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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!
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載入中…

坐标变换 |线性代数的本质,第9节 (Change of basis | Essence of linear algebra, chapter 13)

21 分類 收藏
adam 發佈於 2019 年 10 月 30 日
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