jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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5DUQ3-Y_gX4	I'm not asking for df.
5DUQ3-Y_gX4	Of course, they're related.
5DUQ3-Y_gX4	And let's see if we can move this out of the way.
5DUQ3-Y_gX4	Good.
5DUQ3-Y_gX4	So what was I saying?
5DUQ3-Y_gX4	Yes.
5DUQ3-Y_gX4	So now, what I want to do is basically use
5DUQ3-Y_gX4	this as the definition of a derivative, a more general
5DUQ3-Y_gX4	definition of a derivative.
5DUQ3-Y_gX4	So the linear algebra notion is that we have
5DUQ3-Y_gX4	what's called a linear operator.
5DUQ3-Y_gX4	So basically, the change in the output df
5DUQ3-Y_gX4	is going to be a linear operator.
5DUQ3-Y_gX4	Let me write that as times, which
5DUQ3-Y_gX4	I'm going to call f prime of x times the change in the input.
5DUQ3-Y_gX4	This is our-- whoops.
5DUQ3-Y_gX4	Actually, but my input is in red, right?
5DUQ3-Y_gX4	That's my color code.
5DUQ3-Y_gX4	My color scheme is input is red and output is blue.
5DUQ3-Y_gX4	So this is our dx.
5DUQ3-Y_gX4	So I'm going to interpret this more generally.
5DUQ3-Y_gX4	If x is going to be some kind of vector or matrix or whatever,
5DUQ3-Y_gX4	this is just going to be a linear operation on this.
5DUQ3-Y_gX4	And of course, for numbers, a linear operation,
5DUQ3-Y_gX4	this is just a number.
5DUQ3-Y_gX4	If dx is a number, the only linear operation you can do
5DUQ3-Y_gX4	is multiply by a number.
5DUQ3-Y_gX4	So let me just remind you if you haven't taken linear algebra
5DUQ3-Y_gX4	for a while, a review.
5DUQ3-Y_gX4	Let's talk about what a linear operator is.
5DUQ3-Y_gX4	So suppose we have some given vectors, v,
5DUQ3-Y_gX4	in some vector space.
5DUQ3-Y_gX4	That would be capital V. And remember,
5DUQ3-Y_gX4	a vector space is anything where you can basically
5DUQ3-Y_gX4	add, subtract, and multiply by scalars.
5DUQ3-Y_gX4	We have a plus or minus and times scalar operations
5DUQ3-Y_gX4	that stay in our vector space.
5DUQ3-Y_gX4	That's what the informal definition of a vector space
5DUQ3-Y_gX4	is.
5DUQ3-Y_gX4	You can write out axioms and so forth,
5DUQ3-Y_gX4	but that's basically what it means.
5DUQ3-Y_gX4	A linear operator, this is what we're
5DUQ3-Y_gX4	going to mean by linearization in the derivative.
5DUQ3-Y_gX4	It has to be linear.
5DUQ3-Y_gX4	What does it mean to be linear in general?
5DUQ3-Y_gX4	So we're going to call this--
5DUQ3-Y_gX4	I'm going to denote this by, let's say, L of--
5DUQ3-Y_gX4	let me denote it by square brackets or just by Lv.
5DUQ3-Y_gX4	When it's clear enough, I'll just write it
5DUQ3-Y_gX4	as if it were a multiplication.
5DUQ3-Y_gX4	Often, that'll be clear enough.
5DUQ3-Y_gX4	This is really acting on--
5DUQ3-Y_gX4	when I write Lv, it's not necessarily
5DUQ3-Y_gX4	an ordinary multiplication.
5DUQ3-Y_gX4	This is just going to be acting on v.
5DUQ3-Y_gX4	So a linear operator is a rule that basically takes a vector
5DUQ3-Y_gX4	and gives you a vector out maybe in a different vector space.
5DUQ3-Y_gX4	So this is L takes a vector in, v in,
5DUQ3-Y_gX4	and it gives you a vector--
5DUQ3-Y_gX4	that's a terrible L--
5DUQ3-Y_gX4	Lv out maybe in a different vector space.
5DUQ3-Y_gX4	And linearity means what you think it means.
5DUQ3-Y_gX4	It means if you take, for example,
5DUQ3-Y_gX4	L of v1 plus v2, if you take the sum of the inputs, that's
5DUQ3-Y_gX4	the same thing as L of v1 plus L of v2.
5DUQ3-Y_gX4	So if you add inputs, that's the same thing
5DUQ3-Y_gX4	as adding the outputs or if you multiply by a scalar.
5DUQ3-Y_gX4	And so as usual in linear algebra,
5DUQ3-Y_gX4	Greek letters are going to denote scalars as you would.
5DUQ3-Y_gX4	So that's equal to--
5DUQ3-Y_gX4	you can pull out the scalar.
5DUQ3-Y_gX4	And so the nice thing about linear operators
5DUQ3-Y_gX4	is we can define them on lots of kinds of vector spaces.
5DUQ3-Y_gX4	So let's just do a couple of examples
5DUQ3-Y_gX4	just to make sure we're on the same page here.
5DUQ3-Y_gX4	And so, for example, you could just
5DUQ3-Y_gX4	have L is multiplication by a scalar.
5DUQ3-Y_gX4	So you can have just L of v is just alpha v.
5DUQ3-Y_gX4	That's a perfectly good linear operation.
5DUQ3-Y_gX4	And if your vector space, if your vs are scalars,
5DUQ3-Y_gX4	this is the only option.
5DUQ3-Y_gX4	If vs are, say, real numbers, that's
5DUQ3-Y_gX4	a perfectly good vector space.
5DUQ3-Y_gX4	Another one that you're very familiar with
5DUQ3-Y_gX4	is if L is a multiplication by a matrix.
5DUQ3-Y_gX4	ALAN EDELMAN: Steven, maybe I'll just point out sometimes people
5DUQ3-Y_gX4	like to ask me.
5DUQ3-Y_gX4	Wait, I thought that the linear operators on scalars are,
5DUQ3-Y_gX4	I think in high school notation, y equals mx plus b, right?
5DUQ3-Y_gX4	It's scalar times--
5DUQ3-Y_gX4	STEVEN G. JOHNSON: Yeah, yeah.
5DUQ3-Y_gX4	ALAN EDELMAN: --plus an offset that may not be 0.
5DUQ3-Y_gX4	So what's going on here?
5DUQ3-Y_gX4	Is that linear or not linear?
5DUQ3-Y_gX4	STEVEN G. JOHNSON: Yeah.
5DUQ3-Y_gX4	So what about-- yeah, so let's do that.
5DUQ3-Y_gX4	Let me call it a different thing.
5DUQ3-Y_gX4	What letter should I use?
5DUQ3-Y_gX4	O, let's use O. Ov equals 2v plus 1 for v is real numbers.
5DUQ3-Y_gX4	ALAN EDELMAN: Is that linear or not linear?

5DUQ3-Y_gX4

I'm not asking for df.

5DUQ3-Y_gX4

Of course, they're related.

5DUQ3-Y_gX4

And let's see if we can move this out of the way.

5DUQ3-Y_gX4

Good.

5DUQ3-Y_gX4

So what was I saying?

5DUQ3-Y_gX4

Yes.

5DUQ3-Y_gX4

So now, what I want to do is basically use

5DUQ3-Y_gX4

this as the definition of a derivative, a more general

5DUQ3-Y_gX4

definition of a derivative.

5DUQ3-Y_gX4

So the linear algebra notion is that we have

5DUQ3-Y_gX4

what's called a linear operator.

5DUQ3-Y_gX4

So basically, the change in the output df

5DUQ3-Y_gX4

is going to be a linear operator.

5DUQ3-Y_gX4

Let me write that as times, which

5DUQ3-Y_gX4

I'm going to call f prime of x times the change in the input.

5DUQ3-Y_gX4

This is our-- whoops.

5DUQ3-Y_gX4

Actually, but my input is in red, right?

5DUQ3-Y_gX4

That's my color code.

5DUQ3-Y_gX4

My color scheme is input is red and output is blue.

5DUQ3-Y_gX4

So this is our dx.

5DUQ3-Y_gX4

So I'm going to interpret this more generally.

5DUQ3-Y_gX4

If x is going to be some kind of vector or matrix or whatever,

5DUQ3-Y_gX4

this is just going to be a linear operation on this.

5DUQ3-Y_gX4

And of course, for numbers, a linear operation,

5DUQ3-Y_gX4

this is just a number.

5DUQ3-Y_gX4

If dx is a number, the only linear operation you can do

5DUQ3-Y_gX4

is multiply by a number.

5DUQ3-Y_gX4

So let me just remind you if you haven't taken linear algebra

5DUQ3-Y_gX4

for a while, a review.

5DUQ3-Y_gX4

Let's talk about what a linear operator is.

5DUQ3-Y_gX4

So suppose we have some given vectors, v,

5DUQ3-Y_gX4

in some vector space.

5DUQ3-Y_gX4

That would be capital V. And remember,

5DUQ3-Y_gX4

a vector space is anything where you can basically

5DUQ3-Y_gX4

add, subtract, and multiply by scalars.

5DUQ3-Y_gX4

We have a plus or minus and times scalar operations

5DUQ3-Y_gX4

that stay in our vector space.

5DUQ3-Y_gX4

That's what the informal definition of a vector space

5DUQ3-Y_gX4

is.

5DUQ3-Y_gX4

You can write out axioms and so forth,

5DUQ3-Y_gX4

but that's basically what it means.

5DUQ3-Y_gX4

A linear operator, this is what we're

5DUQ3-Y_gX4

going to mean by linearization in the derivative.

5DUQ3-Y_gX4

It has to be linear.

5DUQ3-Y_gX4

What does it mean to be linear in general?

5DUQ3-Y_gX4

So we're going to call this--

5DUQ3-Y_gX4

I'm going to denote this by, let's say, L of--

5DUQ3-Y_gX4

let me denote it by square brackets or just by Lv.

5DUQ3-Y_gX4

When it's clear enough, I'll just write it

5DUQ3-Y_gX4

as if it were a multiplication.

5DUQ3-Y_gX4

Often, that'll be clear enough.

5DUQ3-Y_gX4

This is really acting on--

5DUQ3-Y_gX4

when I write Lv, it's not necessarily

5DUQ3-Y_gX4

an ordinary multiplication.

5DUQ3-Y_gX4

This is just going to be acting on v.

5DUQ3-Y_gX4

So a linear operator is a rule that basically takes a vector

5DUQ3-Y_gX4

and gives you a vector out maybe in a different vector space.

5DUQ3-Y_gX4

So this is L takes a vector in, v in,

5DUQ3-Y_gX4

and it gives you a vector--

5DUQ3-Y_gX4

that's a terrible L--

5DUQ3-Y_gX4

Lv out maybe in a different vector space.

5DUQ3-Y_gX4

And linearity means what you think it means.

5DUQ3-Y_gX4

It means if you take, for example,

5DUQ3-Y_gX4

L of v1 plus v2, if you take the sum of the inputs, that's

5DUQ3-Y_gX4

the same thing as L of v1 plus L of v2.

5DUQ3-Y_gX4

So if you add inputs, that's the same thing

5DUQ3-Y_gX4

as adding the outputs or if you multiply by a scalar.

5DUQ3-Y_gX4

And so as usual in linear algebra,

5DUQ3-Y_gX4

Greek letters are going to denote scalars as you would.

5DUQ3-Y_gX4

So that's equal to--

5DUQ3-Y_gX4

you can pull out the scalar.

5DUQ3-Y_gX4

And so the nice thing about linear operators

5DUQ3-Y_gX4

is we can define them on lots of kinds of vector spaces.

5DUQ3-Y_gX4

So let's just do a couple of examples

5DUQ3-Y_gX4

just to make sure we're on the same page here.

5DUQ3-Y_gX4

And so, for example, you could just

5DUQ3-Y_gX4

have L is multiplication by a scalar.

5DUQ3-Y_gX4

So you can have just L of v is just alpha v.

5DUQ3-Y_gX4

That's a perfectly good linear operation.

5DUQ3-Y_gX4

And if your vector space, if your vs are scalars,

5DUQ3-Y_gX4

this is the only option.

5DUQ3-Y_gX4

If vs are, say, real numbers, that's

5DUQ3-Y_gX4

a perfectly good vector space.

5DUQ3-Y_gX4

Another one that you're very familiar with

5DUQ3-Y_gX4

is if L is a multiplication by a matrix.

5DUQ3-Y_gX4

ALAN EDELMAN: Steven, maybe I'll just point out sometimes people

5DUQ3-Y_gX4

like to ask me.

5DUQ3-Y_gX4

Wait, I thought that the linear operators on scalars are,

5DUQ3-Y_gX4

I think in high school notation, y equals mx plus b, right?

5DUQ3-Y_gX4

It's scalar times--

5DUQ3-Y_gX4

STEVEN G. JOHNSON: Yeah, yeah.

5DUQ3-Y_gX4

ALAN EDELMAN: --plus an offset that may not be 0.

5DUQ3-Y_gX4

So what's going on here?

5DUQ3-Y_gX4

Is that linear or not linear?

5DUQ3-Y_gX4

STEVEN G. JOHNSON: Yeah.

5DUQ3-Y_gX4

So what about-- yeah, so let's do that.

5DUQ3-Y_gX4

Let me call it a different thing.

5DUQ3-Y_gX4

What letter should I use?

5DUQ3-Y_gX4

O, let's use O. Ov equals 2v plus 1 for v is real numbers.

5DUQ3-Y_gX4

ALAN EDELMAN: Is that linear or not linear?