Projection

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En provenance du cours de Imperial College London

Mathematics for Machine Learning: Linear Algebra

2582 notes

Imperial College London

Mathematics for Machine Learning: Linear Algebra

2582 notes

Cours 1 sur 3 dans 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.

À partir de la leçon

Vectors are objects that move around space

In this module, we look at operations we can do with vectors - finding the modulus (size), angle between vectors (dot or inner product) and projections of one vector onto another. We can then examine how the entries describing a vector will depend on what vectors we use to define the axes - the basis. That will then let us determine whether a proposed set of basis vectors are what's called 'linearly independent.' This will complete our examination of vectors, allowing us to move on to matrices in module 3 and then start to solve linear algebra problems.

Modulus & inner product9:33

Cosine & dot product5:53

Rencontrer les enseignants

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, there's one last thing to talk about in the segment which is called projection.

Projection. And for that,

we'll need to draw a triangle.

So if I've got a vector r and another vector s. Now,

if I take a little right-handed triangle,

drop a little right-handed triangle down here where this angle's 90 degrees,

then I can do the following.

If I can say that, if this angle here is theta,

but cos theta is equal to, from SOHCAHTOA,

is equal to the adjacent length here over the hypotenuse there. Adjacent over the hypotenuse.

That is, and this hypotenuse is the size of s, so it's the adjacent over the size of s, there.

Now, if I compare that to the definition of the dot product,

I can say that r dotted with,

we'll have fun with colors,

dotted with s is equal

to mod r, size of r,

times the size of s, times cos theta.

But the size of s times cos theta if I put s up here,

I just need to put my theta in there,

s cos theta is just the adjacent side,

so that's just that adjacent side here in the triangle.

So, the adjacent side here is just kind of the shadow,

if I had a light coming down from here,

it's the shadow of s on r. That length there,

it's kind of a shadow cast.

If I had a light at 90 degrees to r shining down on s,

and that's called the projection.

So what the dot product gives us,

is it gives us the projection here of s on to r times the size of r.

And one thing to notice here is that if s was perpendicular to r,

if s was pointing this way,

it would have no shadow.

That is if cos theta was 90 degrees that shadow would be nought,

the cos theta would be nought here and I get no projection.

So, the other thing the dot product gives us,

is it gives us the size of r times some idea about

the projection of s on to r. The shadow of s onto r. So,

if I divide the dot product r.s by the length of r,

just bring the r down here,

I get mod s cos theta.

I get that adjacent side,

I get a number which is called,

because r.s is a number, and the size of r is a number,

and that's called the scalar projection.

And that's why the dot product is also called the projection product,

because it takes the projection of one vector onto another.

We just have to divide by the length of r,

and if r happened to be a unit vector or one of

the vectors we used to find the space, of length one,

then that would be of length one and r dot s would just be the scalar projection of

s onto that r, that vector defining the axes or whatever it was.

Now, if I want to remember to encode something about r,

which way r was going into the dot product or into the projection product

I could define something called the vector projection.

And that's defined to be r.s over mod r dotted with itself.

So r.r mod r squared,

that's r.s over r.r if you like because mod r squared is equal to r.r.

And we multiply that by the vector r itself.

So that is, that dot product's just a number,

these sizes are just a number,

and r itself is a vector.

So what we've done here is we've taken the scalar projection r.s over r,

this guy, that's how much s goes along r,

and we've multiplied it by r divided by its length.

So we've multiplied it by a vector going the direction

of r but it's been normalized to have a length one.

So that vector projection is a number,

times a unit vector that goes in the direction

of r. So if r say was some number of lengths,

that would be r divided by its size,

say if that was a unit length vector I've just drawn there,

and the vector projection would be that number s.r, that adjacent side,

times a vector going in the unit length of r. So that's,

if you like the scalar projection also encoded with something about the direction of r,

just a unit vector going in the direction of r.

So we've defined a scalar projection here,

and we've defined a vector projection there.

So, good job! This was really the core video for this week,

we've done some real work here.

We found the size of a vector and we defined the dot projection product.

We've then found out some mathematical operations we can do with the dot product.

That it's distributive over vector addition and

associative with scalar multiplication and that it's commutative.

We then found that it finds the angle between two vectors,

the extent to which they go in the same direction,

and then it finds the projection of one vector onto another.

That's kind of how one vector will collapse onto another,

which is what we'll explore in the next two videos.

So, good work, now's a good time to pause,

and try some examples,

but put all this together and give it all a workout and a bit of a try before we move on.

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