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Linear Algebra

其他書名	An Introduction
出版	Independently Published, 2020-10-11
主題	Education / Schools / Levels / Elementary
ISBN	9798696536941
URL	http://books.google.com.hk/books?id=5oIGzgEACAAJ&hl=&source=gbs_api

註釋Linear Algebra: An Introduction, Second Edition by RICHARD BRONSON and GABRIEL B. COSTAThe second edition of this book presents the fundamental structures of linearalgebra and develops the foundation for using those structures. Many of theconcepts in linear algebra are abstract; indeed, linear algebra introduces studentsto formal deductive analysis. Formulating proofs and logical reasoning are skillsthat require nurturing, and it has been our aim to provide this.Much care has been taken in presenting the concepts of linear algebra in anorderly and logical progression. Similar care has been taken in proving resultswith mathematical rigor. In the early sections, the proofs are relatively simple,not more than a few lines in length, and deal with concrete structures, such asmatrices. Complexity builds as the book progresses. For example, we introducemathematical induction in Appendix A.A number of learning aides are included to assist readers. New concepts arecarefully introduced and tied to the reader's experience. In the beginning, thebasic concepts of matrix algebra are made concrete by relating them to a store'sinventory. Linear transformations are tied to more familiar functions, and vectorspaces are introduced in the context of column matrices. Illustrations givegeometrical insight on the number of solutions to simultaneous linear equations,vector arithmetic, determinants, and projections to list just a few.Highlighted material emphasizes important ideas throughout the text. Compu-tational methods--for calculating the inverse of a matrix, performing a Gram-Schmidt orthonormalization process, or the like--are presented as a sequence ofoperational steps. Theorems are clearly marked, and there is a summary ofimportant terms and concepts at the end of each chapter. Each section endswith numerous exercises of progressive difficulty, allowing readers to gainproficiency in the techniques presented and expand their understanding of theunderlying the Chapter 1 begins with matrices and simultaneous linear equations. The matrix isperhaps the most concrete and readily accessible structure in linear algebra, andit provides a nonthreatening introduction to the subject. Theorems dealing withmatrices are generally intuitive, and their proofs are straightforward. Theprogression from matrices to column matrices and on to general vector spacesis natural and seamless.Separate chapters on vector spaces and linear transformations follow the mater-ial on matrices and lay the foundation of linear algebra. Our fourth chapter dealswith eigenvalues, eigenvectors, and differential equations. We end this chapterwith a modeling problem, which applies previously covered material. With theexception of mentioning partial derivatives in Section 5.2, Chapter 4 is the onlychapter for which a knowledge of calculus is required. The last chapter deals withthe Euclidean inner product; here the concept of least-squares fit is developed inthe context of inner products.We have streamlined this edition in that we have redistributed such topics as theJordan Canonical Form and Markov Chains, placing them in appendices. Ourgoal has been to provide both the instructor and the student with opportunitiesfor further study and reference, considering these topics as additional modules.We have also provided an appendix dedicated to the exposition of determinants,a topic which many, but certainly not all, students have studied.We have two new inclusions: an appendix dealing with the simplex method andan appendix touching upon numerical techniques and the use of technology.Regarding numerical methods, calculations and computations are essential tolinear algebra. Advances in numerical techniques have profoundly altered theway mathematicians approach this subject.Partial pivoting, elementary row operations, and an entire section onLU decomposition are part of Chapter 1.