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Linear Algebra with Applications 7/e (絕)

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ISBN：9781259072994
作者：Keith Nicholson
版次：7
年份：2013
出版商：McGraw-Hill
頁數/規格：596頁/平裝雙色

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Description
Students of linear algebra learn by studying examples and solving problems ' accordingly, the book contains a variety of exercises and solved examples which are used to motivate and illustrate concepts and theorems, carrying the student from concrete to abstract. Chapters 1-4 contain a one-semester course for beginners, and then chapters 5-9 contain a second semester course. Note that the preface contains an updated and helpful guide to assist instructors with different chapter orders depending on their approach to the course

Features

Two stage definition of matrix multiplication: First, in Section 2.2 matrix-vector products are introduced naturally by viewing the left side of a system of linear equations as a product. Second, matrix-matrix products are defined in Section 2.3 by taking the columns of a product AB to be A times the corresponding columns of B. This is motivated by viewing the matrix product as composition of maps. This works pedagogically and the usual dot-product definition follows easily. As a bonus, the proof of associativity of matrix multiplication now takes four lines.
Matrices as transformations: Matrix-column multiplications are viewed (in Section 2.2) as transformations Rn ' Rm. These maps are then used to describe simple geometric reflections and rotations in R2.
Early linear transformations: It has been said that vector spaces exist so that linear transformations can act on them. Consequently these maps are a recurring theme in the text. Motivated by the matrix transformations introduced earlier, linear transformations Rn ' Rm are defined in Section 2.6, their standard matrices are derived, and they are then used to describe rotations, reflections, projections, and other operators on R2.
Early digitalization: As requested by engineers and scientists, this important technique is presented in the first term using only determinants and matrix inverses (before defining independence and dimension). An application to linear recurrences is given.
Early dynamical systems: These are introduced in Chapter 3, and lead (via digitalization) to applications like the possible extinction of species. Beginning students in science and engineering can relate to this because they can see (often for the first time) the relevance of the subject to the real world.
Bridging chapter 5: Chapter 5 lets students deal with tough concepts (like independence, spanning, and basis) in the concrete setting of Rn before having to cope with abstract vector spaces in Chapter 6.
Examples: The text contains over 375 worked examples, which present the main techniques of the subject, illustrate the central ideas, and are keyed to the exercises in each section.
Exercises: The text contains a variety of exercises (nearly 1175, many with multiple parts), starting with computational problems and gradually progressing to more theoretical exercises. Exercises marked with a ' have an answer at the end of the book or in the Students Solution Manual. There is a complete Solution Manual for instructors.
Applications: Reviewers state the type, range, and number of applications is one of Nicholson's strengths.
Appendices: Because they are needed in the text, complex numbers are described in Appendix A, which includes the polar form and roots of unity. Methods of proofs are discussed in Appendix B, followed by mathematical induction in Appendix C. A brief discussion of polynomials is included in Appendix D. All these topics are presented at the high-school level.
Major theorems: Several major results are presented in the book ' proofs are included for the stronger student to be aware of what is involved.
Clear, concise writing style: Nicholson is known for his rigorous, concise approach to Linear Algebra and this text is more rigorous than most. Linear algebra provides one of the best venues for getting the students to begin thinking logically and arguing concisely and to this point, the exercises ask the student to 'show' some simple implication. The book doesn't overcomplicate topics by using descriptive prose. There are many diagrams and examples to help students visualize.

Table of Contents
Chapter 1: Systems of Linear Equations
Chapter 2: Matrix Algebra
Chapter 3: Determinants and Diagonalization
Chapter 4: Vector Geometry
Chapter 5: The Vector Space Rn
Chapter 6: Vector Spaces
Chapter 7: Linear Transformations
Chapter 8: Orthogonality
Chapter 9: Change of Basis
Chapter 10: Inner Product Spaces
Chapter 11: Canonical Forms

Keith Nicholson, University of Calgary