Basic Matrix Algebra with Algorithms and Applications

Robert A. Liebler

• Publisher: Chapman and Hall/CRC
• Year: 2018
• ISBN:
  9781351989619
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Clear prose, tight organization, and a wealth of examples and computational techniques make Basic Matrix Algebra with Algorithms and Applications an outstanding introduction to linear algebra. The author designed this treatment specifically for freshman majors in mathematical subjects and upper-level students in natural resources, the social sciences, business, or any discipline that eventually requires an understanding of linear models. With extreme pedagogical clarity that avoids abstraction wherever possible, the author emphasizes minimal polynomials and their computation using a Krylov algorithm. The presentation is highly visual and relies heavily on work with a graphing calculator to allow readers to focus on concepts and techniques rather than on tedious arithmetic. Supporting materials, including test preparation Maple worksheets, are available for download from the Internet.This unassuming but insightful and remarkably original treatment is organized into bite-sized, clearly stated objectives. It goes well beyond the LACSG recommendations for a first course while still implementing their philosophy and core material. Classroom tested with great success, it prepares readers well for the more advanced studies their fields ultimately will require.

Chapter 1

Systems of linear equations and their solution

Mathematics, as you know, is full of symbols, and much of this notation does not appear in other subjects. In this course you will see a lot of new notation. This notation is really a carefully constructed language.

You may not be aware of the importance of notation in mathematics. Good notation is designed to accentuate the significant aspects of the problem being studied and to suppress its insignificant aspects. Because what is significant is sometimes a matter of judgment, our notation (and language) influences our perception.

For example, situations need to be discussed in which a great many variables are present. The exact number of variables n is really not important. What should the variables be named? The usual choice of letters of the alphabet starting with x has the disadvantage that there may not be enough letters. A more serious disadvantage is that you can’t even write them all down without implicitly specifying their number. This dilemma is solved with new notation. Then we choose a compound notation and write the variables x₁, x₂,…,x_n. The x part of the notation tells you it is (most likely) a variable and the subscript distinguishes the different variables without determining n.

Notice also the ellipsis “…”. In this course, you will always be safe if you think of “…” as shorthand for “and some omitted terms.” A consequence of the power of this notion is ambiguity. The exact nature of these terms is not specified and must be determined by context.

This chapter builds on your familiarity with elementary functions, analytic ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Examples
• Major results/proofs
• 1 Systems of linear equations and their solution
• 2 Matrix number systems
• 3 Diagonalizable matrices
• 4 Determinants
• A pedagogical postscript
• Appendix: The abstract setting
• Selected practice problem answers
• Index