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Matrices and Linear Transformations: Second Edition (Dover Books on Mathematics)
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"Comprehensive . . . an excellent introduction to the subject." — Electronic Engineer's Design Magazine.This introductory textbook, aimed at sophomore- and junior-level undergraduates in mathematics, engineering, and the physical sciences, offers a smooth, in-depth treatment of linear algebra and matrix theory. The major objects of study are matrices over an arbitrary field. Contents include Matrices and Linear Systems; Vector Spaces; Determinants; Linear Transformations; Similarity: Part I and Part II; Polynomials and Polynomial Matrices; Matrix Analysis; and Numerical Methods. The first seven chapters, which require only a first course in calculus and analytic geometry, deal with matrices and linear systems, vector spaces, determinants, linear transformations, similarity, polynomials, and polynomial matrices. Chapters 8 and 9, parts of which require the student to have completed the normal course sequence in calculus and differential equations, provide introductions to matrix analysis and numerical linear algebra, respectively. Among the key features are coverage of spectral decomposition, the Jordan canonical form, the solution of the matrix equation AX = XB, and over 375 problems, many with answers.
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Product details
Series: Dover Books on Mathematics
Paperback: 336 pages
Publisher: Dover Publications; 2nd edition (May 1, 1990)
Language: English
ISBN-10: 0486663280
ISBN-13: 978-0486663289
Product Dimensions:
5.4 x 0.7 x 8.5 inches
Shipping Weight: 12 ounces (View shipping rates and policies)
Average Customer Review:
4.0 out of 5 stars
25 customer reviews
Amazon Best Sellers Rank:
#209,824 in Books (See Top 100 in Books)
I got this book to brush up on my linear algebra. I'm looking to get into digital signal processing, and matrix operations are absolutely vital to pretty much everything a computer does. I approached this book as an engineer, meaning I just wanted a couple cheat sheets and quick algorithms to get out of it that I could then apply to a program, while chucking all the mathematical formality and notation out the window.And because of that, by the second chapter I was failing miserably. I thought I could skim over the first chapter because I legitimately did know and remember all the concepts in it. I know matrix multiplication, I know what row reduced echelon form is, I know the basics of solving systems of equations with matrices etc. As I went on, I found myself unable to understand any of the book. And then I went back, and right there in chapter 1.1, it explains exactly what every little notation is. I had to get myself into a mathematical mindset. There's all sorts of seemingly trivial little details, little caveats and exceptions and specific wording that's actually crucial.Trust me, if you use this book, you will understand every single little tiny detail of matrices so incredibly well. But! You have to work through it. Don't skip anything. If some proof looks scary and complex, it's because you skipped something earlier in the book. Sometimes it will go "And now from this equation we go to this equation" and some notation that you thought you knew what it meant but you really don't, and you'll wonder why he skipped like 10 steps. Then you go back and realize those 10 steps could all be condensed into one simple theorem explained a couple subchapters back. If you have to reread a proof 5 times to get it, do it. If all you know is arithmetic, algebra, and some basic calculus (high school math pretty much), you can become a master of matrices by the end of this book.This book is for the patient, that is for sure. Stick with it and you'll be amazed at how far you get.
This book is a very good book for someone wanting to dive into Linear algebra. The author does a massive number of the proofs in the book and is fairly good at explaining the subject matter in each chapter. Given that every person learns a little differently, I'd still suggest using online resources to supplement the readings and exercises, especially for practical applications.
Charles Cullen's original goal was to "provide a text, requiring a minimum number of prerequisites, which would enable me, in a one-term course, to cover...linear algebra and matrix theory." This volume does so in the first five chapters and continues with four additional chapters designed to support a second term. He suggests as background "only" an introductory course in calculus and analytic geometry.The first chapter introduces basic matrix operations such as addition, multiplication, transposition and inversion. Chapter 2 covers vector spaces and the concepts of linear independence and rank. Chapters 3 and 4 introduce and elaborate matrix determinants and linear transformations. Chapter 5 develops the Jordan canonical form using invariant subspaces and direct sum decompositions. Chapters 6 and 7 take an alternative path to explaining the Jordan canonical form. Chapters 8 and 9 use concepts and tools from previous chapters to introduce matrix analysis and numerical analysis.I bought this book for an online matrix algebra course when I found the required text, Matrix Algebra: An Introduction, less than helpful. Even though the course text was targeted toward social scientists like myself, its explanations and formula derivations were less helpful than its examples. I found Cullen's book and it got me through the course. In addition to clear explanations, practice exercises with answers, and helpful organization, it was also less expensive and covered more material.I recommend this book to those needing a refresher in matrix methods or approaching them for the first time.
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