le01(2010)

the importance of a proper understanding of these more theoretical aspects of the subject. Some of the theory you learn could prove useful in other applications of mathematics. ∙ Lectures I recommend that you take some notes during the lectures, if only to capture the examples I give. You might also find it helpful to read the relevant sections of the textbook even before the lecture. The order of the course will be almost the same as the schedule on the textbook, so you should know what’s coming. ∙ Exercises In this course, as in other courses in Mathematics, it is very important that you finish all the exercises specified, and hand them in for grading by every Monday. There is a big difference between watching other people carry out calculations and being able to do them yourself, and it is vital that you get practice in the various techniques covered in the course. It is also important that you hand the work in, so that you can get feedback on whether your work is presented in an appropriate way. Exercises will be set every week. They will be taken directly from the course textbook. About one-third of the exercises on the textbook will be specified, but students are expected to attempt all the exercises behind each chapter. ∙ Teaching and Learning Methods Two lectures each week, one on Monday and one on Wednesday from 15:50-18:15. Total learning hours is about 72. The learning methods dependent on you. Any methods which has be proven to be successful to you is still applicable. ∙ Learning Materials There are many books on Advanced Algebra most of which have much the same title and contain much the same material. Our Textbook: Guohua Peng and Delang Li, Linear Algebra, Higher Education Press, 2006 The following books are recommended for references 1. Beida, Advanced Algebra (Chinese), Higher Education Press 2. B. Jacob, Linear Algebra, W.H. Freeman and Company, 1990 3. N. Jacobson, Basic Algebra, W.H. Freeman and Company, 1985 4. L.W. Johnson, R.D. Riess and J.T. Arnold, Introduction to Linear Algebra (5th Edition), Prentice-Hall Inc. and China Machine Press, 2002 5. D.C. Lay, Linear Algebra and Its Applications (3rd Edition), Pearson Addison Wesley Asia Limited and Publishing House of Electronics Industry, 2003 6. G. Strang, Linear Algebra and Its Applications, International Thomson Publishing, 1988 Some materials online ♥ Some English e-books are available at /book/mp103.pdf /linearalgebra/ 2
♥ You can enter some related areas on Linear Algebra by the following web-page: /Science/Mathematics/Linear Algebra/ ♥ Related history of Linear algebra may be available at /library/topics/linear ♥ Some online modules designed to help you learn and practice basic linear algebra procedures such as Gauss-Jordan reduction, calculating the determinant, or checking for linear independence may be available at / bogacki/cgi-bin/lat.cgi ♥ You may obtain a lot of Chinese materials on Linear Algebra by searching “高等代数,精品 课程” through Google. ∙ Grading Your grade will be based on your accomplishment of homework (30%) and a final exam (70%). ∙ Credits 5, each semester ∙ Communication Since the teachers will leave Jiang’an campus, the students will have to ask questions during class. Make use of my class hours and class break to ask more questions, make comments, and just converse. If you prefer to contact with me by internet, please send email to me. My email address is peng@ And my Skype contact address is peng1728 You are also encouraged to see me in my office: Room South-211, Management Building, Wangjiang Campus. My office hour is 8:30-10:30 on Tuesday. I’m looking forward to getting to know you.
Lecture 1
Monday September 13, 2010 Guohua Peng
email: peng@
Contents
1 About The Course 2 Integers 1 3

About The Course
∙ Course Title: Linear Algebra I ∙ General Information Advanced Algebra is a vital, widely-used area of mathematics. It is a core mathematics course required of all the various mathematics majors and it is for students working on a degree in mathematics. The course material can be regarded as an elaboration of theory on polynomials and linear algebra. However, the level of abstraction and the emphasis on proof technique make this a difficult course for many students. Successful students emerge from the experience not only with a better understanding of advanced algebra, but also with a higher level of mathematical maturity, better equipped to deal with abstract concepts. The material covered in Advanced Algebra includes general theory on polynomials, systems of linear equations, matrices, determinants, linear spaces, linear maps, ������-matrices, Euclidean spaces, quadratic forms, and basic theory on multilinear algebra. These topics will be discussed in two successive semesters. Generally a subsequent course Abstract Algebra will be offered in the fifth semester. ∙ Aims The aim of this course is to introduce basic theory on polynomials and the basic ideas of linear algebra which are used throughout mathematics. A variety of key techniques are introduced for solving problems in the area. ∙ Learning Outcomes On successful completion of the course the student will have had considerable experience in solving linear equations using Gaussian elimination. The student should also have developed the ability to handle matrices of arbitrary size and to appreciate their geometrical significance. The calculation and use of determinants of square matrices should also be mastered as should the elementary aspects of eigenvalues and eigenvectors. The concept of linear spaces and linear maps are also very important in Advanced Algebra. The student should master how to describe them in terms of matrices. The accomplish of the course will involve learning some new practical skills, such as how to apply basic theory to symmetric matrices, and gaining an appreciation of how mathematical knowledge is extended through the notion of proof. The students will also learn some of the principles underlying these techniques, such as why linear spaces are useful and along the way you’ll see examples showing 1