jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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5DUQ3-Y_gX4	That is the question.
5DUQ3-Y_gX4	The graph is a line.
5DUQ3-Y_gX4	So we all think of it as linear, but go ahead.
5DUQ3-Y_gX4	STEVEN G. JOHNSON: Yeah.
5DUQ3-Y_gX4	So does it satisfy the rules?
5DUQ3-Y_gX4	That's the question.
5DUQ3-Y_gX4	So if I multiply the input by 2, does it
5DUQ3-Y_gX4	multiply the output by 2?
5DUQ3-Y_gX4	No.
5DUQ3-Y_gX4	So if we do O of 2--
5DUQ3-Y_gX4	no, let's do 3.
5DUQ3-Y_gX4	3v, that's, what, 6v plus 1.
5DUQ3-Y_gX4	And that's very much not equal to 3Ov,
5DUQ3-Y_gX4	which that would be 6v plus 3.
5DUQ3-Y_gX4	So this one, it does have a name.
5DUQ3-Y_gX4	It's related.
5DUQ3-Y_gX4	These are sometimes called affine.
5DUQ3-Y_gX4	ALAN EDELMAN: Affine, but not linear even if the graph
5DUQ3-Y_gX4	is demonstrably aligned?
5DUQ3-Y_gX4	STEVEN G. JOHNSON: Yeah.
5DUQ3-Y_gX4	ALAN EDELMAN: They're not linear in the sense of linear algebra.
5DUQ3-Y_gX4	STEVEN G. JOHNSON: Right?
5DUQ3-Y_gX4	So another one is clearly multiplication by a matrix.
5DUQ3-Y_gX4	That's why we do matrices in linear algebra.
5DUQ3-Y_gX4	Because they're a nice way of writing down a linear operation
5DUQ3-Y_gX4	if your vs are column vectors.
5DUQ3-Y_gX4	They're not the only way of writing down linear operation.
5DUQ3-Y_gX4	So for example, if you take a column vector
5DUQ3-Y_gX4	and multiply it by 3, you could write
5DUQ3-Y_gX4	that down as a matrix with all 3s along the diagonal.
5DUQ3-Y_gX4	But it's a lot easier to write that down as 3,
5DUQ3-Y_gX4	I'd say, as a scalar, than to write it down as a matrix.
5DUQ3-Y_gX4	So another example, just to be more--
5DUQ3-Y_gX4	so another vector space.
5DUQ3-Y_gX4	If you took 1806, you learned that we can have a-- whoops,
5DUQ3-Y_gX4	I have to get my color scheme.
5DUQ3-Y_gX4	Yeah, so my vectors are red.
5DUQ3-Y_gX4	Suppose the vector space V is the set of functions f of x
5DUQ3-Y_gX4	that take real numbers in and give you real numbers out.
5DUQ3-Y_gX4	Those are a perfectly good vector space.
5DUQ3-Y_gX4	I can take two functions.
5DUQ3-Y_gX4	I can add them or subtract them, get another function.
5DUQ3-Y_gX4	I can take a function and multiply by 2,
5DUQ3-Y_gX4	get another function.
5DUQ3-Y_gX4	ALAN EDELMAN: Wait, how do we get sine plus cosine?
5DUQ3-Y_gX4	STEVEN G. JOHNSON: I get sine x plus cosine x.
5DUQ3-Y_gX4	It's just got some other function.
5DUQ3-Y_gX4	So if you take sine x plus cosine x,
5DUQ3-Y_gX4	that's the function f of x equals sine x plus cosine x.
5DUQ3-Y_gX4	It's another rule that gives you-- takes
5DUQ3-Y_gX4	real numbers to real numbers.
5DUQ3-Y_gX4	And so what would be your linear operators on this?
5DUQ3-Y_gX4	Well, multiplication by a scalar, that, of course, works.
5DUQ3-Y_gX4	So let's think of L on a function f of x is just 2 f
5DUQ3-Y_gX4	of x.
5DUQ3-Y_gX4	That takes a function in, function out.
5DUQ3-Y_gX4	That's linear.
5DUQ3-Y_gX4	What about a linear operator on a function of f of x?
5DUQ3-Y_gX4	Again, that gives you the derivative, just
5DUQ3-Y_gX4	the ordinary 18.01 derivative.
5DUQ3-Y_gX4	This is the 18.01 derivative.
5DUQ3-Y_gX4	Obviously, that only works if the function is differentiable.
5DUQ3-Y_gX4	So maybe we can look at the subspace
5DUQ3-Y_gX4	of differentiable functions.
5DUQ3-Y_gX4	That's also a vector space.
5DUQ3-Y_gX4	Because if I take two differentiable functions
5DUQ3-Y_gX4	and add or subtract or multiply by constants,
5DUQ3-Y_gX4	they're still differentiable.
5DUQ3-Y_gX4	I could also do integration.
5DUQ3-Y_gX4	So if f of x that takes a function f of x in
5DUQ3-Y_gX4	and gives you the integral from, I don't know,
5DUQ3-Y_gX4	0 to x of f prime--
5DUQ3-Y_gX4	no, so f of x prime and dx prime if they're integrable.
5DUQ3-Y_gX4	Again, we need to restrict what functions
5DUQ3-Y_gX4	are allowed if we're taking derivatives or integrals, so
5DUQ3-Y_gX4	things where these exist.
5DUQ3-Y_gX4	But this is perfectly linear.
5DUQ3-Y_gX4	Why?
5DUQ3-Y_gX4	Because if I take the function and I double it,
5DUQ3-Y_gX4	if I double the integrand, it doubles the integrals.
5DUQ3-Y_gX4	If I add two integrands, you add the integrals.
5DUQ3-Y_gX4	Integration is a linear operation.
5DUQ3-Y_gX4	Derivative is a linear operation.
5DUQ3-Y_gX4	Another fun one is suppose we take L of f of x.
5DUQ3-Y_gX4	And the output is the function f of x squared.
5DUQ3-Y_gX4	So this doesn't look linear.
5DUQ3-Y_gX4	I have a square there.
5DUQ3-Y_gX4	But why is this linear?
5DUQ3-Y_gX4	Why?
5DUQ3-Y_gX4	Because, let's see, if I take L of two functions,
5DUQ3-Y_gX4	if I have f of x plus g of x, that should be f of x squared
5DUQ3-Y_gX4	plus g of x squared.
5DUQ3-Y_gX4	I'm squaring the input, not the output.
5DUQ3-Y_gX4	So that's equal to L of f plus L of g.
5DUQ3-Y_gX4	ALAN EDELMAN: So I'll just comment.
5DUQ3-Y_gX4	Leave it to mathematicians to take what most people would
5DUQ3-Y_gX4	think of as just a column of numbers
5DUQ3-Y_gX4	and abstract it out and say that this finite dimensional column
5DUQ3-Y_gX4	of numbers is somehow the same as continuous functions
5DUQ3-Y_gX4	or differentiable functions, satisfies the same axioms.

5DUQ3-Y_gX4

That is the question.

5DUQ3-Y_gX4

The graph is a line.

5DUQ3-Y_gX4

So we all think of it as linear, but go ahead.

5DUQ3-Y_gX4

STEVEN G. JOHNSON: Yeah.

5DUQ3-Y_gX4

So does it satisfy the rules?

5DUQ3-Y_gX4

That's the question.

5DUQ3-Y_gX4

So if I multiply the input by 2, does it

5DUQ3-Y_gX4

multiply the output by 2?

5DUQ3-Y_gX4

No.

5DUQ3-Y_gX4

So if we do O of 2--

5DUQ3-Y_gX4

no, let's do 3.

5DUQ3-Y_gX4

3v, that's, what, 6v plus 1.

5DUQ3-Y_gX4

And that's very much not equal to 3Ov,

5DUQ3-Y_gX4

which that would be 6v plus 3.

5DUQ3-Y_gX4

So this one, it does have a name.

5DUQ3-Y_gX4

It's related.

5DUQ3-Y_gX4

These are sometimes called affine.

5DUQ3-Y_gX4

ALAN EDELMAN: Affine, but not linear even if the graph

5DUQ3-Y_gX4

is demonstrably aligned?

5DUQ3-Y_gX4

STEVEN G. JOHNSON: Yeah.

5DUQ3-Y_gX4

ALAN EDELMAN: They're not linear in the sense of linear algebra.

5DUQ3-Y_gX4

STEVEN G. JOHNSON: Right?

5DUQ3-Y_gX4

So another one is clearly multiplication by a matrix.

5DUQ3-Y_gX4

That's why we do matrices in linear algebra.

5DUQ3-Y_gX4

Because they're a nice way of writing down a linear operation

5DUQ3-Y_gX4

if your vs are column vectors.

5DUQ3-Y_gX4

They're not the only way of writing down linear operation.

5DUQ3-Y_gX4

So for example, if you take a column vector

5DUQ3-Y_gX4

and multiply it by 3, you could write

5DUQ3-Y_gX4

that down as a matrix with all 3s along the diagonal.

5DUQ3-Y_gX4

But it's a lot easier to write that down as 3,

5DUQ3-Y_gX4

I'd say, as a scalar, than to write it down as a matrix.

5DUQ3-Y_gX4

So another example, just to be more--

5DUQ3-Y_gX4

so another vector space.

5DUQ3-Y_gX4

If you took 1806, you learned that we can have a-- whoops,

5DUQ3-Y_gX4

I have to get my color scheme.

5DUQ3-Y_gX4

Yeah, so my vectors are red.

5DUQ3-Y_gX4

Suppose the vector space V is the set of functions f of x

5DUQ3-Y_gX4

that take real numbers in and give you real numbers out.

5DUQ3-Y_gX4

Those are a perfectly good vector space.

5DUQ3-Y_gX4

I can take two functions.

5DUQ3-Y_gX4

I can add them or subtract them, get another function.

5DUQ3-Y_gX4

I can take a function and multiply by 2,

5DUQ3-Y_gX4

get another function.

5DUQ3-Y_gX4

ALAN EDELMAN: Wait, how do we get sine plus cosine?

5DUQ3-Y_gX4

STEVEN G. JOHNSON: I get sine x plus cosine x.

5DUQ3-Y_gX4

It's just got some other function.

5DUQ3-Y_gX4

So if you take sine x plus cosine x,

5DUQ3-Y_gX4

that's the function f of x equals sine x plus cosine x.

5DUQ3-Y_gX4

It's another rule that gives you-- takes

5DUQ3-Y_gX4

real numbers to real numbers.

5DUQ3-Y_gX4

And so what would be your linear operators on this?

5DUQ3-Y_gX4

Well, multiplication by a scalar, that, of course, works.

5DUQ3-Y_gX4

So let's think of L on a function f of x is just 2 f

5DUQ3-Y_gX4

of x.

5DUQ3-Y_gX4

That takes a function in, function out.

5DUQ3-Y_gX4

That's linear.

5DUQ3-Y_gX4

What about a linear operator on a function of f of x?

5DUQ3-Y_gX4

Again, that gives you the derivative, just

5DUQ3-Y_gX4

the ordinary 18.01 derivative.

5DUQ3-Y_gX4

This is the 18.01 derivative.

5DUQ3-Y_gX4

Obviously, that only works if the function is differentiable.

5DUQ3-Y_gX4

So maybe we can look at the subspace

5DUQ3-Y_gX4

of differentiable functions.

5DUQ3-Y_gX4

That's also a vector space.

5DUQ3-Y_gX4

Because if I take two differentiable functions

5DUQ3-Y_gX4

and add or subtract or multiply by constants,

5DUQ3-Y_gX4

they're still differentiable.

5DUQ3-Y_gX4

I could also do integration.

5DUQ3-Y_gX4

So if f of x that takes a function f of x in

5DUQ3-Y_gX4

and gives you the integral from, I don't know,

5DUQ3-Y_gX4

0 to x of f prime--

5DUQ3-Y_gX4

no, so f of x prime and dx prime if they're integrable.

5DUQ3-Y_gX4

Again, we need to restrict what functions

5DUQ3-Y_gX4

are allowed if we're taking derivatives or integrals, so

5DUQ3-Y_gX4

things where these exist.

5DUQ3-Y_gX4

But this is perfectly linear.

5DUQ3-Y_gX4

Why?

5DUQ3-Y_gX4

Because if I take the function and I double it,

5DUQ3-Y_gX4

if I double the integrand, it doubles the integrals.

5DUQ3-Y_gX4

If I add two integrands, you add the integrals.

5DUQ3-Y_gX4

Integration is a linear operation.

5DUQ3-Y_gX4

Derivative is a linear operation.

5DUQ3-Y_gX4

Another fun one is suppose we take L of f of x.

5DUQ3-Y_gX4

And the output is the function f of x squared.

5DUQ3-Y_gX4

So this doesn't look linear.

5DUQ3-Y_gX4

I have a square there.

5DUQ3-Y_gX4

But why is this linear?

5DUQ3-Y_gX4

Why?

5DUQ3-Y_gX4

Because, let's see, if I take L of two functions,

5DUQ3-Y_gX4

if I have f of x plus g of x, that should be f of x squared

5DUQ3-Y_gX4

plus g of x squared.

5DUQ3-Y_gX4

I'm squaring the input, not the output.

5DUQ3-Y_gX4

So that's equal to L of f plus L of g.

5DUQ3-Y_gX4

ALAN EDELMAN: So I'll just comment.

5DUQ3-Y_gX4

Leave it to mathematicians to take what most people would

5DUQ3-Y_gX4

think of as just a column of numbers

5DUQ3-Y_gX4

and abstract it out and say that this finite dimensional column

5DUQ3-Y_gX4

of numbers is somehow the same as continuous functions

5DUQ3-Y_gX4

or differentiable functions, satisfies the same axioms.