Start Date: 07/05/2020
Course Type: Common Course |
Course Link: https://www.coursera.org/learn/linear-algebra-machine-learning
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.
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.
Mathematics for Machine Learning: Linear Algebra Mathematics for Machine Learning is the entry point to understanding any linear algebra problem. We assume that you have already acquired basic knowledge in computer science, and that you are willing to take a small mathematical test. We hope that you will take it; otherwise, you will probably get annoyed and frustrated. In this course, we will build a linear model for an input vector x such that a(x) < b(x) We assume that we have determined the linear relationship between a and b. We will assume that we have also determined the conservation of mass, and that we have used vector calculus. We will assume that we have also learned the necessary calculus, as discussed in the course material. If you have not, please take a look at our introductory course on linear algebra and vector spaces. If you have a basic knowledge in linear algebra, you will be able to follow along with the course material. If not, please check out our other courses on linear algebra and data science. When we first introduce the class, we assume that you already have these prerequisites. If not, please read our disclaimer at the beginning of this course. Linear models are very powerful machine learning algorithms. They allow us to solve complex machine learning problems very quickly, and in many cases, we can use them to introduce new problems. In this course, we will focus on applications of linear models, introducing you to the basics of
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Linear algebra | Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. |
Linear algebra | Because of the ubiquity of vector spaces, linear algebra is used in many fields of mathematics, natural sciences, computer science, and social science. Below are just some examples of applications of linear algebra. |
Linear algebra | Linear algebra first appeared in American graduate textbooks in the 1940s and in undergraduate textbooks in the 1950s. Following work by the School Mathematics Study Group, U.S. high schools asked 12th grade students to do "matrix algebra, formerly reserved for college" in the 1960s. In France during the 1960s, educators attempted to teach linear algebra through finite-dimensional vector spaces in the first year of secondary school. This was met with a backlash in the 1980s that removed linear algebra from the curriculum. In 1993, the U.S.-based Linear Algebra Curriculum Study Group recommended that undergraduate linear algebra courses be given an application-based "matrix orientation" as opposed to a theoretical orientation. |
Linear Algebra and its Applications | Linear Algebra and its Applications is a biweekly peer-reviewed mathematics journal published by Elsevier and covering matrix theory and finite-dimensional linear algebra. |
Linear algebra | Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. |
Special linear Lie algebra | In mathematics, the special linear Lie algebra of order n (denoted formula_1) is the Lie algebra of formula_2 matrices with trace zero and with the Lie bracket formula_3. This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. The Lie group that it generates is the special linear group. |
Linear algebra | In 1882, Hüseyin Tevfik Pasha wrote the book titled "Linear Algebra". The first modern and more precise definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The use of matrices in quantum mechanics, special relativity, and statistics helped spread the subject of linear algebra beyond pure mathematics. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. |
Basic Linear Algebra Subprograms | Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication. They are the "de facto" standard low-level routines for linear algebra libraries; the routines have bindings for both C and Fortran. Although the BLAS specification is general, BLAS implementations are often optimized for speed on a particular machine, so using them can bring substantial performance benefits. BLAS implementations will take advantage of special floating point hardware such as vector registers or SIMD instructions. |
Quadratic-linear algebra | In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by . An example is the universal enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form "XY" – "YX" – ["X", "Y"] = 0. |
Linear algebra | Linear algebra provides the formal setting for the linear combination of equations used in the Gaussian method. Suppose the goal is to find and describe the solution(s), if any, of the following system of linear equations: |
Quantum machine learning | Quantum matrix inversion can be applied to machine learning methods in which the training reduces to solving a linear system of equations, for example in least-squares linear regression, the least-squares version of support vector machines, and Gaussian processes. |
Linear algebra | Reviews of the teaching of linear algebra call for stress on visualization and geometric interpretation of theoretical ideas, and to include the "jewel in the crown" of linear algebra, the singular value decomposition (SVD), as 'so many other disciplines use it'. To better suit 21st century applications, such as data mining and uncertainty analysis, linear algebra can be based upon the SVD instead of Gaussian Elimination. |
Quantitative analyst | Because of their backgrounds, quantitative analysts draw from various forms of mathematics: statistics and probability, calculus centered around partial differential equations, linear algebra, discrete mathematics, and econometrics. Some on the buy side may use machine learning. The |
Automatically Tuned Linear Algebra Software | Automatically Tuned Linear Algebra Software (ATLAS) is a software library for linear algebra. It provides a mature open source implementation of BLAS APIs for C and Fortran77. |
Algebra | Some areas of mathematics that fall under the classification abstract algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name. |
Linear Lie algebra | In algebra, a linear Lie algebra is a subalgebra formula_1 of the Lie algebra formula_2 consisting of endomorphisms of a vector space "V". In other words, a linear Lie algebra is the image of a Lie algebra representation. |
Kernel (linear algebra) | In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces "V" and "W", is the set of all elements v of "V" for which , where 0 denotes the zero vector in "W". That is, in set-builder notation, |
Algebra II | Algebra II is generally the first mathematics course that deeply involves abstract and non-linear thinking. |
Projection (linear algebra) | Projections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems: |
Linear algebra | The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, which was initially listed as an advancement in geodesy. |