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14 of 18 people found the following review helpful:
4.0 out of 5 stars
a fine book telling you what the numbers mean, September 20, 2004
This review is from: Linear Algebra: A Geometric Approach (Hardcover)
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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5 of 6 people found the following review helpful:
4.0 out of 5 stars
Very suitable for use as a text in a linear algebra course, December 7, 2006
This review is from: Linear Algebra: A Geometric Approach (Hardcover)
When looking over the reviews of a college textbook, one must take care not to fall into the fallacy of accepting a study where the selection is extremely non-random. Students who take a course where a specific book is used and have difficulty with the material tend to be the ones who try to get even by writing horrific reviews. This book is nowhere near as bad as the comments of other reviewers would lead you to believe.
I teach mathematics at the college level and examined this book for possible adoption as the text for a course in linear algebra. While my teaching assignment was changed so I was no longer teaching the course, there is no question that this book would have been suitable.
There are many worked examples and they are clear, thorough and yet concise. A diagram is included when necessary but there are no cases where a diagram is superfluous. The coverage is that of a traditional linear algebra course and there are special sections on:
*) Complex eigenvalues and Jordan canonical form
*) Computer graphics and geometry
*) Matrix exponentials and differential equations
Solutions to the majority of the exercises are included in an appendix.
Linear algebra is the traditional transition course in the math major, where the student bridges from what is sometimes called the "plug and chug" level of mathematics to the "theorem-proof" level. In this book, the authors take an appropriate approach to this transition, using geometry as much as possible to aid in the understanding of what the constructs of linear algebra are.
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1 of 1 people found the following review helpful:
2.0 out of 5 stars
Not worthless, but inadequate, September 2, 2010
This review is from: Linear Algebra: A Geometric Approach (Hardcover)
Let me first say that I attend Duke, enjoy math, get good grades in math, and am willing to put in the hours to understand something. But I think that it is very telling that those who teach linear algebra and those who learn linear algebra have very different attitudes towards this book.
The parts where the book explains something with diagrams, explains a proof, or puts forth a definition are generally quite good. The concepts are exacty what I want to learn. I would ask that the book be a little more directed towards beginners and not assume that repitition of concepts is unnecessary, but I have no major complaints about that part.
The huge, glaring, fatal weakness of this book is that there are nowhere near enough examples and practice problems. I took notes on each section, worked through each example multiple times, and make a sincere effort to do every single one of the exercises (don't tell me I haven't put in enough effort, because I've been nearly ignorning my other classes to spend hours every day trying to puzzle out this book).
I consider myself a math and science person, and I am doing everything I can. But the simple fact is that I need more examples. More explanations of problems. More ways to practice what I'm trying to learn.
I'm fine with a challenging text, but there's a difference between challenging and simply frustrating. I don't want the text to become any easier, I just want a way to improve my understanding of it, and I don't think the current amount of exercises and examples are anywhere near adequate.
To be perfectly clear, DO NOT purchase this book for self-study unless you already are quite familiar with linear algebra. If you know and love linear algebra pretty well already, I can see how this book might be useful.
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