24
I'll add 3 times row one to row
two.
748
00:46:24 --> 00:46:28
That's a case where the inverse
is clear.
749
00:46:28 --> 00:46:35
It's clear in words what to do,
because what this did was
750
00:46:35 --> 00:46:38
simple to express.
751
00:46:38 --> 00:46:42
It just changed row two by
subtracting 3 of row one.
752
00:46:42 --> 00:46:44
So to invert it,
I go that way.
753
00:46:44 --> 00:46:47
And if you -- if we do that
calculation, 3 times this row
754
00:46:47 --> 00:46:51
plus 1 times this row,
comes out the right row of the
755
00:46:51 --> 00:46:52
identity.
756
00:46:52 --> 00:46:56
Okay, so inverses are an -- so
if this matrix was E and this
757
00:46:56 --> 00:46:59.98
matrix is I for identity,
then what's the notation for
758
00:46:59.98 --> 00:47:01
this guy?
759
00:47:01 --> 00:47:02
E to the minus one.
760
00:47:02 --> 00:47:03
E inverse.
761
00:47:03 --> 00:47:03
Okay.
762
00:47:03 --> 00:47:05
Let's stop there for today.
763
00:47:05 --> 00:47:09
That's a little jump on what's
coming on Monday.
764
00:47:09 --> 00:47:12
So, see you Monday.