If I have a vector sitting here in 2D space

we have a standard way to describe it with coordinates.

In this case, the vector has coordinates [3, 2],

which means going from its tail to its tip

involves moving 3 units to the right and 2 units up.

Now, the more linear-algebra-oriented way to describe coordinates

is to think of each of these numbers as a scalar

a thing that stretches or squishes vectors.

You think of that first coordinate as scaling i-hat

the vector with length 1, pointing to the right

while the second coordinate scales j-hat

the vector with length 1, pointing straight up.

The tip to tail sum of those two scaled vectors

is what the coordinates are meant to describe.

You can think of these two special vectors

as encapsulating all of the implicit assumptions of our coordinate system.

The fact that the first number indicates rightward motion

that the second one indicates upward motion

exactly how far unit of distances.

All of that is tied up in the choice of i-hat and j-hat

as the vectors which are scalar coordinates are meant to actually scale.

Anyway to translate between vectors and sets of numbers

is called a coordinate system

and the two special vectors, i-hat and j-hat, are called the basis vectors

of our standard coordinate system.

What I'd like to talk about here

is the idea of using a different set of basis vectors.

For example, let's say you have a friend, Jennifer

who uses a different set of basis vectors which I'll call b1 and b2

Her first basis vector b1 points up into the right a little bit

and her second vector b2 points left and up

Now, take another look at that vector that I showed earlier

The one that you and I would describe using the coordinates [3, 2]

using our basis vectors i-hat and j-hat.

Jennifer would actually describe this vector with the coordinates [5/3, 1/3]

What this means is that the particular way to get to that vector

using her two basis vectors

is to scale b1 by 5/3, scale b2 by 1/3

then add them both together.

In a little bit, I'll show you how you could have figured out those two numbers 5/3 and

1/3.

In general, whenever Jennifer uses coordinates to describe a vector

she thinks of her first coordinate as scaling b1

the second coordinate is scaling b2

and she adds the results.

What she gets will typically be completely different

from the vector that you and I would think of as having those coordinates.

To be a little more precise about the setup here

her first basis vector b1

is something that we would describe with the coordinates [2, 1]

and her second basis vector b2

is something that we would describe as [-1, 1].

But it's important to realize from her perspective in her system

those vectors have coordinates [1, 0] and [0, 1]

They are what define the meaning of the coordinates [1, 0] and [0, 1] in her world.

So, in effect, we're speaking different languages

We're all looking at the same vectors in space

but Jennifer uses different words and numbers to describe them.

Let me say a quick word about how I'm representing things here

when I animate 2D space

I typically use this square grid

But that grid is just a construct

a way to visualize our coordinate system

and so it depends on our choice of basis.

Space itself has no intrinsic grid.

Jennifer might draw her own grid

which would be an equally made-up construct

meant is nothing more than a visual tool

to help follow the meaning of her coordinates.

Her origin, though, would actually line up with ours

since everybody agrees on what the coordinates [0, 0] should mean.

It's the thing that you get

when you scale any vector by 0.

But the direction of her axes

and the spacing of her grid lines

will be different, depending on her choice of basis vectors.

So, after all this is set up

a pretty natural question to ask is

How we translate between coordinate systems?

If, for example, Jennifer describes a vector with coordinates [-1, 2]

what would that be in our coordinate system?

How do you translate from her language to ours?

Well, what our coordinates are saying

is that this vector is -1 b1 + 2 b2.

And from our perspective

b1 has coordinates [2, 1]

and b2 has coordinates [-1, 1]

So we can actually compute -1 b1 + 2 b2

as they're represented in our coordinate system

And working this out

you get a vector with coordinates [-4, 1]

So, that's how we would describe the vector that she thinks of as [-1, 2]

This process here of scaling each of her basis vectors

by the corresponding coordinates of some vector

then adding them together

might feel somewhat familiar

It’s matrix-vector multiplication

with a matrix whose columns represent Jennifer's basis vectors in our language

In fact, once you understand matrix-vector multiplication

as applying a certain linear transformation

say, by watching what I've you to be the most important video in this series, chapter 3.

There's a pretty intuitive way to think about what's going on here.

A matrix whose columns represent Jennifer's basis vectors

can be thought of as a transformation

that moves our basis vectors, i-hat and j-hat

the things we think of when we say [1,0] and [0, 1]

to Jennifer's basis vectors

the things she thinks of when she says [1, 0] and [0, 1]

To show how this works

let's walk through what it would mean

to take the vector that we think of as having coordinates [-1, 2]

and applying that transformation.

Before the linear transformation

we’re thinking of this vector

as a certain linear combination of our basis vectors -1 x i-hat + 2 x j-hat.

And the key feature of a linear transformation

is that the resulting vector will be that same linear combination

but of the new basis vectors

-1 times the place where i-hat lands + 2 times the place where j-hat lands.

So what this matrix does

is transformed our misconception of what Jennifer means

into the actual vector that she's referring to.

I remember that when I was first learning this

it always felt kind of backwards to me.

Geometrically, this matrix transforms our grid into Jennifer's grid.

But numerically, it's translating a vector described in her language to our language.

What made it finally clicked for me

was thinking about how it takes our misconception of what Jennifer means

the vector we get using the same coordinates but in our system

then it transforms it into the vector that she really meant.

What about going the other way around?

In the example I used earlier this video

when I have the vector with coordinates [3, 2] in our system

How did I compute that it would have coordinates [5/3, 1/3] in Jennifer system?

You start with that change of basis matrix

that translates Jennifer's language into ours

then you take its inverse.

Remember, the inverse of a transformation

is a new transformation that corresponds to playing that first one backwards.

In practice, especially when you're working in more than two dimensions

you'd use a computer to compute the matrix that actually represents this inverse.

In this case, the inverse of the change of basis matrix

that has Jennifer's basis as its columns

ends up working out to have columns [1/3, -1/3] and [1/3, 2/3]

So, for example

to see what the vector [3, 2] looks like in Jennifer's system

we multiply this inverse change of basis matrix by the vector [3, 2]

which works out to be [5/3, 1/3]

So that, in a nutshell

is how to translate the description of individual vectors

back and forth between coordinate systems.

The matrix whose columns represent Jennifer's basis vectors

but written in our coordinates

translates vectors from her language into our language.

And the inverse matrix does the opposite.

But vectors aren't the only thing that we describe using coordinates.

For this next part

it's important that you're all comfortable

representing transformations with matrices

and that you know how matrix multiplication

corresponds to composing successive transformations.

Definitely pause and take a look at chapters 3 and 4

if any of that feels uneasy.

Consider some linear transformation

like a 90°counterclockwise rotation.

When you and I represent this with the matrix

we follow where the basis vectors i-hat and j-hat each go.

i-hat ends up at the spot with coordinates [0, 1]

and j-hat end up at the spot with coordinates [-1, 0]

So those coordinates become the columns of our matrix

but this representation

is heavily tied up in our choice of basis vectors

from the fact that we're following i-hat and j-hat in the first place

to the fact that we're recording their landing spots

in our own coordinate system.

How would Jennifer describe this same 90°rotation of space?

You might be tempted to just

translate the columns of our rotation matrix into Jennifer's language.

But that's not quite right.

Those columns represent where our basis vectors i-hat and j-hat go.

But the matrix that Jennifer wants

should represent where her basis vectors land

and it needs to describe those landing spots in her language.

Here's a common way to think of how this is done.

Start with any vector written in Jennifer's language.

Rather than trying to follow what happens to it in terms of her language

first, we're going to translate it into our language

using the change of basis matrix

the one whose columns represent her basis vectors in our language.

This gives us the same vector

but now written in our language.

Then, apply the transformation matrix to what you get

by multiplying it on the left.

This tells us where that vector lands

but still in our language.

So as a last step

apply the inverse change of basis matrix

multiplied on the left as usual

to get the transformed vector

but now in Jennifer's language.

Since we could do this

with any vector written in her language

first, applying the change of basis

then, the transformation

then, the inverse change of basis

That composition of three matrices

gives us the transformation matrix in Jennifer's language.

it takes in a vector of her language

and spits out the transformed version of that vector in her language

For this specific example

when Jennifer's basis vectors look like [2, 1] and [-1, 1] in our language

and when the transformation is a 90°rotation

the product of these three matrices

if you work through it

has columns [1/3, 5/3] and [-2/3, -1/3]

So if Jennifer multiplies that matrix

by the coordinates of a vector in her system

it will return the 90°rotated version of that vector

expressed in her coordinate system.

In general, whenever you see an expression like A^(-1) M A

it suggests a mathematical sort of empathy.

That middle matrix represents a transformation of some kind, as you see it

and the outer two matrices represent the empathy, the shift in perspective

and the full matrix product represents that same transformation

but as someone else sees it.

For those of you wondering why we care about alternate coordinate systems

the next video on eigen vectors and eigen values

will give a really important example of this.

See you then!