Doing a transformation in a changed basis

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Del curso dictado por Imperial College London

Mathematics for Machine Learning: Linear Algebra

2604 calificaciones

Imperial College London

Mathematics for Machine Learning: Linear Algebra

2604 calificaciones

Curso 1 de 3 en SpecializationMathematics for Machine Learning

In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we're aiming at data-driven applications, we'll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you'll write code blocks and encounter Jupyter notebooks in Python, but don't worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning.

De la lección

Matrices make linear mappings

In Module 4, we continue our discussion of matrices; first we think about how to code up matrix multiplication and matrix operations using the Einstein Summation Convention, which is a widely used notation in more advanced linear algebra courses. Then, we look at how matrices can transform a description of a vector from one basis (set of axes) to another. This will allow us to, for example, figure out how to apply a reflection to an image and manipulate images. We'll also look at how to construct a convenient basis vector set in order to do such transformations. Then, we'll write some code to do these transformations and apply this work computationally.

Matrices changing basis11:14

Doing a transformation in a changed basis6:30

Conoce a los instructores

David Dye
Professor of Metallurgy
Department of Materials
Samuel J. Cooper
Lecturer
Dyson School of Design Engineering
A. Freddie Page
Strategic Teaching Fellow
Dyson School of Design Engineering

Now, let's do one last thing.

For this example, let's go back to the first transformation of Bear's,

the first set of Bear's axis which were three, one

was his first axis,

and one, one, was his second axis.

And we said that was the matrix B which described

Bear's basis in my world.

Now, let's say I have a vector,

which I want to do something to.

I want to rotate it, or shift it,

or transform it, or send it somewhere.

And I only know that vector in Bear's world.

So relative to Bear's coordinate system.

And I will call that vector x, y in Bear's world.

Now my problem is,

I don't know what this transformation,

rotations say, could be a reflection, whatever it is.

I don't know what that transformation matrix is in Bear's funny coordinate system.

I only know it in my nice sensible coordinate system, because Bear is crazy,

he has these crazy normal orthogonal non-unit vectors he's using to describe his axis.

And I only know the transformation in my system.

So the first thing I'm going to do actually is

I am going to take Bear's vector and I'm going to put it in my world.

And then, I can apply the rotation in my world.

So first, I'm going to do this multiplication,

and that gives me the vector in my world, right?

That's nice. Then what I'm going to do I'm is going to do this rotation.

So let's call this rotation R. I will say that, in my world,

I know that that rotation is one over root two times one minus one, one, one.

What that does is that it actually rotates,

if I've got some basis vectors like that,

it turns them to one, one,

goes over there, and this one goes to minus one, one.

So they both get rotated.

So this is a rotation actually of 45 degrees.

So then, I've got my vector when I multiply it by R,

which is my 45 degree rotation.

I've then got the vector

rotated and in my description, in my basis.

So then, how do I get a vector in my basis back into Bear's basis?

Because Bear doesn't care.

Bear isn't interested in my basis.

He's interested in what this rotation is and what

the vector is when it's rotated in his basis.

Because he only really wants to know about

his coordinate system because he's a Bear of little brain.

So, I need to then turn it back into a vector in Bear's basis.

So I'm got to have to use B to the minus one,

which is the thing that does that.

And that is the vector three, one,

leave the ones on the diagonal,

flip the sign if the other two.

It's easy for two by two.

And the determinant of that is three minus one.

So, I've got to divide by the determinant of two.

So that, when I do that whole set,

I've then got the vector back in Bear's basis after the rotation.

So I figured out how to do this rotation in Bear's world.

So if I multiply those three out,

then I just get what that rotation matrix is in Bear's coordinate system.

So now let's just do the sums in order to make ourselves happy that we can all do this.

So I've got one times three minus one times one,

so I've got two.

And then if I do this one,

I've got one times one minus one times one.

So I've got nought,

then I've got one times three plus one times one, I've got four.

And I've got one times one plus one plus one times one it's going to give me two.

And I've got to keep my one over root two in there.

So that's what RB is.

Then if I do B to the minus one times RB,

I'm going to multiply that guy by that guy,

and I get it back in Bear's basis.

So, then I've got to do that.

So I've got three times two minus four is six minus four gives

me two divided by two gives me minus one.

So, three times two is six,

minus four is minus two divided by two is minus one.

Then I do it again,

And as I go through I will get the answer which I will just write out for you,

which is one over root two,

timse minus one, minus one, five, three. So that's what

a 45 degree rotation looks like in Bear's coordinate system.

Notice it's completely different to the one in my standard basis.

It isn't very easy necessarily or obvious to intuit just from your head,

you have to do the calculation.

So, if you want to do some kind of transformation but in some funny basis,

this equation, B to the minus one times R times B is gonna be very useful.

So to step back here,

the point here is that when we transform to non-orthonormal coordinate systems,

then the transformation matrices also change.

And we have to be mindful of that.

And this is sort of algebra you see all the time,

we've got the transformation matrix R and B inverse, B

wrapping round it to do

the translation from my world into the world of the new basis system.

So thanks Bear. That was really useful.

So what we've done in this video is we've looked at how the numbers in the matrix were

used to describe the vector change when we change the basis,

and we thought about coordinate systems.

It's a bit counter-intuitive,

but Bear helped us out.

We also saw that if we have an orthonormal basis,

then we can just use the dot product to do the projections like we said before.

And we also saw that if we use an arbitrary basis,

then the transformations themselves will change.

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