Mathematics for Machine Learning: Linear Algebra

Go to Course: https://www.coursera.org/learn/linear-algebra-machine-learning

Introduction

### Course Review: Mathematics for Machine Learning: Linear Algebra on Coursera **Overview:** The course "Mathematics for Machine Learning: Linear Algebra" is an exquisite gateway for learners eager to delve into the mathematical foundations critical to machine learning and data science. This course focuses on linear algebra, covering essential concepts such as vectors, matrices, eigenvalues, and eigenvectors. Throughout the course, students are guided through clear, intuitive explanations, practical applications, and computational implementations, all of which are designed to establish a robust understanding of how these mathematical principles are leveraged in real-world data problems. **Course Structure:** The curriculum is well-structured and spans five significant modules: 1. **Introduction to Linear Algebra and Mathematics for Machine Learning:** This module sets the stage by contextualizing linear algebra within the broader scope of machine learning. The emphasis is placed on gaining mathematical intuition rather than rote calculations. Here, students learn to appreciate the functionality of callable Python functions that simplify algebraic operations, thus preparing them for the challenges that arise when things don’t go as planned. 2. **Vectors: Objects that Move Around Space:** In this segment, learners engage with vector operations, including measuring their magnitude, calculating angles between them (using the dot product), and projecting one vector onto another. A profound understanding of vector representation and linear independence is fostered, equipping students with the foundational knowledge necessary to tackle matrices. 3. **Matrices in Linear Algebra: Objects that Operate on Vectors:** This module introduces matrices as transformative tools within linear algebra. Students will explore how matrices can be used to solve systems of linear equations, not forgetting the significance of inverse matrices and determinants. Intuitive insights are provided regarding special matrices that affect computing tasks, thereby building a toolkit for recognizing when certain algorithms may falter. 4. **Matrices Make Linear Mappings:** Continuing with matrices, this module navigates the complexities of matrix multiplication and operations through advanced notations like the Einstein Summation Convention. Additionally, there’s practical work on how transformations can be applied to images, thus linking theoretical concepts with programming applications. 5. **Eigenvalues and Eigenvectors: Application to Data Problems:** The final module focuses on the pivotal concepts of eigenvalues and eigenvectors, exploring their utility in real-world applications such as the famous PageRank algorithm. Students will engage in coding exercises, bringing theoretical knowledge full circle into practical implementation. **Why Recommend This Course?** 1. **Intuitive Learning Approach:** The course is highly approachable, emphasizing the importance of understanding over memorization, which is essential for beginners and those who may feel intimidated by mathematical concepts. 2. **Accessible Programming Integration:** By weaving Python programming into the lessons, learners not only grasp the theory but also see how it is applied in practice. This is invaluable, as it arms students with coding skills that parlay directly into data science. 3. **Well-rounded Content:** The course covers foundational concepts comprehensively while culminating in advanced applications, catering to a diverse audience from novices to those looking to strengthen their existing knowledge. 4. **Relevance to Current Technologies:** Given the course's focus on topics like the PageRank algorithm and image manipulation, it aligns well with ongoing advancements in technology and data analysis, ensuring learners are acquainted with relevant frameworks. 5. **Collaborative Environment:** Coursera's platform enables interaction with a community of learners, aiding in networking and collaborative study, which enriches the learning experience. **Conclusion:** If you're venturing into the fields of machine learning or data science and want to build a solid mathematical foundation, "Mathematics for Machine Learning: Linear Algebra" is an essential course. It not only equips you with the knowledge of linear algebra but also fosters critical thinking about the mathematical machinery behind algorithms. Whether you're looking to start a career in data science or simply want to understand the mathematical concepts that underpin modern technology, this course is highly recommended.

Syllabus

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.

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.

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.

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.

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.