Example: Reflecting in a plane

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Из курса от партнера Imperial College London

Mathematics for Machine Learning: Linear Algebra

1254 оценки

Imperial College London

Mathematics for Machine Learning: Linear Algebra

1254 оценки

Курс 1 из 3 — Specialization Mathematics 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.

Из урока

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.

The Gram–Schmidt process6:07

Example: Reflecting in a plane14:10

Познакомьтесь с преподавателями

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

[MUSIC]

Okay, so let's put all this together.

Let's use our transformations knowledge and

our basis knowledge in order to do something quite tricky.

And see if we can't actually make our life quite simple.

What I want to do here is know what a vector looks like when I reflect it in

some funny plane.

For example, the way this board works, when I write on the whiteboard here,

if you're looking at it, all the writing would appear mirrored.

But what we do to make that work is we reflect everything in post production,

left-right, and then everything comes out okay.

The example we're going to do here asks what the reflection of Bear, say,

something in a mirror would look like to me if

the mirror was off at some funny angle.

Now my first challenge is going to be that I don't know the plane of the mirror

very well.

But I do know two vectors in the mirror, (1, 1, 1) and (2, 0, 1).

And I've got a third vector, which is out of the plane of the mirror,

which is at (3, 1, -1), that's my third vector.

So I've got vectors v1, v2, and v3,

and these two guys are in the plane of the mirror.

We could draw it something like v1 and v2, and

they're in some plane like this, and v3 is out of the plane.

So I have got v3 there, v1, and v2.

So first let us do the Gram-Schmidt process and

find some orthonormal vectors describing this plane and its normal v3.

So my first vector e1 is going to be just the normalised version of v1.

v1 here is of length root 3,

1 squared plus 1 squared plus 1 squared all square rooted.

So it's going to be over root 3 times (1, 1, 1), that's a normalised version of v1.

So then we can carry on, and

I can find u2 = v2- some number of e1's.

So then the sum number's going to be a projection of v2 onto

e1, times e1.

So that's going to be (2, 0, 1)

-(2, 0,1) dotted with e1, which is 1 over root 3,

(1,1,1) times 1 over root 3 (1,1,1), because that's e1.

So that's (2, 0, 1) minus, the root 3s are going to come outside,

so I can just have them being a third.

(2, 0, 1) dotted with (1, 1, 1) is 2 plus 0 plus 1 is 3, so

that actually goes and has a party and becomes 1.

And yeah, okay, I confess, I fixed the example.

So it's (2, 0, 1)-(1,1,1),

which is going to give me (1, -1, 1-1 is 0).

So (1 -1, 0), that's u2.

Now if I want to normalize u2,

I can say e2 is equal to the normalised version of u2.

Which is just 1 over root 2 times (1 -1, 0).

So then I just need to find u3, and I can do the same again with u3.

I can say that that's equal to v3- (the projection of v3 onto e1),

-(the projection of v3 onto e2).

So that's going to be (3, 1, -1)-

(3,1,-1) dotted with 1 over root 3 (1, 1, 1)],

and that's a number.

And it's going in the direction of the unit

vector v1- v3 dotted with e2.

So that's (3, 1, -1) dotted with 1 over root 2

(1, -1, 0), times e2,

which is 1 over root 2 (1, -1, 0).

So it's quite a complicated sum.

But I've got an answer here which I can do.

(3, 1, -1) minus,

the 1 over root 3s come out again,

3 plus 1 minus 1 is 3, so that goes.

Then I've got (1, 1, 1) there, so that becomes 1.

Minus, the halves are going to come out, the 1 over root 2s.

I've got 3 minus 1 minus 0, so

that's 2, so they cancel and become one again.

As I said, I fixed the example to make my life easy.

So then I've got (3, 1, -1) -(1, 1, 1)-(1, -1, 0).

So therefore, I get an answer for u2 being

(3, -1, -1), so that's 1.

1 minus 1 is 0, minus -1 is plus 1,

-1 minus 1 is -2.

And so I can then normalize that and get e3.

So e3 is just the normalised version of that,

which is going to be 1 over root 6 of (1, 1, -2).

Now let's just check, so (1, 1, -2) is normal to

(1, -1, 0), it is normal to (1,1,1).

Those two are normal to each other, so they are a normal basis set.

Just need to make sure that's 1 over root 6, they are all of unit length.

So I can write down my new transformation matrix, which I'm going to call e.

It's the transformation matrix described by the basis vectors e1, e2, e3.

So I've got e1, e2, e3 all written down as column vectors.

And that's going to be my transformation matrix, that first contains the plane,

notice, and then contains the normal to the plane.

So I've redrawn everything just to get it all more compact so we can carry on.

We've got our original two vectors v1 and v2, and

we've defined e1 to be the normalised version of v1.

And we've defined e2 to be the perpendicular part of v2 to e1,

normalised to be of unit length.

So these all are in a plane, and then e3 is normal to that plane.

It's the bit of v3 that we can't make by projecting on to v1 and v2,

then of unit length.

Now say I've got a vector r over here, r.

Now what I want to do is reflect r down through this plane.

So I'm going to drop r down through this plane.

There he is when he intersects the plane.

And then out the other side to get a vector r prime.

And let's say that r has some number like, I don't know, (2, 3, 5), (2, 3, 5).

Now this is going to be really awkward,

this plane's off at some funny angle composed of these vectors.

And even these basis vectors adds up some funny angle, and

then how do I drop it down and do the perpendicular?

There's going to be a lot of trigonometry.

But the neat thing here is that I can think of r as being composed of

a vector that's in the plane, so some vector that's composed of e1s and e2s.

Sorry, so here's e1 here.

And some vector that's normal, so some vector that's some number of e3's.

And when I reflect it through the bit that's in the plane is going to be

the same.

But this bit that's some number of e3's,

this bit here, I'm just going to make into minus this bit here.

So if I wrote that down as a transformation matrix,

the transformation matrix in my basis E, is going to be to keep

the e1 bit the same, keep the e2 bit the same.

So that's the e1 bit, that's the e2 bit, and

then reflect the e3 bit from being up to being down.

(0, 0, -1), so that's a reflection matrix

in e3, so that's a reflection in the plane.

So just by thinking about it quite carefully,

I can think about what the reflection is.

And that T_E is in the basis of the plane, not in my basis, but

in the basis of the plane.

So that is easy to define.

And therefore, if I can get the vector r defined in the plane's

basis vector set, in the E basis, I can then do the reflection.

And then I can put it back into my basis vector set.

And then I have the complete transformation.

So a way of thinking about that is that if I've got my vector r,

and I'm trying to get it through some transformation matrix to r prime.

But that's going to be hard, that's going to be tricky, but

I can transform it, into the basis of the plane.

So I can make an r in the basis of the plane, and

I'm going to do that using E to the -1.

E to the -1 is the thing that got me into,

I've got a vector of mine, into Bear's basis, remember.

Then I can do my transformation, I can do my reflection in the basis of the plane.

I can do my reflection transformation and then I will get r,

That's been transformed, primed, in the basis of the plane.

Then I can read that back into my basis by doing E,

because E is the thing that takes Bear's vector and puts it back into my basis.

So I can avoid the hard thing by going round, doing these three operations.

So r, E to the minus 1, E inverse, T in the E basis, E.

If I do those three things, I've done the complete transformation, and

I get r prime.

So this problem reduces to doing that matrix multiplication.

So we've got that, we've got that, so we can just do the math now,

and then we'll be done.

So I've just put the logic up there so

that I can have the space down here for later.

And I've put the transformation we're going to do there.

A couple of things to note, one is because E, we've carefully constructed by our

Gram–Schmidt process to be orthonormal, we know that E transpose is the inverse.

So calculating the inverse here isn't going to be a pain in the neck.

The other thing is, compared to the situation with Bear where we're

changing bases, here we're changing from our vector r to Bear's,

or actually the plane's coordinate system.

Then we're doing the transformation of the reflection in the plane, and

then we're coming back to our basis.

So the E and and the E to the -1 are flipped compared to the last video,

because we're doing the logic the other way around.

It's quite neat, right?

The actual multiplication of doing this isn't awfully edifying,

it doesn't build you up very much, it's just doing some arithmetic.

So I'm just going to write it down here, and if you want to verify it you can

pause the video, and then we'll come back and we'll comment on it.

So this is TE times the transpose of E, then I take that and

multiply it by E itself.

And I get E TE E transpose, which is this guy, simplifies to this guy, so that's T.

All comes out quite nicely, so that's very, very nice.

So then we can apply that to r.

So we can say that T times r is equal

to T times our vector (2, 3, 5), and that's going to give us r prime.

And that gives us an answer of one-third of (11, 14, 5).

So r prime here is equal to one third of (11, 14, 5).

So that's a process that would've been very,

very difficult to do with trigonometry.

But actually, with transformations, once we get into the plane of the mirror, and

the normal to the mirror,

then it all becomes very easy to do that reflection operation.

And it's quite magical, it's really amazing.

So that's really nice, right?

It's really cool.

So what we've done here is we've done an example where we've put

everything we've learned about matrices and vectors together

to describe how to do something fun like reflect a point in space in a mirror.

This might be useful, for instance, if you want to transform images of faces for

the purpose of doing facial recognition.

Transform my face from being like to being like that.

And then we could use our neural networks,

our machine learning to do that facial recognition part.

In summary, this week we've gone out into the world with matrices and

learned about constructing orthogonal bases, changing bases, and

we've related that back to vectors and projections.

So it's been a lot of fun.

And it sets us up for the next topic, which Sam is going to lead,

on eigenvalues and eigenvectors.

[MUSIC]

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