Reformation Through Merging Higher Algebra and Analytic Geometry

Abstract

For undergraduates majoring in mathematics and applied mathematics and even computer science, linear algebra and analytic geometry are two required courses in all Chinese universities and Colleges. Actually, the course "higher algebra" is used instead of ``linear algebra" in China. The most important difference is the theory of polynomials on number fields added in higher algebra. And in some textbooks of higher algebra in China contains the theory of $\lambda-$Matrices, such as textbooks used in Beijing University and Southwest University, in which some versions contain introductions to number theory and use the terms of "ring" and "field" as well, of coursed, this is helpful for students conceive of the idea of modern algebra and number theory, but the practical reason to introduce such concepts to first year students is to make the system more general. For example, after students understand concepts of "ring" and "field", we can make the definition of a vector space and a polynomial over an arbitrary number fields not only rational, real and complex number field. In SWU, we use this kind of system many years, it is successful, especially it looks better connected to modern algebra.

Generally, in linear algebra, discussion of diagonalizing matrices stops after giving the (necessary and) sufficient conditions on diagonalizability of matrices. So there is a question how about the matrices who cannot be diagonalized, if they are similar to some kinds of simpler matrices, especially if we can make criteria for similarity of matrices through pure matrices language. This is done by introduce the theory of $\lambda -$matrices, which some textbook named ``Jordan Canonical Form", which is introduced by few textbook "linear algebra". Reformation Through Merging Higher Algebra and Analytic Geometry: About 6 years ago, a new round reformation of system of curriculums started in universities with purpose to promote the quality of teaching and studying through shorten teaching time and make each curriculum adapted to both theoretical and practical usage. In such background, several universities, such as Nankai University, East China Normal University and Southwest China Normal University, made experimental reform to merging higher algebra and analytic geometry. This reform was started in 1998 in SWU by the section of algebra of School of Mathematics and Finance(former name, now is School of Mathematics and Statistics), and teaching material was first used in 1999, the textbook named "higher algebra and analytic geometry" was published in 2002. Now I would like to introduce the system of this book.

This book contains 11 chapters, which rooted in [czm] and [LHW]. It succeeded in connecting contents of two books. In the chapter 4 named "Vector Space", we define vectors and its operation in a plane and three dimension space and summarize properties what such vectors have before the definition of vector space over an arbitrary number field. In such way the students can understand a vector space of dimension n is a kind of generation of a plane and three dimension space. In chapter 8 named "Euclid space", after introducing "inner product", definition of a Euclid space, orthogonality of vectors, and classification of Euclid spaces, we turn to analytic geometry again to introduce the exterior product and mixed product of vectors in three dimension space, properties and relations of planes and lines. Then we solve the distance from a point to a subspace of a $n-$ dimension space, as its application we give the special case in three dimension space and Least-squares solution. Last section of this chapter introduces classification of orthogonal transformations and affine transformations. In chapter 9 named "Quadratic Forms", we first explain the classifications of quadratic forms over real number field and complex number field, then as an application we list the classification of quadratic surfaces. In chapter 10 named "Familiar Surfaces", we explain the graphs and geometrical properties of familiar quadratic curves and surfaces, especially how to judge the region surrounded by several surfaces, which plays an very important role in calculus of several variables. At last chapter of this book named "General Theory of Quadratic Curves", we explain simplifying quadratic equations of quadratic curves and properties of quadratic curves including its tangent line, center, diameter and asymptotic line. Effect Analysis of the Reformation: Recent five years, we have been using the merged textbook for first year students and canceled independent coursed "analytic geometry". In mainland China, most universities schedules 'higher algebra" as one-year course by 5 lessons per week. We used the same time to finish this book, which was divided into two courses several years ago, one is two semesters course "higher algebra" by 5 lessons per week, another is one semester course "analytic geometry" by 4 lessons per week. So we gave more free time to students than before. The students can understand better and easier the connection of algebra and geometry. Hence we appreciate this merged textbook is a successful one. But a little shortcoming is this textbook requires teachers not only familiar to algebra but also geometry, especially being able to draw various kinds of graphs of quadratic surfaces and curves. Otherwise, the students cannot master the region surrounded by several surfaces when they need to use it in calculus. This textbook also requires the coursed "calculus on several variables" is arranged in third semester, otherwise students don't have the knowledge to fix the region surrounded by several surfaces.