À propos de ce cours : 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.
Les destinataires de ce cours : This course is for people who want to refresh their maths skills in linear algebra, particularly for the purposes of doing data science and machine learning, or learning about data science and machine learning. We look at vectors, matrices and how to apply these to solve linear systems of equations, and how to apply these to computational problems.
Créé par : Imperial College London
Enseigné par : David Dye, Professor of Metallurgy
Department of Materials
Enseigné par : Samuel J. Cooper, Lecturer
Dyson School of Design Engineering
Enseigné par : A. Freddie Page, Strategic Teaching Fellow
Dyson School of Design Engineering
| Informations de base | Cours 1 de 3 dans Mathematics for Machine Learning Specialization |
| Niveau | Beginner |
| Engagement | 5 weeks of study, 2-5 hours/week |
| Langue | English |
| Matériel requis | No additional hardware is needed to complete this course |
| Comment réussir | Réussissez tous les devoirs notés pour terminer le cours. |
| Notes des utilisateurs |
Programme de cours
SEMAINE 1
Introduction to Linear Algebra and to Mathematics for Machine Learning
In this first module we look at how linear algebra is relevant to machine learning and data science. Then we'll wind up the module with an initial introduction to vectors. Throughout, we're focussing on developing your mathematical intuition, not of crunching through algebra or doing long pen-and-paper examples. For many of these operations, there are callable functions in Python that can do the adding up - the point is to appreciate what they do and how they work so that, when things go wrong or there are special cases, you can understand why and what to do.
5 vidéos, 4 lectures, 3 quiz pour s'exercer
- Reading: About Imperial College & the team
- Reading: How to be successful in this course
- Reading: Grading policy
- Reading: Additional readings & helpful references
- Discussion Prompt: Nice to meet you!
- Ungraded Plugin: Complete our short pre-course survey
- Vidéo: Motivations for linear algebra
- Practice Quiz: Solving some simultaneous equations
- Vidéo: Getting a handle on vectors
- Practice Quiz: Exploring parameter space
- Vidéo: Operations with vectors
- Practice Quiz: Doing some vector operations
- Vidéo: Summary
SEMAINE 2
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.
8 vidéos, 3 quiz pour s'exercer
- Vidéo: Modulus & inner product
- Vidéo: Cosine & dot product
- Vidéo: Projection
- Practice Quiz: Dot product of vectors
- Vidéo: Changing basis
- Practice Quiz: Changing basis
- Vidéo: Basis, vector space, and linear independence
- Vidéo: Applications of changing basis
- Practice Quiz: Linear dependency of a set of vectors
- Vidéo: Summary
Noté: Vector operations assessment
SEMAINE 3
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.
8 vidéos, 2 quiz pour s'exercer
- Vidéo: How matrices transform space
- Vidéo: Types of matrix transformation
- Vidéo: Composition or combination of matrix transformations
- Practice Quiz: Using matrices to make transformations
- Vidéo: Solving the apples and bananas problem: Gaussian elimination
- Vidéo: Going from Gaussian elimination to finding the inverse matrix
- Practice Quiz: Solving linear equations using the inverse matrix
- Vidéo: Determinants and inverses
- Notebook: Identifying special matrices
- Vidéo: Summary
Noté: Identifying special matrices
SEMAINE 4
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.
6 vidéos, 2 quiz pour s'exercer
- Practice Quiz: Non-square matrix multiplication
- Vidéo: Matrices changing basis
- Vidéo: Doing a transformation in a changed basis
- Practice Quiz: Mappings to spaces with different numbers of dimensions
- Vidéo: Orthogonal matrices
- Vidéo: The Gram–Schmidt process
- Notebook: Gram-Schmidt process
- Vidéo: Example: Reflecting in a plane
- Notebook: Reflecting Bear
Noté: Gram-Schmidt Process
Noté: Reflecting Bear
SEMAINE 5
Eigenvalues and Eigenvectors: Application to Data Problems
Eigenvectors are particular vectors that are unrotated by a transformation matrix, and eigenvalues are the amount by which the eigenvectors are stretched. These special 'eigen-things' are very useful in linear algebra and will let us examine Google's famous PageRank algorithm for presenting web search results. Then we'll apply this in code, which will wrap up the course.
9 vidéos, 1 lecture, 3 quiz pour s'exercer
- Vidéo: What are eigenvalues and eigenvectors?
- Practice Quiz: Selecting eigenvectors by inspection
- Vidéo: Special eigen-cases
- Vidéo: Calculating eigenvectors
- Practice Quiz: Characteristic polynomials, eigenvalues and eigenvectors
- Vidéo: Changing to the eigenbasis
- Vidéo: Eigenbasis example
- Practice Quiz: Diagonalisation and applications
- Vidéo: Introduction to PageRank
- Notebook: PageRank
- Vidéo: Summary
- Vidéo: Wrap up of this linear algebra course
- Reading: Did you like the course? Let us know!
- Ungraded Plugin: Post-Course Survey
Noté: Page Rank
Noté: Eigenvalues and eigenvectors
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Créateurs
Imperial College London
Imperial College London is a world top ten university with an international reputation for excellence in science, engineering, medicine and business. located in the heart of London. Imperial is a multidisciplinary space for education, research, translation and commercialisation, harnessing science and innovation to tackle global challenges.
Imperial students benefit from a world-leading, inclusive educational experience, rooted in the College’s world-leading research. Our online courses are designed to promote interactivity, learning and the development of core skills, through the use of cutting-edge digital technology.
Tarification
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|---|---|
| Accéder au contenu du cours | Disponible |
| Accéder au contenu noté | Disponible |
| Recevoir une Note Finale | Disponible |
| Obtenir un Certificat de Cours partageable | Disponible |
Notation et examens
nice
Excellent introduction. For me, as someone who had studied vectors and matrices at school, decades ago, it was wonderful to go back and re-learn this stuff in a different way. This course is much more focused on the meaning and usefulness of these things, rather than just learning how to do the maths. The first 3 minutes of the session on eigenvectors brilliantly showed in graphical form what they really are, something I'd never really grasped at school. Recommended.
I enjoyed the presenter's. The course helps you see how linear algebra is connected to data science.
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EA
Very nice