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Linear Algebra: A Geometric Approach, by Theodore Shifrin, Malcolm Adams
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Introducing students to a subject that lies at the foundations of modern mathematics, physics, statistics, and many other disciplines, Linear Algebra: A Geometric Approach appeals to science and engineering students as well as mathematics students making the transition to more abstract advanced courses.� One of the goals of this text is to help students learn to think about mathematical concepts and to write rigorous mathematical arguments. The authors do not presuppose any exposure to vectors or vector algebra, and only a passing acquaintance with the derivative and integral is required for certain (optional) topics.�Linear Algebra, First Edition is now available exclusively at CourseSmart, as a digital eTextbook.
- Sales Rank: #1155124 in Books
- Brand: Brand: W. H. Freeman
- Published on: 2001-08-24
- Original language: English
- Number of items: 1
- Dimensions: 9.22" h x 1.12" w x 7.16" l, 1.92 pounds
- Binding: Hardcover
- 464 pages
- Used Book in Good Condition
Most helpful customer reviews
8 of 9 people found the following review helpful.
Truncated Explanations
By KPL
I gave this book a poor review not because it shows little examples and applications and demands very theoretical work without much background (all of which are true), but because it does not do a good job with what it claims to be its mission: to foster a good intuition of the geometric implications. If you read the reviews of the earlier edition, "Twit" writes what this book never does. In all truth, this would be an excellent book if you could sit for two hours a day with a tutor and slowly see the big picture, as Shifrin wants. It is a book that intentionally makes things laborious. I have always seen the example section as a place to READ problems, not to DO them. Perhaps a very similar problem might follow an example, but Shifrin and Adams leave even the teaching to the reader.
As a mathematical treatise, this book is excellent; as a pedagogical device, it is unhelpful. The "big picture" is always expected to be gleaned from the exercises; other books will state it off the bat, then let the student gain a deeper appreciation through the exercises.
The authors take pride in the book's being advanced; that said, I am a good math student and am happy to work through difficult problems - but I want to be grounded in the fundamentals before I do the abstract proofs he demands. It always just hints at what should really be understood without ever explicitly communicating it.
The advanced level is excellent and should be preserved, but Shifrin and Adams MUST add more grounding exercises in the next edition, which I had to look to other books to find.
4 of 4 people found the following review helpful.
Graduate Math Student Perspective
By N. Martin
Original Review (2/2013):
Having taken linear algebra several years ago with a 2002 printing of Shifrin's book, I'd have to say it was a difficult text to learn by. I did well due to much rote memorization. I kept expecting that toward the semester's end all things would come together and make sense, but they never did. At the time I had a vague sense of unease by how muddy the waters seemed after the class was over, but heck - an A's an A, right? Fast forward now to graduate linear algebra. I'm using this book as a reference guide to fill in the gaps of my forgotten knowledge (since I keep all my textbooks) and realize more now than I did then how much it really lacks.
The good: The proofs are great. Even as an undergrad, I think there was only one I had to really work at to grasp the steps in logic. The blue boxes detailing important guidelines and tips are quite helpful.
The bad: The pictures aren't clear enough to give you an idea of what's going on without a better corresponding explanation in the text. (The pictures/figures may have changed in a more recent edition, but in the 2002 edition it's hard to tell what plane the different vectors are supposed to be in or coming out of due to the blue on black with no shading/shadow/light source rendering. Some were confusing enough I gave up trying to make sense of the pictures and just read the text. That's bad for a book that claims to be a geometric approach.) Even with better pictures, a little text box representing what you're supposed to be looking at and why it's important would be useful to a new student. While this is more the publisher's fault than the author's, other readers have commented and it seems to be an issue: I bought my book new and the pages fell out of the binding 2/3 of the way through the semester. I replaced the binding with a 3-ring binder.
The ugly: Somehow I managed to get through the entire linear algebra course using this book without realizing how the dimension of a subspace related to the real number space. Previous comments are right on the mark about the examples. Instead of using the examples to motivate a difficult concept, the author may give a generalization of a new topic for an mxn matrix in R^(mn) and afterward give an example of how this concept works with a 3x2 matrix you can actually work with. A student new to the material will likely miss the point of the example and have a hard time understanding the meaning of the new concept's generalization. In my opinion, it's easier to understand a concrete example and then move onto a generalization. The examples that are given are shown as computational exercises. Why they're important isn't really explained in the text. Notes in the margin detailing the steps would be helpful for a student new to the material to fill in the missing gaps.
Bottom line: You can succeed with this textbook (my rote memorization/computation approach to the class years ago worked) but you may lack in basic understanding of the concepts. If you have a good instructor or supplement with MIT's Open CourseWare lecture notes and the short lecture revision videos online, you'll be fine. However, if you would like a thorough knowledge of the material for further studies, I strongly recommend you buy another textbook to learn the material from and keep as a reference after your course. You can rent this one for a semester or get the homework problems from a classmate if you're short on cash.
Update (5/2013):
I read through the book again, side-by-side with Hoffman/Kunze for a graduate course in Advanced Linear Algebra to refresh and fill in the gaps in my knowledge. I explored the application sections, which my undergraduate instructor skipped, and was pleased with how much it added to my understanding. This time around, I noticed why things are shown in 2 or n dimensions and how that relates to what is going on in the various subspaces. Even though this was lost on me as an undergraduate because the authors never really explicitly addressed or explained it, I'm upgrading to 3 stars.
The chapters vs. section labeling was still confusing. The authors frequently referenced previous problems in the examples, and I spent a great deal of wasted time just trying to figure out where the example was. Last, calculations in the examples aren't intuitive. While I figured out the examples this time around, I remain baffled at how I managed to learn the material as an undergrad. In the last half of the book in particular, the authors often just present a basis for a space or transformation without explaining how they arrived at it. This would be fine if they had actually explicitly explained how to do the task in section xyz, but they never really did. They implied how, but let the reader figure it out on their own. While that makes for intriguing reading of leisure material in your free time, it is poor exposition for text book purportedly aimed at reaching students in a classroom setting.
In short, the book was much better the second time through, but primarily due to greater mathematical maturity I brought to the table. It is a fine book to prepare you for further work in the subject, but you should purchase a supplemental text if this will be your first foray into Linear Algebra. Hopefully the authors will introduce a second edition in the near future to address some of these issues.
24 of 29 people found the following review helpful.
a fine book telling you what the numbers mean
By twit
Again I was puzzled that such a fine book by such fine authors could receive several pans here. Looking at it again I see why. As usual it is because the book gives the reader more than some of them want, and hence expects more from them in turn.
Instead of merely exhibiting pages of sterile computations with rows of matrices and linear equations, but no visible meaning, the authors begin with a short and useful review of the geometry of vectors in the plane, including ways of computing angles via dot products. Using the ideas developed there, they expand to discuss n dimensional vector geometry, and pose the problem of describing hyperplanes in n space, i.e. copies of n-1 dimensional subspaces embedded linearly in n space. Of course these ideas are already challenging.
Why do they do this? Because this is exactly what the solutions of a linear equation in n variables represents. One equation represents one hyperplane. Hence several simultaneous equations represent the intersection of several hyperplanes. that's all folks.
The accompanying geometry reveals exactly why 2 equations in 3 variables are expected to have infinitely many solutions: it is because the two planes represented by the two equations, intersect generally along a line in 3 space. But the uncurious student who does not care what solutions of equations mean, is annoyed rather than enlightened.
This is unfortunate, but the authors are rather to be complemented for explaining not only how calculations in the subject are carried out, but what they mean geometrically, and also how they can be applied in many situations. Perhaps the deepest applications, to differential operators, occurs as well at the end of the book.
All in all a fine book for some one who wants to understand not just the numerology, but also the geometry of linear algebra, i.e. the interpretation that gives intuitive substance to all the theorems.
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