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.
Ce cours fait partie de la Spécialisation Mathematics for Machine Learning
Mathematics for Machine Learning: Linear Algebra
proposé par
À propos de ce cours
4.6
2,183 notes
Compétences que vous acquerrez
Eigenvalues And EigenvectorsBasis (Linear Algebra)Transformation MatrixLinear Algebra
Programme du cours : ce que vous apprendrez dans ce cours
Semaine
12 heures pour terminer
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 (Total 31 min), 4 lectures, 3 quiz
5 vidéos
Motivations for linear algebra3 min
Getting a handle on vectors9 min
Operations with vectors11 min
Summary1 min
4 lectures
About Imperial College & the team5 min
How to be successful in this course5 min
Grading policy5 min
Additional readings & helpful references10 min
3 exercices pour s'entraîner
Solving some simultaneous equations15 min
Exploring parameter space20 min
Doing some vector operations12 min
Semaine
22 heures pour terminer
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 (Total 44 min), 4 quiz
8 vidéos
Modulus & inner product9 min
Cosine & dot product5 min
Projection6 min
Changing basis11 min
Basis, vector space, and linear independence4 min
Applications of changing basis3 min
Summary1 min
4 exercices pour s'entraîner
Dot product of vectors15 min
Changing basis15 min
Linear dependency of a set of vectors15 min
Vector operations assessment15 min
Semaine
33 heures pour terminer
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 (Total 58 min), 3 quiz
8 vidéos
How matrices transform space5 min
Types of matrix transformation8 min
Composition or combination of matrix transformations7 min
Solving the apples and bananas problem: Gaussian elimination8 min
Going from Gaussian elimination to finding the inverse matrix8 min
Determinants and inverses12 min
Summary59s
2 exercices pour s'entraîner
Using matrices to make transformations12 min
Solving linear equations using the inverse matrix16 min
Semaine
46 heures pour terminer
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 (Total 56 min), 4 quiz
6 vidéos
Matrices changing basis11 min
Doing a transformation in a changed basis6 min
Orthogonal matrices8 min
The Gram–Schmidt process6 min
Example: Reflecting in a plane14 min
2 exercices pour s'entraîner
Non-square matrix multiplication10 min
Mappings to spaces with different numbers of dimensions12 min
4.6
382 avis27%
a commencé une nouvelle carrière après avoir terminé ces cours
27%
a bénéficié d'un avantage concret dans sa carrière grâce à ce cours
Meilleurs avis
par PL•Aug 26th 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.
par NS•Dec 23rd 2018
Professors teaches in so much friendly manner. This is beginner level course. Don't expect you will dive deep inside the Linear Algebra. But the foundation will become solid if you attend this course.
Enseignants
David Dye
Professor of MetallurgyDepartment of Materials
Samuel J. Cooper
LecturerDyson School of Design Engineering
A. Freddie Page
Strategic Teaching FellowDyson School of Design Engineering
À propos de 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.
À propos de la Spécialisation Mathematics for Machine Learning
For a lot of higher level courses in Machine Learning and Data Science, you find you need to freshen up on the basics in mathematics - stuff you may have studied before in school or university, but which was taught in another context, or not very intuitively, such that you struggle to relate it to how it’s used in Computer Science. This specialization aims to bridge that gap, getting you up to speed in the underlying mathematics, building an intuitive understanding, and relating it to Machine Learning and Data Science.
In the first course on Linear Algebra we look at what linear algebra is and how it relates to data. Then we look through what vectors and matrices are and how to work with them.
The second course, Multivariate Calculus, builds on this to look at how to optimize fitting functions to get good fits to data. It starts from introductory calculus and then uses the matrices and vectors from the first course to look at data fitting.
The third course, Dimensionality Reduction with Principal Component Analysis, uses the mathematics from the first two courses to compress high-dimensional data. This course is of intermediate difficulty and will require basic Python and numpy knowledge.
At the end of this specialization you will have gained the prerequisite mathematical knowledge to continue your journey and take more advanced courses in machine learning.
Foire Aux Questions
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