Modern Matrix Algebra

PREFACE Linear algebra is an important course for an increasing number of students in diverse disciplines. It is the course in which the majority of students have the opportunity to learn abstract concepts and deal with a wide variety of applications that can be drawn from fields such as physics, chemistry, biology, geology, economics, engineering, computer science, psychology, and sociology. This book provides an introduction to the basic algebraic, geometric, and computational tenets of linear algebra at the sophomore level. It also stresses the abstract foundations of the topics involved, using a spiral approach. To this end we introduce many of the foundations of linear algebra by first dealing with matrices, their properties, and the algebra associated with them, while showing the geometric foundations of the topics. It is important for students to be able to associate a visual perception with certain topics, and we try to emphasize that algebra and geometry are complementary to one another. We next use a linear system of equations to develop the basic abstract notions of linear algebra. Rather than use abstract vector spaces, we concentrate on the subspaces associated with a linear system and its coefficient matrix. Chapters 1 through 4 establish these foundations and provide the opportunity to explore a variety of applications that illustrate the underlying concepts. The second level of the spiral involves Chapters 5 through 7, in which the major ideas of beginning linear algebra are developed from a more abstract point of view. At this point the student has worked with the notions and jargon in the special cases of matrices and the subspaces associated with a linear system. Hence the reformulation of the topics to the general setting of abstract vector spaces is not as intimidating from either a conceptual or algebraic standpoint. We feel that the experience obtained in Chapters 1 through 4 will make the assimilation of ideas in the more abstract setting easier since we have established conceptual hooks for all the major concepts. Throughout the text there as the opportunity to use MATLAB to complement the topics.It is not required, but in today's computer-oriented society and workplace our experience is that it adds a level of understanding through hands-on involvement with computations and concepts. This certainly is in line with the Linear Algebra Curriculum Study Group recommendations and is supported by the experiences of a wide segment of mathematical educators. The idea is to use software tools within MATLAB to enhance the presentation of topics and provide opportunities for students to explore topics without the algebraic overhead involved in solving linear systems. To this end there are tools available for both the instructor and the student. Our approach is somewhat different from that developed elsewhere. However,the major objective is to present the basic ideas of linear algebra in a manner that the student will understand by utilizing both algebraic and geometric constructs. EXERCISES The exercises are an integral part of this text. Many of them are numerical in nature whereas others are more conceptually oriented. The conceptual exercises often cal!: for an explanation, description, or supporting reasoning to establish credence of a statement. It is extremely important in our technological age to be able to communicate with precision and to express ideas clearly. Thus exercises of this type are; designed to sharpen such skills. An instructor can choose whether to incorporate complex numbers and matrices as part of the course. Whilethe use of MATLAB is optional, we highly encourage both instructors and students to incorporate it in some way within the instructional and study processes associated with this text.. Hence, where appropriate we have included MATLAB exercises, denoted ML exercises, in many sectKolman, Bernard is the author of 'Modern Matrix Algebra', published 2000 under ISBN 9780139488528 and ISBN 0139488529.