Introduction to Linear Algebra

I write as a 35-year veteran teacher of mathematics and statistics, at Mount Holyoke College. This semester I am teaching two sections of linear algebra, from Gilbert Strang's Introduction to Linear Algebra, 4th edition. I understand that I'm one of the first, perhaps the very first, to teach from this edition, scooping even the author himself, whose spring semester at MIT began a week after Mount Holyoke's. In choosing a book for my course, I reviewed more than a dozen choices. In what follows, I'll try to set out why, looking back on the first two-thirds of the semester, I'm firmly convinced that I chose the right book to teach from. But first, here's an excerpt from an e-mail I sent the author a few weeks ago: I've admired your book ever since the first edition came out, but in our department we have to wait in line to teach linear algebra, and this is my first chance to teach from your book. It's hard to put into words how much I'm enjoying it. In 35 years, I've nearly always ended up feeling deeply disappointed with almost any textbook I've tried to teach from. However I've had the good fortune to find two books I really admire. Yours is one of those two inspiring books. Thanks to you, I'm having a blast! Enthusiasm aside, I'll start the substance of my review with four questions, aimed both at students and at teachers. These questions highlight the features I find inspiring - but they are not merely rhetorical: I've tried to formulate questions that should be helpful to anyone trying to decide whether Strang's Introduction to Linear Algebra is the right choice for them. In each instance, although my own answer is a resounding "Yes to choice one!" I can imagine teachers and readers whose preference would go the other way. * Do you want a book that puts a top priority on the substantive content of linear algebra as a subject in its own right, or a book that uses linear algebra as vehicle for teaching formal proofs? * Do you want a book whose exercises are imaginative, minimize unnecessary computation, challenge the reader to think about core concepts, and anticipate the content to come, or do you want exercises that closely track the pattern of worked examples du jour, with multiple instances of each type? * Do you want a book that works hard and thoughtfully to communicate the ideas that unite linear algebra, or do you want a book that has thinned and linearized the content in order to make teaching and learning go more smoothly? * Do you want a book written by a mathematician with a lifetime experience using linear algebra to understand important, authentic, applied problems, a former president of the Society for Industrial and Applied Mathematics, or do you want a book shaped mainly by the esthetics of pure mathematicians with only a weak, theoretical connection to how linear algebra is used in the natural and social sciences? To get more technical: The order of topics in a linear al

People say that mathematical truths never change, and that's true enough. New concepts, applications, and techniques keep emerging, though, so math teaching needs to keep up with the times. Strang has done an outstanding job of keeping this book current and relevant. It's not a mathematician's math book - this is aimed at people who need results and needs computational techniques more than they need crystalline theorems. That's why it's so helpful to see applications like Markov models, Kirchoff's laws, and Google's analyses of the web. It's also helpful to see examples worked in Mathematica and MATLAB, the tools of choice for desktop exploration of numerical systems. It's startlingly easy to come up with a 100x100 system of equations, and just nuts to try to solve it by hand. Strang assumes some amount of calculus in this book, something that other books on linear algebra sometimes skip. That raises the bar for the readership, but also opens up topics like change-of-basis in function space, including Fourier analysis. It also allows differential equations to be addressed as linear systems. Even without calculus, though, a reader is exposed to the singular value decompostion, QR and other matrix decompositions, and considerations in performing the computations. I found a few oddities, such as the description of a matrix's condition number. That has great physical meaning when it's taken as the ratio of the matrix's highest and lowest eigenvalues, but Strang gives a definition that I found less intuitive. Such oddities are rare, though. Even though this book covers many topics, its emphasis is on clear and applicable presentation. I recommend this to anyone studying linear algebra or who, like me, has to brush up on basics not used in many years. //wiredweird

As someone who already knows the basics of the subject, I guess I'm looking at things with the benefit of hindsight. However, I needed to shore up my own knowledge of Linear Algebra and thought I might as well turn to Strang for a refresher and a different approach. The result is that I am truly pleased with this book. His writing is lively and engaging. Linear Algebra has a phenomenal tendency to get dry and Strang does an excellent job of turning the subject this way and that so that one can admire it from every angle. In particular, there are three major approaches in this book that make it stand out. 1. Strang places heavy emphasis on vectors, vector spaces and transformations. This is good preparation for future study in Linear Algebra. This will provide an intuitive understanding of linear operators on vector spaces later. 2. Another reviewer mentioned that the book utilises a discovery-based approach. While this might be a disadvantage when you're in a hurry, the approach prepares one well for learning more theoretically oriented subjects where self-guided discovery is imperative. In this sense, I think the discovery approach is far superior to others and prepares the reader well for future studies. The problems are really fun (although I personally think they are much too easy). Many of the questions require light-weight proofs without undue formalism (not really required at this level). These pseudo-proofs really do help build understanding of the subject. Maths-phobes will not even realise that they're fleshing out the subject themselves. 3. The didactic approach taken in the book is conversational and informal. When added to the freely available video lectures at the OCW site, given by Strang himself, you really have a perfect introduction to the subject of Linear Algebra. The lectures are superb and Strang is an excellent teacher. His enthusiasm and passion for the subject is obvious and infectious. I really wish I had learned Linear Algebra from this book initially. The book does a good job of encouraging geometric intuition and visualisation. That said, I do not think the book is an ideal book for maths majors. The primary problems being too little exposure to abstraction and problems which are too easy. However, I do believe that the book can be used in conjunction with a more rigorous approach in cases where the latter gets just a touch too dry. There is time to develop the rigour in theoretical Algebra courses at a later stage, with the added benefit that the reader will have learned the experimental approach to learning taken in the book. I suppose some will find Strang's excitement over Linear Algebra a bit of a pain, but personally I think this conveys the sheer joy of pursuing an intellectual endeavour. I've always bordered on disinterest with Linear Algebra and this has been very much dispelled. I like to be reminded why I chose to study mathematics in the first place sometimes. While I can se

I used the book for self-study in combination with Strang's freely available video lectures. The book's emphasis is on understanding and appreciating concepts, rather than on formal proofs. As I have experienced him, Strang is very good at explaining things. He uses lots of examples, and, the textbook as well as the video lectures are very easy to comprehend. His informal writing helps to make the book an interesting read. For me, having had some linear algebra in high school, most ideas were familiar. However, it was always delightful for me to look at Strang's perspective on things. One clearly sees that Strang has a lot of experience in teaching linear algebra. He introduces new ideas in a very natural way, emphasizing what they are good for beforehand. I particularly liked the chapter on determinants. The book doesn't require any prior knowledge besides very basic high school math. I used it in my year of civilian national service, before starting to go to college. Compared to other college level books for math majors I have looked at, this book works great for self-study. The level of difficulty seems to be something inbetween high school and university level (math major).

Gilbert Strang is a very experienced teacher of Linear Algebra, and this book is written as a text based on his MIT linear algebra class. Math majors will not find the 'definition-proposition-lemma-theorem-proof-corollary' treatment here. Instead Strang, aware of the need to teach non-math majors the subject, explains linear algebra in a simple but effective way --examples, diagrams, motivations. This book is one of those with which you can skip class the whole semester and get good grades (but don't do it! get your education in the classroom).