jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

video_id stringclasses 7 values	text stringlengths 2 29.3k
ZK3O402wf1c	Do we go through the origin or not?
ZK3O402wf1c	In this case, yes, because there's a zero over there.
ZK3O402wf1c	In this case we don't go through the origin,
ZK3O402wf1c	because if x and y are zero, we don't get three.
ZK3O402wf1c	So, let me again say suppose y is zero,
ZK3O402wf1c	what x do we actually get?
ZK3O402wf1c	If y is zero, then I get x is minus three.
ZK3O402wf1c	So if y is zero, I go along minus three.
ZK3O402wf1c	So there's one point on this second line.
ZK3O402wf1c	Now let me say, well, suppose x is minus one --
ZK3O402wf1c	just to take another x.
ZK3O402wf1c	If x is minus one, then this is a one
ZK3O402wf1c	and I think y should be a one, because if x is minus one,
ZK3O402wf1c	then I think y should be a one and we'll get that point.
ZK3O402wf1c	Is that right?
ZK3O402wf1c	If x is minus one, that's a one.
ZK3O402wf1c	If y is a one, that's a two and the one
ZK3O402wf1c	and the two make three and that point's on the equation.
ZK3O402wf1c	Okay.
ZK3O402wf1c	Now, I should just draw the line, right,
ZK3O402wf1c	connecting those two points at --
ZK3O402wf1c	that will give me the whole line.
ZK3O402wf1c	And if I've done this reasonably well,
ZK3O402wf1c	I think it's going to happen to go through -- well,
ZK3O402wf1c	not happen -- it was arranged to go through that point.
ZK3O402wf1c	So I think that the second line is this one,
ZK3O402wf1c	and this is the all-important point that lies on both lines.
ZK3O402wf1c	Shall we just check that that point which
ZK3O402wf1c	is the point x equal one and y was two, right?
ZK3O402wf1c	That's the point there and that, I believe,
ZK3O402wf1c	solves both equations.
ZK3O402wf1c	Let's just check this.
ZK3O402wf1c	If x is one, I have a minus one plus four equals three, okay.
ZK3O402wf1c	Apologies for drawing this picture
ZK3O402wf1c	that you've seen before.
ZK3O402wf1c	But this -- seeing the row picture --
ZK3O402wf1c	first of all, for n equal 2, two equations and two unknowns,
ZK3O402wf1c	it's the right place to start.
ZK3O402wf1c	Okay.
ZK3O402wf1c	So we've got the solution.
ZK3O402wf1c	The point that lies on both lines.
ZK3O402wf1c	Now can I come to the column picture?
ZK3O402wf1c	Pay attention, this is the key point.
ZK3O402wf1c	So the column picture.
ZK3O402wf1c	I'm now going to look at the columns of the matrix.
ZK3O402wf1c	I'm going to look at this part and this part.
ZK3O402wf1c	I'm going to say that the x part is really x times --
ZK3O402wf1c	you see, I'm putting the two --
ZK3O402wf1c	I'm kind of getting the two equations at once --
ZK3O402wf1c	that part and then I have a y and in the first equation
ZK3O402wf1c	it's multiplying a minus one and in the second equation a two,
ZK3O402wf1c	and on the right-hand side, zero and three.
ZK3O402wf1c	You see, the columns of the matrix, the columns of A
ZK3O402wf1c	are here and the right-hand side b is there.
ZK3O402wf1c	And now what is the equation asking for?
ZK3O402wf1c	It's asking us to find -- somehow to combine that vector
ZK3O402wf1c	and this one in the right amounts to get that one.
ZK3O402wf1c	It's asking us to find the right linear combination --
ZK3O402wf1c	this is called a linear combination.
ZK3O402wf1c	And it's the most fundamental operation in the whole course.
ZK3O402wf1c	It's a linear combination of the columns.
ZK3O402wf1c	That's what we're seeing on the left side.
ZK3O402wf1c	Again, I don't want to write down a big definition.
ZK3O402wf1c	You can see what it is.
ZK3O402wf1c	There's column one, there's column two.
ZK3O402wf1c	I multiply by some numbers and I add.
ZK3O402wf1c	That's a combination -- a linear combination and I want to make
ZK3O402wf1c	those numbers the right numbers to produce zero three.
ZK3O402wf1c	Okay.
ZK3O402wf1c	Now I want to draw a picture that, represents what this --
ZK3O402wf1c	this is algebra.
ZK3O402wf1c	What's the geometry, what's the picture that goes with it?
ZK3O402wf1c	Okay.
ZK3O402wf1c	So again, these vectors have two components,
ZK3O402wf1c	so I better draw a picture like that.
ZK3O402wf1c	So can I put down these columns?
ZK3O402wf1c	I'll draw these columns as they are,
ZK3O402wf1c	and then I'll do a combination of them.
ZK3O402wf1c	So the first column is over two and down one, right?
ZK3O402wf1c	So there's the first column.
ZK3O402wf1c	The first column.
ZK3O402wf1c	Column one.
ZK3O402wf1c	It's the vector two minus one.
ZK3O402wf1c	The second column is --
ZK3O402wf1c	minus one is the first component and up two.
ZK3O402wf1c	It's here.
ZK3O402wf1c	There's column two.
ZK3O402wf1c	So this, again, you see what its components are.
ZK3O402wf1c	Its components are minus one, two.
ZK3O402wf1c	Good.
ZK3O402wf1c	That's this guy.
ZK3O402wf1c	Now I have to take a combination.
ZK3O402wf1c	What combination shall I take?
ZK3O402wf1c	Why not the right combination, what the hell?
ZK3O402wf1c	Okay.
ZK3O402wf1c	So the combination I'm going to take
ZK3O402wf1c	is the right one to produce zero three
ZK3O402wf1c	and then we'll see it happen in the picture.
ZK3O402wf1c	So the right combination is to take x as one of those
ZK3O402wf1c	and two of these.

ZK3O402wf1c

Do we go through the origin or not?

ZK3O402wf1c

In this case, yes, because there's a zero over there.

ZK3O402wf1c

In this case we don't go through the origin,

ZK3O402wf1c

because if x and y are zero, we don't get three.

ZK3O402wf1c

So, let me again say suppose y is zero,

ZK3O402wf1c

what x do we actually get?

ZK3O402wf1c

If y is zero, then I get x is minus three.

ZK3O402wf1c

So if y is zero, I go along minus three.

ZK3O402wf1c

So there's one point on this second line.

ZK3O402wf1c

Now let me say, well, suppose x is minus one --

ZK3O402wf1c

just to take another x.

ZK3O402wf1c

If x is minus one, then this is a one

ZK3O402wf1c

and I think y should be a one, because if x is minus one,

ZK3O402wf1c

then I think y should be a one and we'll get that point.

ZK3O402wf1c

Is that right?

ZK3O402wf1c

If x is minus one, that's a one.

ZK3O402wf1c

If y is a one, that's a two and the one

ZK3O402wf1c

and the two make three and that point's on the equation.

ZK3O402wf1c

Okay.

ZK3O402wf1c

Now, I should just draw the line, right,

ZK3O402wf1c

connecting those two points at --

ZK3O402wf1c

that will give me the whole line.

ZK3O402wf1c

And if I've done this reasonably well,

ZK3O402wf1c

I think it's going to happen to go through -- well,

ZK3O402wf1c

not happen -- it was arranged to go through that point.

ZK3O402wf1c

So I think that the second line is this one,

ZK3O402wf1c

and this is the all-important point that lies on both lines.

ZK3O402wf1c

Shall we just check that that point which

ZK3O402wf1c

is the point x equal one and y was two, right?

ZK3O402wf1c

That's the point there and that, I believe,

ZK3O402wf1c

solves both equations.

ZK3O402wf1c

Let's just check this.

ZK3O402wf1c

If x is one, I have a minus one plus four equals three, okay.

ZK3O402wf1c

Apologies for drawing this picture

ZK3O402wf1c

that you've seen before.

ZK3O402wf1c

But this -- seeing the row picture --

ZK3O402wf1c

first of all, for n equal 2, two equations and two unknowns,

ZK3O402wf1c

it's the right place to start.

ZK3O402wf1c

Okay.

ZK3O402wf1c

So we've got the solution.

ZK3O402wf1c

The point that lies on both lines.

ZK3O402wf1c

Now can I come to the column picture?

ZK3O402wf1c

Pay attention, this is the key point.

ZK3O402wf1c

So the column picture.

ZK3O402wf1c

I'm now going to look at the columns of the matrix.

ZK3O402wf1c

I'm going to look at this part and this part.

ZK3O402wf1c

I'm going to say that the x part is really x times --

ZK3O402wf1c

you see, I'm putting the two --

ZK3O402wf1c

I'm kind of getting the two equations at once --

ZK3O402wf1c

that part and then I have a y and in the first equation

ZK3O402wf1c

it's multiplying a minus one and in the second equation a two,

ZK3O402wf1c

and on the right-hand side, zero and three.

ZK3O402wf1c

You see, the columns of the matrix, the columns of A

ZK3O402wf1c

are here and the right-hand side b is there.

ZK3O402wf1c

And now what is the equation asking for?

ZK3O402wf1c

It's asking us to find -- somehow to combine that vector

ZK3O402wf1c

and this one in the right amounts to get that one.

ZK3O402wf1c

It's asking us to find the right linear combination --

ZK3O402wf1c

this is called a linear combination.

ZK3O402wf1c

And it's the most fundamental operation in the whole course.

ZK3O402wf1c

It's a linear combination of the columns.

ZK3O402wf1c

That's what we're seeing on the left side.

ZK3O402wf1c

Again, I don't want to write down a big definition.

ZK3O402wf1c

You can see what it is.

ZK3O402wf1c

There's column one, there's column two.

ZK3O402wf1c

I multiply by some numbers and I add.

ZK3O402wf1c

That's a combination -- a linear combination and I want to make

ZK3O402wf1c

those numbers the right numbers to produce zero three.

ZK3O402wf1c

Okay.

ZK3O402wf1c

Now I want to draw a picture that, represents what this --

ZK3O402wf1c

this is algebra.

ZK3O402wf1c

What's the geometry, what's the picture that goes with it?

ZK3O402wf1c

Okay.

ZK3O402wf1c

So again, these vectors have two components,

ZK3O402wf1c

so I better draw a picture like that.

ZK3O402wf1c

So can I put down these columns?

ZK3O402wf1c

I'll draw these columns as they are,

ZK3O402wf1c

and then I'll do a combination of them.

ZK3O402wf1c

So the first column is over two and down one, right?

ZK3O402wf1c

So there's the first column.

ZK3O402wf1c

The first column.

ZK3O402wf1c

Column one.

ZK3O402wf1c

It's the vector two minus one.

ZK3O402wf1c

The second column is --

ZK3O402wf1c

minus one is the first component and up two.

ZK3O402wf1c

It's here.

ZK3O402wf1c

There's column two.

ZK3O402wf1c

So this, again, you see what its components are.

ZK3O402wf1c

Its components are minus one, two.

ZK3O402wf1c

Good.

ZK3O402wf1c

That's this guy.

ZK3O402wf1c

Now I have to take a combination.

ZK3O402wf1c

What combination shall I take?

ZK3O402wf1c

Why not the right combination, what the hell?

ZK3O402wf1c

Okay.

ZK3O402wf1c

So the combination I'm going to take

ZK3O402wf1c

is the right one to produce zero three

ZK3O402wf1c

and then we'll see it happen in the picture.

ZK3O402wf1c

So the right combination is to take x as one of those

ZK3O402wf1c

and two of these.