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• 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

• 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?

• 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.

• 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!

If I have a vector sitting here in 2D space

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B2 H-INT UK vector matrix jennifer basis hat coordinate

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