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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

  • 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!

If I have a vector sitting here in 2D space

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B2 H-INT UK vector matrix jennifer basis hat coordinate

Change of basis | Essence of linear algebra, chapter 13

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    adam   posted on 2019/10/30
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