Doing a transformation in a changed basis

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Del curso dictado por Imperial College London

Mathematics for Machine Learning: Linear Algebra

3020 calificaciones

Imperial College London

Mathematics for Machine Learning: Linear Algebra

3020 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 basis4:37

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 look at how we do the transformation of

a vector that's already in a change basis.

Last time, we looked at bears basis.

Bears basis had a first axis of three,

one and a second axis of one, one.

Let's say I have a vector x,

y defined in bears basis.

As I want to transform it by doing something

like a rotation of 45 degrees.

But the problem is, I don't know how to write

a 45 degree rotation in bears funny coordinate system.

I only know how to write down

a 45 degree rotation in my normal 1,

0, 0, 1 system.

So in my system which is 1, 0,

0, 1 a 45 degree rotation rotates 1,

0 up like that.

So if it's still a unit vector at 1,

0 over root 2, 1, over root 2,

that is a normalized 1, 1.

It takes 0, 1 round to minus 1 over root 2,

1 over root 2.

That is that 45 degrees there,

that 45 degrees there.

So I can write it down the rotation in my notation.

Let's call it R being a 45 degree rotation,

as being 1 over root 2 times 1,

1 minus 1, 1.

That's what a 45 degree rotation is in my world.

So what I need to do is first transform the vector x,

y into my basis.

I do that by multiplying it by B.

Then I can apply

my nice sensible rotation R

to that vector that's now in my basis.

So what I get here when I do B,

I've got the vector in my coordinate frame.

Now the problem is bear doesn't care about my world,

he wants to get the rotation in his basis.

So then I have to transform the resulting vector,

back into bears basis.

I do that by applying B to the minus 1.

B to the minus 1, I get by

flipping the terms on the leading diagonal,

taking minus the off-diagonal terms,

and dividing by the determinant here

which is 2 so I multiply it by a half.

So that then gives me,

the vector back

in bears frame.

So overall what I've done,

is I've done B to the minus 1 times R, times B.

What that's giving me,

is it's given me the rotation,

in bears coordinate system. Which is really neat.

So now what we have to do is do the sums.

When we do that R, B gives us this,

and B to the -1 RB gives us this.

I've written them down there, so pause and ponder if

you want to verify those on your own.

So this is what

a 45 degree rotation looks

like in bears coordinate system.

Notice that it's completely

different to the one in my standard basis.

It isn't very easy

necessarily or obvious to intuit just out of your head,

you have to do the calculation.

So if you want to do some translation

but in some funny basis,

this equation B to the minus 1 RB,

is going to be very useful.

To step back here, the point is that if we want to

transform to normal form normal coordinate systems,

then the translation matrices also change.

We have to be mindful of that.

This is the algebra you see all the time.

We've got the transformation matrix R,

wrapped around by B,

B is the minus 1, that does

the translation from my world

to the world of the new basis system.

So thanks a lot bear, you really

helped us out to understand all this stuff.

So what we've done in these two videos is we've looked at

how the numbers in a vector change,

where we change the basis.

We thought about the coordinate systems and how to do

transformations in non-orthogonal coordinate systems.

It's been quite hard work,

but this really sets us up for

example in principal component analysis,

to operate in different basis systems.

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