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How matrices transform space

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Mathematics for Machine Learning: Linear Algebra

Imperial College London

4.7 (4,360 évaluations) | 99K étudiants inscrits

Cours 1 de 3 dans Mathématiques pour l'apprentissage automatique Spécialisation

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.

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Compétences que vous apprendrez

Eigenvalues And Eigenvectors, Basis (Linear Algebra), Transformation Matrix, Linear Algebra

Avis

4.7 (4,360 évaluations)

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EC

Sep 10, 2019

Excellent review of Linear Algebra even for those who have taken it at school. Handwriting of the first instructor wasn't always legible, but wasn't too bad. Second instructor's handwriting is better.

PL

Aug 26, 2018

Great way to learn about applied Linear Algebra. Should be fairly easy if you have any background with linear algebra, but looks at concepts through the scope of geometric application, which is fresh.

À partir de la leçon

Matrices in Linear Algebra: Objects that operate on Vectors

Now that we've looked at vectors, we can turn to matrices. First we look at how to use matrices as tools to solve linear algebra problems, and as objects that transform vectors. Then we look at how to solve systems of linear equations using matrices, which will then take us on to look at inverse matrices and determinants, and to think about what the determinant really is, intuitively speaking. Finally, we'll look at cases of special matrices that mean that the determinant is zero or where the matrix isn't invertible - cases where algorithms that need to invert a matrix will fail.

How matrices transform space5:41

Types of matrix transformation8:38

Composition or combination of matrix transformations8:59

Enseigné par

David Dye
Professor of Metallurgy
Samuel J. Cooper
Lecturer
A. Freddie Page
Strategic Teaching Fellow

Transcription

[MUSIC] Previously we introduced the idea of a matrix and related it to the problem of solving simultaneous equations. And we showed that the columns of a matrix just read us what it does to the unit vector along each axis. Now we'll look at different types of matrices and what they do to space. And what happens if we apply one matrix transformation and then another, which is termed composition. Now because we could make any vector out of a vector sum of the scaled versions of e1 hat and e2 hat. Then what that means is the result of the transformation is just going to be some sum of the transform vectors, which I'm calling e1 hat and e2 hat. This is a bit hard to see but what it means is that the grid lines of our space stay parallel and evenly spaced. They might be stretched or sheared, but the origin stays where it is and there isn't any curviness to the space, it doesn't get warped. And that's a consequence of our scalar addition and multiplication rules for vectors. So that is if I write down the matrix as capital A and the vector its transforming is r, whatever it was. a b in our apples and bananas problem. And that gives me some altered version. We said it was 8 13 before, but I'm going to call it r transformed or r prime. Then we can look at what happens when I do algebra with it. So if I multiply r by some number, just a number, let's call it n. And if I apply A to (nr), what I'm saying is that I will get nr prime. And hopefully you can see if I put an n in there, when I multiply it all out I'm going to get an n in there. Similarly, if I multiply A by the vector (r+s), then I will get Ar + As. So if I get that multiplication, get that, do the whole thing again with another vector, and get another vector s, and add those two, that will be true. So what I'm saying here is that if I can then think of these as being the original basis vectors, e1 hat and e2 hat, I'll then get some addition of e2 and e1 primed. So if I say that's ne1 hat + me2 hat, I'll get nAe1 hat + mAe2 hat, which is ne1 prime + me2 primed. Which is just the vector sum of some multiple of those. So the space gets transformed, e1 and e2 get moved. And then I can just add up vectors with them. So that's very nice, that means that all of our vector sum rules work. Now maybe that's a bit confusing so let's try it with an example. So I've got my matrix A here from my apples bananas problem. So I have 2 3 10 1, or if we like, the vector 2 10 and the vector 3 1. Now let's try an example like a vector 3 2. Now if I multiply that out just straightforwardly, as we probably did at school, I've got 2 times 3 plus 3 times 2. So that's 6 plus 6, that's 12. And I've got 10 times 3, which is 30, plus 1 times 2, which is 2. So that's 32. But I could think of that as being 2 3 10 1, times (3 times 1 0 + 2 times 0 1). That is, 3 of our e1 hat and 2 of e2 hat in a vector sum. Now I can take the 3 out so that's 3 times (2 3 10 1 times 1 0). + 2 times (2 3 10 1 times 0 1). And this, we know that this is what happens from e1 hat to get to e1 prime here, so that's 3 times 2 10. And that's 2 times what happens to e2 hat, and that goes to 3 1. So that gives us 6 plus 6 is 12 and 30 plus 2 is 32. So it really is true, these rules really do work. We can think of a matrix multiplication as just being the multiplication of the vector sum of the transformed basis vectors. So pause now for a moment and try that maybe with an example of your own. And verify that that really does work because that's really quite deep. This matrix just tells us where the basis vectors go. That's the transformation it does. It's not a complicated, multiplying out thing. We don't need to worry about the mechanics of doing the sum. We can just think of it in terms of what it does to vectors in the space. [MUSIC]

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