Advanced Algebra and Analytic Geometry

Author: Niu Xingwen
Publisher:
Publish Date: 2005-12-01
Features: The main part of advanced algebra, linear algebra, originated from solving systems of linear equations. Space analytical geometry transforms geometric problems of planes and quadratic surfaces into linear algebra problems through coordinate systems. Linear algebra studies these problems, leading to the theories of matrices and linear spaces, which, together with polynomial algebra, constitute advanced algebra. Combining space analytical geometry with advanced algebra as a course not only helps in understanding and mastering abstract algebraic concepts but also cultivates the ability to solve geometric problems using algebraic methods. This book is divided into three parts. Part 1, consisting of the first two chapters, introduces the basic concepts of logic and set theory, reviewing plane analytical geometry from the perspectives of vectors and matrices. Part 2, covering Chapters 3 to 7, poses problems from geometry and solves them using matrix methods, then returns to the geometric meaning of the solutions, introducing projective geometry and metric geometry respectively. Part 3, comprising Chapters 8 to 12, introduces the theories of linear spaces and Euclidean spaces, with Chapter 8 featuring polynomials as a linear space as an example beyond geometric vector spaces and n-dimensional vector spaces. This book is compiled based on the author's extensive mathematical practice and meets the teaching requirements of Advanced Algebra and Space Analytical Geometry in terms of depth and breadth. It is written in a detailed and fluent manner, with rigorous arguments, accompanied by a substantial number of examples and exercises of varying difficulty, making it suitable as a textbook or for self-study in mathematics, applied mathematics, and information and computational science at universities and colleges.