jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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ZK3O402wf1c	And I want to take those equations one at a time and ask
ZK3O402wf1c	--
ZK3O402wf1c	and make a picture of all the points that satisfy --
ZK3O402wf1c	let's take equation number two.
ZK3O402wf1c	If I make a picture of all the points that satisfy --
ZK3O402wf1c	all the x, y, z points that solve this equation --
ZK3O402wf1c	well, first of all, the origin is not one of them.
ZK3O402wf1c	x, y, z -- it being 0, 0, 0 would not solve that equation.
ZK3O402wf1c	So what are some points that do solve the equation?
ZK3O402wf1c	Let's see, maybe if x is one, y and z could be zero.
ZK3O402wf1c	That would work, right?
ZK3O402wf1c	So there's one point.
ZK3O402wf1c	I'm looking at this second equation,
ZK3O402wf1c	here, just, to start with.
ZK3O402wf1c	Let's see.
ZK3O402wf1c	Also, I guess, if z could be one,
ZK3O402wf1c	x and y could be zero, so that would just
ZK3O402wf1c	go straight up that axis.
ZK3O402wf1c	And, probably I'd want a third point here.
ZK3O402wf1c	Let me take x to be zero, z to be zero,
ZK3O402wf1c	then y would be minus a half, right?
ZK3O402wf1c	So there's a third point, somewhere -- oh my -- okay.
ZK3O402wf1c	Let's see.
ZK3O402wf1c	I want to put in all the points that satisfy that equation.
ZK3O402wf1c	Do you know what that bunch of points will be?
ZK3O402wf1c	It's a plane.
ZK3O402wf1c	If we have a linear equation, then, fortunately,
ZK3O402wf1c	the graph of the thing, the plot of all the points that solve it
ZK3O402wf1c	are a plane.
ZK3O402wf1c	These three points determine a plane,
ZK3O402wf1c	but your lecturer is not Rembrandt
ZK3O402wf1c	and the art is going to be the weak point here.
ZK3O402wf1c	So I'm just going to draw a plane, right?
ZK3O402wf1c	There's a plane somewhere.
ZK3O402wf1c	That's my plane.
ZK3O402wf1c	That plane is all the points that solves this guy.
ZK3O402wf1c	Then, what about this one?
ZK3O402wf1c	Two x minus y plus zero z.
ZK3O402wf1c	So z actually can be anything.
ZK3O402wf1c	Again, it's going to be another plane.
ZK3O402wf1c	Each row in a three by three problem
ZK3O402wf1c	gives us a plane in three dimensions.
ZK3O402wf1c	So this one is going to be some other plane --
ZK3O402wf1c	maybe I'll try to draw it like this.
ZK3O402wf1c	And those two planes meet in a line.
ZK3O402wf1c	So if I have two equations, just the first two
ZK3O402wf1c	equations in three dimensions, those give me a line.
ZK3O402wf1c	The line where those two planes meet.
ZK3O402wf1c	And now, the third guy is a third plane.
ZK3O402wf1c	And it goes somewhere.
ZK3O402wf1c	Okay, those three things meet in a point.
ZK3O402wf1c	Now I don't know where that point is, frankly.
ZK3O402wf1c	But -- linear algebra will find it.
ZK3O402wf1c	The main point is that the three planes, because they're not
ZK3O402wf1c	parallel, they're not special.
ZK3O402wf1c	They do meet in one point and that's the solution.
ZK3O402wf1c	But, maybe you can see that this row picture is
ZK3O402wf1c	getting a little hard to see.
ZK3O402wf1c	The row picture was a cinch when we looked at two lines
ZK3O402wf1c	meeting.
ZK3O402wf1c	When we look at three planes meeting,
ZK3O402wf1c	it's not so clear and in four dimensions probably
ZK3O402wf1c	a little less clear.
ZK3O402wf1c	So, can I quit on the row picture?
ZK3O402wf1c	Or quit on the row picture before I've successfully
ZK3O402wf1c	found the point where the three planes meet?
ZK3O402wf1c	All I really want to see is that the row picture consists
ZK3O402wf1c	of three planes and, if everything works right,
ZK3O402wf1c	three planes meet in one point and that's a solution.
ZK3O402wf1c	Now, you can tell I prefer the column picture.
ZK3O402wf1c	Okay, so let me take the column picture.
ZK3O402wf1c	That's x times --
ZK3O402wf1c	so there were two xs in the first equation minus one x is,
ZK3O402wf1c	and no xs in the third.
ZK3O402wf1c	It's just the first column of that.
ZK3O402wf1c	And how many ys are there?
ZK3O402wf1c	There's minus one in the first equations, two in the second
ZK3O402wf1c	and maybe minus three in the third.
ZK3O402wf1c	Just the second column of my matrix.
ZK3O402wf1c	And z times no zs minus one zs and four zs.
ZK3O402wf1c	And it's those three columns, right,
ZK3O402wf1c	that I have to combine to produce the right-hand side,
ZK3O402wf1c	which is zero minus one four.
ZK3O402wf1c	Okay.
ZK3O402wf1c	So what have we got on this left-hand side?
ZK3O402wf1c	A linear combination.
ZK3O402wf1c	It's a linear combination now of three vectors,
ZK3O402wf1c	and they happen to be -- each one is a three dimensional
ZK3O402wf1c	vector, so we want to know what combination of those three
ZK3O402wf1c	vectors produces that one.
ZK3O402wf1c	Shall I try to draw the column picture, then?
ZK3O402wf1c	So, since these vectors have three components --
ZK3O402wf1c	so it's some multiple -- let me draw in the first column
ZK3O402wf1c	as before --
ZK3O402wf1c	x is two and y is minus one.
ZK3O402wf1c	Maybe there is the first column.
ZK3O402wf1c	y -- the second column has maybe a minus one and a two and the y
ZK3O402wf1c	is a minus three, somewhere, there possibly, column two.
ZK3O402wf1c	And the third column has --
ZK3O402wf1c	no zero minus one four, so how shall I draw that?

ZK3O402wf1c

And I want to take those equations one at a time and ask

ZK3O402wf1c

--

ZK3O402wf1c

and make a picture of all the points that satisfy --

ZK3O402wf1c

let's take equation number two.

ZK3O402wf1c

If I make a picture of all the points that satisfy --

ZK3O402wf1c

all the x, y, z points that solve this equation --

ZK3O402wf1c

well, first of all, the origin is not one of them.

ZK3O402wf1c

x, y, z -- it being 0, 0, 0 would not solve that equation.

ZK3O402wf1c

So what are some points that do solve the equation?

ZK3O402wf1c

Let's see, maybe if x is one, y and z could be zero.

ZK3O402wf1c

That would work, right?

ZK3O402wf1c

So there's one point.

ZK3O402wf1c

I'm looking at this second equation,

ZK3O402wf1c

here, just, to start with.

ZK3O402wf1c

Let's see.

ZK3O402wf1c

Also, I guess, if z could be one,

ZK3O402wf1c

x and y could be zero, so that would just

ZK3O402wf1c

go straight up that axis.

ZK3O402wf1c

And, probably I'd want a third point here.

ZK3O402wf1c

Let me take x to be zero, z to be zero,

ZK3O402wf1c

then y would be minus a half, right?

ZK3O402wf1c

So there's a third point, somewhere -- oh my -- okay.

ZK3O402wf1c

Let's see.

ZK3O402wf1c

I want to put in all the points that satisfy that equation.

ZK3O402wf1c

Do you know what that bunch of points will be?

ZK3O402wf1c

It's a plane.

ZK3O402wf1c

If we have a linear equation, then, fortunately,

ZK3O402wf1c

the graph of the thing, the plot of all the points that solve it

ZK3O402wf1c

are a plane.

ZK3O402wf1c

These three points determine a plane,

ZK3O402wf1c

but your lecturer is not Rembrandt

ZK3O402wf1c

and the art is going to be the weak point here.

ZK3O402wf1c

So I'm just going to draw a plane, right?

ZK3O402wf1c

There's a plane somewhere.

ZK3O402wf1c

That's my plane.

ZK3O402wf1c

That plane is all the points that solves this guy.

ZK3O402wf1c

Then, what about this one?

ZK3O402wf1c

Two x minus y plus zero z.

ZK3O402wf1c

So z actually can be anything.

ZK3O402wf1c

Again, it's going to be another plane.

ZK3O402wf1c

Each row in a three by three problem

ZK3O402wf1c

gives us a plane in three dimensions.

ZK3O402wf1c

So this one is going to be some other plane --

ZK3O402wf1c

maybe I'll try to draw it like this.

ZK3O402wf1c

And those two planes meet in a line.

ZK3O402wf1c

So if I have two equations, just the first two

ZK3O402wf1c

equations in three dimensions, those give me a line.

ZK3O402wf1c

The line where those two planes meet.

ZK3O402wf1c

And now, the third guy is a third plane.

ZK3O402wf1c

And it goes somewhere.

ZK3O402wf1c

Okay, those three things meet in a point.

ZK3O402wf1c

Now I don't know where that point is, frankly.

ZK3O402wf1c

But -- linear algebra will find it.

ZK3O402wf1c

The main point is that the three planes, because they're not

ZK3O402wf1c

parallel, they're not special.

ZK3O402wf1c

They do meet in one point and that's the solution.

ZK3O402wf1c

But, maybe you can see that this row picture is

ZK3O402wf1c

getting a little hard to see.

ZK3O402wf1c

The row picture was a cinch when we looked at two lines

ZK3O402wf1c

meeting.

ZK3O402wf1c

When we look at three planes meeting,

ZK3O402wf1c

it's not so clear and in four dimensions probably

ZK3O402wf1c

a little less clear.

ZK3O402wf1c

So, can I quit on the row picture?

ZK3O402wf1c

Or quit on the row picture before I've successfully

ZK3O402wf1c

found the point where the three planes meet?

ZK3O402wf1c

All I really want to see is that the row picture consists

ZK3O402wf1c

of three planes and, if everything works right,

ZK3O402wf1c

three planes meet in one point and that's a solution.

ZK3O402wf1c

Now, you can tell I prefer the column picture.

ZK3O402wf1c

Okay, so let me take the column picture.

ZK3O402wf1c

That's x times --

ZK3O402wf1c

so there were two xs in the first equation minus one x is,

ZK3O402wf1c

and no xs in the third.

ZK3O402wf1c

It's just the first column of that.

ZK3O402wf1c

And how many ys are there?

ZK3O402wf1c

There's minus one in the first equations, two in the second

ZK3O402wf1c

and maybe minus three in the third.

ZK3O402wf1c

Just the second column of my matrix.

ZK3O402wf1c

And z times no zs minus one zs and four zs.

ZK3O402wf1c

And it's those three columns, right,

ZK3O402wf1c

that I have to combine to produce the right-hand side,

ZK3O402wf1c

which is zero minus one four.

ZK3O402wf1c

Okay.

ZK3O402wf1c

So what have we got on this left-hand side?

ZK3O402wf1c

A linear combination.

ZK3O402wf1c

It's a linear combination now of three vectors,

ZK3O402wf1c

and they happen to be -- each one is a three dimensional

ZK3O402wf1c

vector, so we want to know what combination of those three

ZK3O402wf1c

vectors produces that one.

ZK3O402wf1c

Shall I try to draw the column picture, then?

ZK3O402wf1c

So, since these vectors have three components --

ZK3O402wf1c

so it's some multiple -- let me draw in the first column

ZK3O402wf1c

as before --

ZK3O402wf1c

x is two and y is minus one.

ZK3O402wf1c

Maybe there is the first column.

ZK3O402wf1c

y -- the second column has maybe a minus one and a two and the y

ZK3O402wf1c

is a minus three, somewhere, there possibly, column two.

ZK3O402wf1c

And the third column has --

ZK3O402wf1c

no zero minus one four, so how shall I draw that?