Linear Algebra: A First Course in Pure and Applied Math

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9780130470171: Linear Algebra: A First Course in Pure and Applied Math

This innovative book features an “Active Reading” theme, stressing the learning of proofs by first focusing on reading mathematics. This helps users understand that linear algebra is not just another course in computation. A secondary theme on Least Squares and the “best” solution to Ax = b adds a modern computational flavor that readers will welcome. Key ideas are revisited & reinforced throughout—Linear independence/dependence; eigenvalues/vectors; projection of one vector on another; the plane spanned by vectors.

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Why yet another book in linear algebra? At many universities, calculus, the usual prerequisite for linear algebra, has become a "course without proofs," with minimal reading requirements. In fact, students who are "successful" in calculus have developed the wonderful ability to match examples to homework with almost no reading. So, if one wishes to keep the algebra (vector spaces, linear transformations) in linear algebra, students need to be taught not only how to handle proofs, but more fundamentally they need to be taught how to read mathematics. My goal in this book is to use an intrinsically interesting subject to show students that there is more to mathematics than numbers.

The presentation in this text is frequently interrupted by "Reading Challenges," little questions a student can use to see if she really does understand the point just made. These Reading Challenges promote active reading. The goal is to get students reading actively by questioning the material as they read. Every section in this book concludes with answers to these Reading Challenges followed by a number of "True/False" questions that test definitions and concepts, not dexterity with numbers or buttons. Every exercise set concludes with "Critical Reading" exercises, which include harder problems, yes, but also problems that are easily solved if understood. Before the review exercises at the end of a chapter, there is a list of key words and concepts that have arisen in the chapter that I hope students will use as a mental check that they have understood the chapter.

There are several reasons for the "pure & applied" subtitle. While a pure mathematician myself, 1 have lost sympathy with the dry abstract vector space/linear transformation approach of many books that requires maturity which the typical sophomore does not have. My approach to linear algebra is through matrices. Essentially all vector spaces are (subspaces of) Euclidean n-space. Other standard examples, like polynomials and matrices, are merely mentioned. In my opinion, when students are thoroughly comfortable with Rn, they have little trouble transferring concepts to general vector spaces. The emphasis on matrices and matrix factorizations, the recurring theme of projections starting in Chapter 1, and the idea of the "best (least squares) solution" to Ax = b around which Chapter 6 revolves, make this book more applied than most of its competition.

However, it does not take most readers long to discover the rigor and maturity that is required to move through the pages of this text. Whether a mathematical argument is always labeled "proof," virtually no statements are made without justification and a large number of exercises are of the "show or prove" variety. Since students tend to find such exercises difficult, I have included an appendix entitled "Show and Prove" designed to teach students how to write a mathematical proof and I tempt them actually to read this section by promising that they will find there the solutions to a number of exercises in the text itself! So despite its applied flavor, I believe that a linear algebra course that uses this book would serve well as the "introduction to proof' transition course that is now a part of many degree programs.

Organization, Philosophy, Style

It is somewhat atypical to begin with, even to include in a linear algebra course, a chapter on vector geometry of the plane and 3-space. My purpose is twofold. Such an introduction allows the immediate introduction of the terminology of linear algebra—linear combination, span (the noun and the verb), linear dependence and independence—in concrete settings that the student can readily understand. Furthermore, the student is quickly alerted to the fact that this course is not about manipulating numbers, that serious thinking and critical reading will be required to succeed. Many exercise sets contain writing exercises, marked as such by a symbol, that ask for explanations, and other questions that can be answered very briefly, if understood.

The inclusion of vector geometry does not come at the expense of other more standard topics. In fact, some nonstandard topics like the pseudoinverse and singular value decomposition are even included. How has this efficiency been achieved?

While a great fan of technology myself, and after years of experimenting and delivering courses entirely via MAPLE worksheets, I now believe that technology can quickly become more distracting than useful for novice linear algebra students. The use of technology is certainly time-consuming and it gives students more to learn in addition to the subject itself. Of course, if you wish to integrate technology with this book, you can certainly do so. Prentice Hall can make available with this text various MATLAB and MAPLE technology manuals at relatively little cost. For more details, see the section on "Supplements".

Applications of linear algebra appear in the final chapter, leaving the instructor free to import a favorite application after earlier sections wherever appropriate or desired. (I think this is preferable to optional—and oft omitted—sections on applications interspersed throughout the text.) Again, it is my experience that there is not enough time in a short semester to do justice to the fundamental concepts and ideas of linear algebra, while at the same time exposing students to a wide variety of applications. Interestingly, linear algebra is the one course I teach nowadays in which I am not regularly asked "What is this used for?" As a matter of fact, I am often approached by students who want to tell me how linear algebra is making appearances in their other courses, from computer science to geology.

Students usually find the start of their linear algebra course easy with its typical heavy emphasis on matrix manipulation and linear systems, while comfort levels take a deep dive the day vector spaces are introduced. Sometimes, within a couple of lectures, terms like "linear combination," "span," "spanning set," "linear independence," "basis," and "dimension" are introduced and sophisticated theorems discussed, before the definitions have really settled in students' minds. Even the word "trivial," as in "trivial linear combination," is not so trivial for some!

To ease the transition from linear systems to vector spaces, I have moved a lot of the terminology of vector spaces to Chapter 1. Readers encounter and see in use the language of linear algebra throughout the first three chapters, before they meet the formal definition of "vector space." They see lots of examples that illustrate concepts usually not introduced until the typical "vector space" chapter. The idea of a plane being "spanned" by two vectors is not hard for beginning students; neither is the idea of "linear combination" or the fact that linear dependence of three or more vectors (in R3) means that the vectors all lie in a plane. That Ax is a linear combination of the columns of A, surely one of the most useful ideas of linear algebra, is introduced early in Chapter 2 and used time and time again. Long before we introduce vector spaces, all the terminology and techniques of proof have been at work for a long time.

I endeavor to introduce new concepts only when they are needed, not just for their own sake. We solve linear systems by using Gaussian elimination in order to achieve row echelon form, postponing the definition of reduced row echelon form to where it is of benefit, in the calculation of the inverse of a matrix. The notion of the "transpose" of a matrix is introduced in the section on matrix multiplication because it is important for students to know that the dot product of two column vectors u and v is the matrix product uTv. Symmetric matrices, whose LDU factorizations are so easy, appear for the first time in the section on LDU, and reappear as part of the characterization of a projection matrix.

There can be few aspects of linear algebra more useful, practical, and interesting than eigenvectors, eigenvalues, and diagonalizability. Moreover, these topics provide an excellent opportunity to discuss linear independence in a nonthreatening manner. Why do these ideas appear so late in so many books?

My writing style may be less formal than that of other authors, but my students quickly discover that informality is not synonymous with lack of rigor or demand for understanding. There is far less emphasis on computation and far more on mathematical reasoning in this book. I repeatedly ask students to explain "why." Already in Chapter 1, students are asked to show that two vectors are linearly dependent if and only if one is a scalar multiple of another. The concept of matrix inverse appears early, as well as its utility in solving matrix equations, long before we discuss how actually to find the inverse of a matrix (a skill arguably not as important as it once was). The fact that students find the early sections of this book quite difficult is evidence to my mind that I-have succeeded in emphasizing the importance of asking "Why," discovering "Why," and then clearly communicating the reason "Why."

A Course Outline

A few comments about what I actually include in courses based on this book may be helpful. To achieve economy within the first three chapters, I omit Section 2.6 on LDU factorization (never the section on LU) and discuss only a few of the properties of determinants (Section 3.2), most of which are used, primarily, to assist in finding determinants, a task few people do by hand any more.

There is more material in Chapters 4 to 7 than I can ever manage. Thus I often discuss the matrix of a linear transformation only with respect to standard bases, omitting Sections 5.3 and 5.4. The material of Section 6.1, which is centered around the best least squares solution to overdetermined linear systems, is nontraditional, but try to resi...

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