The Chinese Roots Of Linear Algebra

A monumental accomplishment in the history of non-Western mathematics, The Chinese Roots of Linear Algebra explains the fundamentally visual way Chinese mathematicians understood and solved mathematical problems. It argues convincingly that what the West "discovered" in the sixteenth and seventeenth centuries had already been known to the Chinese for 1,000 years. Accomplished historian and Chinese-language scholar Roger Hart examines Nine Chapters of Mathematical Arts—the classic ancient Chinese mathematics text—and the arcane art of fangcheng, one of the most significant branches of mathematics in Imperial China. Practiced between the first and seventeenth centuries by anonymous and most likely illiterate adepts, fangcheng involves manipulating counting rods on a counting board. It is essentially equivalent to the solution of systems of N equations in N unknowns in modern algebra, and its practice, Hart reveals, was visual and algorithmic. Fangcheng practitioners viewed problems in two dimensions as an array of numbers across counting boards. By "cross multiplying" these, they derived solutions of systems of linear equations that are not found in ancient Greek or early European mathematics. Doing so within a column equates to Gaussian elimination, while the same operation among individual entries produces determinantal-style solutions. Mathematicians and historians of mathematics and science will find in The Chinese Roots of Linear Algebra new ways to conceptualize the intellectual development of linear algebra.

Roger Hart debunks the long-held belief that linear algebra developed independently in the West. Accounts of the seventeenth-century Jesuit Mission to China have often celebrated it as the great encounter of two civilizations. The Jesuits portrayed themselves as wise men from the West who used mathematics and science in service of their mission. Chinese literati-official Xu Guangqi (1562–1633), who collaborated with the Italian Jesuit Matteo Ricci (1552–1610) to translate Euclid’s Elements into Chinese, reportedly recognized the superiority of Western mathematics and science and converted to Christianity. Most narratives relegate Xu and the Chinese to subsidiary roles as the Jesuits' translators, followers, and converts. Imagined Civilizations tells the story from the Chinese point of view. Using Chinese primary sources, Roger Hart focuses in particular on Xu, who was in a position of considerable power over Ricci. The result is a perspective startlingly different from that found in previous studies. Hart analyzes Chinese mathematical treatises of the period, revealing that Xu and his collaborators could not have believed their declaration of the superiority of Western mathematics. Imagined Civilizations explains how Xu’s West served as a crucial resource. While the Jesuits claimed Xu as a convert, he presented the Jesuits as men from afar who had traveled from the West to China to serve the emperor.

The purpose of the edited volume is to provide an international lens to examine evidence-based investigations in Ethno-STEM research: Ethno-science, Ethno-technology, Ethno-engineering, and Ethno-mathematics. These themes grew out of multi-national, multi-institutional and multi-disciplinary efforts to preserve as well as epitomize the role that Indigenous Knowledge Systems (IKS) play in cognitive development and its vital contributions to successful and meaningful learning in conventional and non-conventional contexts. Principled by the Embodied, Situated, and Distributed Cognition (ESDC), this innovative book will provide evidence supporting the embeddedness of a thinking-in-acting model as a fundamental framework that explains and supports students’ acquisition of scientific knowledge. So often ‘western’ science curricula are experienced as irrelevant, since it does not take cognizance of the daily experiences and world in which the learner finds himself. This book takes a socio-cultural look at IKS and applies research in neuroscience to make a case its incorporation in the STEM (Science, Technology, Engineering and Mathematics) classroom. We use the Embodied Situated Distributed Cognition (ESDC) Model as conceptual framework in this book. Although the value of IKS is often acknowledged in curriculum policy documents, teachers are most often not trained in incorporating IK in the classroom. Teachers’ lack of the necessary pedagogical content knowledge (PCK) in effectively incorporating IK in their classrooms is a tremendous problem internationally. Another problem is that IK is often perceived as “pseudo-science”, and scholars advocating for the incorporation of IK in the school curriculum often do not contextualize their arguments within a convincing theoretical and conceptual framework.

A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples.

This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the underlying linear algebraic techniques, thereby enabling students not only to learn how to apply the mathematical tools in routine contexts, but also to understand what is required to adapt to unusual or emerging problems. No previous knowledge of linear algebra is needed to approach this text, with single-variable calculus as the only formal prerequisite. However, the reader will need to draw upon some mathematical maturity to engage in the increasing abstraction inherent to the subject. Once equipped with the main tools and concepts from this book, students will be prepared for further study in differential equations, numerical analysis, data science and statistics, and a broad range of applications. The first author’s text, Introduction to Partial Differential Equations, is an ideal companion volume, forming a natural extension of the linear mathematical methods developed here.

This expansive volume describes the history of numerical methods proposed for solving linear algebra problems, from antiquity to the present day. The authors focus on methods for linear systems of equations and eigenvalue problems and describe the interplay between numerical methods and the computing tools available at the time. The second part of the book consists of 78 biographies of important contributors to the field. A Journey through the History of Numerical Linear Algebra will be of special interest to applied mathematicians, especially researchers in numerical linear algebra, people involved in scientific computing, and historians of mathematics.

This textbook demonstrates the strong interconnections between linear algebra and group theory by presenting them simultaneously, a pedagogical strategy ideal for an interdisciplinary audience. Being approached together at the same time, these two topics complete one another, allowing students to attain a deeper understanding of both subjects. The opening chapters introduce linear algebra with applications to mechanics and statistics, followed by group theory with applications to projective geometry. Then, high-order finite elements are presented to design a regular mesh and assemble the stiffness and mass matrices in advanced applications in quantum chemistry and general relativity. This text is ideal for undergraduates majoring in engineering, physics, chemistry, computer science, or applied mathematics. It is mostly self-contained—readers should only be familiar with elementary calculus. There are numerous exercises, with hints or full solutions provided. A series of roadmaps are also provided to help instructors choose the optimal teaching approach for their discipline. The second edition has been revised and updated throughout and includes new material on the Jordan form, the Hermitian matrix and its eigenbasis, and applications in numerical relativity and electromagnetics.

This book is a biography of a remarkable Scottish missionary worker, Alexander Wylie, a classical nineteenth century artisan and autodidact with a gift and passion for languages and mathematics. He made significant contributions to knowledge transfer, both to and from China: in missionary work as a printer, playing an important role in the production and distribution of a new Chinese translation of the Bible; as a teacher, translating into Chinese key western texts in science and mathematics including Newton and Euclid and publishing the first Chinese textbooks on modern symbolic algebra, calculus and astronomy; and as a writer in English and an internationally recognised major sinologist, bringing to the West much knowledge of China and contributing extensively to the development of British sinology. The book concludes with an overall evaluation of Wylie’s contribution to knowledge transfer to and from China, noting the imbalance between the significant corpus of scholarly work specifically on Wylie by Chinese scholars in Chinese and the lack of academic studies by western scholars in English.