- [Narrator] In the last video we were

looking at this particular function.

It's a very non linear function.

And we were picturing it as a transformation

that takes every point x, y in space to the point

x plus sign y, y plus sign of x.

And moreover, we zoomed in on a specific point.

And let me actually write down what

point we zoomed in on, it was (-2,1).

That's something we're gonna want to record here (-2,1).

And I added couple extra grid lines around it

just so we can see in detail what the transformation

does to points that are in the neighborhood of that point.

And over here, this square shows the zoomed

in version of that neighborhood.

And what we saw is that even though the

function as a whole, as a transformation,

looks rather complicated, around that one point,

it looks like a linear function.

It's locally linear so what I'll show you here

is what matrix is gonna tell you the linear

function that this looks like.

And this is gonna be kind of two by two matrix.

I'll make a lot of room for ourselves here.

It'll be a two by two matrix and the way

to think about it is to first go back

to our original setup before the transformation.

And think of just a tiny step to the right.

What I'm gonna think of as a little, partial x.

A tiny step in the x direction.

And what that turns into after the transformation

is gonna be some tiny step in the output space.

And here let me actually kind of draw on

what that tiny step turned into.

It's no longer purely in the x direction.

It has some rightward component.

But now also some downward component.

And to be able to represent this in a nice way,

what I'm gonna do is instead of writing the

entire function as something with

a vector valued output, I'm gonna go ahead

and represent this as a two separate scalar value functions.

I'm gonna write the scalar value functions f1 of x, y.

So I'm just giving a name to x plus sign y.

And f2 of x, y, again all I'm doing is

giving a name to the functions we already have written down.

When I look at this vector, the

consequence of taking a tiny d, x step

in the input space that corresponds to

some two d movement in the output space.

And the x component of that movement.

Right if I was gonna draw this out

and say hey, what's the x component of that movement.

That's something we think of as a little

partial change in f1, the x component of our output.

And if we divide this, if we take you know

partial f1 divided by the size of that

initial tiny change, it basically scales

it up to be a normal sized vector.

Not a tiny nudge but something that's more

constant that doesn't shrink as we

zoom in further and further.

And then similarly the change in the y direction,

right the vertical component of that step

that was still caused by the dx.

Right, it's still caused by that initial

step to the right, that is gonna be

the tiny, partial change in f2.

The y component of the output cause

here we're all just looking in the output space

that was caused by a partial change in the x direction.

And again I kind of like to think about this

we're dividing by a tiny amount.

This partial f2 is really a tiny, tiny nudge.

But by dividing by the size of the initial

tiny nudge that caused it, we're getting

something that's basically a number.

Something that doesn't shrink when

we consider more and more zoomed in versions.

So that, that's all what happens when

we take a tiny step in the x direction.

But another thing you could do, another thing you can

consider is a tiny step in the y direction.

Right cause we wanna know, hey, if

you take a single step some tiny unit upward,

what does that turn into after the transformation.

And what that looks like is this vector

that still has some upward component.

But it also has a rightward component.

And now I'm gonna write its components

as the second column of the matrix.

Because as we know when you're representing

a linear transformation with a matrix,

the first column tells you where the first

basis vector goes and the second column

shows where the second basis vector goes.

If that feels unfamiliar, either

check out the refresher video or

maybe go and look at some of the linear algebra content.

But to figure out the coordinates of this guy,

we do basically the same thing.

Let's say first of all, the change in the x direction

here, the x component of this nudge vector.

That's gonna be given as a partial change to f1, right,

to the x component of the output.

Here we're looking in the outputs base.

We're dealing with f1, f1 and f2

and we're asking what that change was

that was caused by a tiny change in the y direction.

So the change in f1 caused by some tiny step in the y

direction divided by the size of that tiny step.

And then the y component of our output here.

The y component of the step in the outputs base

that was caused by the initial tiny

step upward in the input space.

Well that is the change of f2,

second component of our output as caused by dy.

As caused by that little partial y.

And of course all of this is very specific

to the point that we started at right.

We started at the point (-2,1).

So each of these partial derivatives

is something that really we're saying,

don't take the function, evaluate it at the point (2,-1),

and when you evaluate each one of these

at the point (2,-1) you'll get some number.

And that will give you a very

concrete two by two matrix that's gonna

represent the linear transformation that this

guy looks like once you've zoomed in.

So this matrix here that's full of all

of the partial derivatives has a very special name.

It's called as you may have guessed, the Jacobian.

Or more fully you'd call it the Jacobian Matrix.

And one way to think about it is that it

carries all of the partial differential information right.

It's taking into account both of these components

of the output and both possible inputs.

And giving you a kind of a grid of

what all the partial derivatives are.

But as I hope you see, it's much

more than just a way of recording

what all the partial derivatives are.

There's a reason for organizing it

like this in particular and it really

does come down to this idea of local linearity.

If you understand that the Jacobian Matrix

is fundamentally supposed to represent

what a transformation looks like when you zoom

in near a specific point, almost everything else

about it will start to fall in place.

And in the next video, I'll go ahead

and actually compute this just to

show you what the process looks like.

And how the result we get kind of

matches with the picture we're

looking at, see you then.