About this course: 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.
Who is this class for: 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.
Created by: Imperial College London
Taught by: David Dye, Professor of Metallurgy
Department of MaterialsTaught by: Samuel J. Cooper, Lecturer
Dyson School of Design EngineeringTaught by: A. Freddie Page, Strategic Teaching Fellow
Dyson School of Design Engineering
| Basic Info | Course 1 of 3 in the Mathematics for Machine Learning Specialization |
| Level | Beginner |
| Commitment | 5 weeks of study, 2-5 hours/week |
| Language | English |
| Hardware Req | No additional hardware is needed to complete this course |
| How To Pass | Pass all graded assignments to complete the course. |
| User Ratings |
Syllabus
WEEK 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 videos, 4 readings, 3 practice quizzes
- 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
- Video: Motivations for linear algebra
- Practice Quiz: Solving some simultaneous equations
- Video: Getting a handle on vectors
- Practice Quiz: Exploring parameter space
- Video: Operations with vectors
- Practice Quiz: Doing some vector operations
- Video: Summary
WEEK 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 videos, 3 practice quizzes
- Video: Modulus & inner product
- Video: Cosine & dot product
- Video: Projection
- Practice Quiz: Dot product of vectors
- Video: Changing basis
- Practice Quiz: Changing basis
- Video: Basis, vector space, and linear independence
- Video: Applications of changing basis
- Practice Quiz: Linear dependency of a set of vectors
- Video: Summary
Graded: Vector operations assessment
WEEK 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 videos, 2 practice quizzes
- Video: How matrices transform space
- Video: Types of matrix transformation
- Video: Composition or combination of matrix transformations
- Practice Quiz: Using matrices to make transformations
- Video: Solving the apples and bananas problem: Gaussian elimination
- Video: Going from Gaussian elimination to finding the inverse matrix
- Practice Quiz: Solving linear equations using the inverse matrix
- Video: Determinants and inverses
- Notebook: Identifying special matrices
- Video: Summary
Graded: Identifying special matrices
WEEK 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 videos, 2 practice quizzes
- Practice Quiz: Non-square matrix multiplication
- Video: Matrices changing basis
- Video: Doing a transformation in a changed basis
- Practice Quiz: Mappings to spaces with different numbers of dimensions
- Video: Orthogonal matrices
- Video: The Gram–Schmidt process
- Notebook: Gram-Schmidt process
- Video: Example: Reflecting in a plane
- Notebook: Reflecting Bear
Graded: Gram-Schmidt Process
Graded: Reflecting Bear
WEEK 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 videos, 1 reading, 3 practice quizzes
- Video: What are eigenvalues and eigenvectors?
- Practice Quiz: Selecting eigenvectors by inspection
- Video: Special eigen-cases
- Video: Calculating eigenvectors
- Practice Quiz: Characteristic polynomials, eigenvalues and eigenvectors
- Video: Changing to the eigenbasis
- Video: Eigenbasis example
- Practice Quiz: Diagonalisation and applications
- Video: Introduction to PageRank
- Notebook: PageRank
- Video: Summary
- Video: Wrap up of this linear algebra course
- Ungraded Plugin: Post-Course Survey
- Reading: Congratulations! Ready for course 2?
Graded: Page Rank
Graded: Eigenvalues and eigenvectors
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Ratings and Reviews
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.
A Very good course to get knowledge of linear algebra and its applications to ML and data science
great course. gave me lots of intuition of linear algebra
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A fun way to review Linear Algebra basics focused on its applications on Machine Learning.
Good structure, nice pace, solid content.