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Linear Algebra, by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence
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This book presents a treatment of principal topics of linear algebra and illustrates the importance of the subject through a variety of applications. The topics include vector spaces, linear transformations and matrices, elementary matrix operations and systems of linear equations, determinants, diagonalization, inner product spaces, canonical forms. In addition there are applications to differential equations, economics, geometry, physics and probability. The second edition uses Gaussian elimination instead of Gauss-Jordan method, reduces the dependence of direct sums, shortens and reorganizes the coverage of diagonalization, develops unitary diagonalization via Schur's theorem as opposed to the more abstract invariant subspaces, interchanges canonical forms and inner product spaces, and features a new subsection which develops motions in the plane.
- Sales Rank: #3113892 in Books
- Published on: 1989-02
- Original language: English
- Number of items: 1
- Dimensions: 9.50" h x 6.25" w x 1.00" l,
- Binding: Hardcover
- 528 pages
Most helpful customer reviews
35 of 36 people found the following review helpful.
Good alternative to Hoffman and Kunze
By Kindle Customer
A few introductory comments are in order: (1) This is *not* intended to be a first look at the subject of linear algebra, at least from the "computational side". (2) This is an undergraduate level text, though typically students will not encounter this material before their junior or senior years. (3) There is some overlap with a graduate level course in linear algebra, though this book is not comprehensive enough for a course at that level.
Ok, now that we've gotten that out of the way...
We used this as the primary textbook as a cross-listed advanced undergraduate/beginning graduate course I took in linear algebra. I had to supplement this book with outside reading/assignments to fulfill the balance of the course requirements. Contrary to what you might expect, you do not need an "introductory linear algebra course" (read that as "linear algebra for engineers") to successfully navigate this book. Actually, much (not all) of the material covered in this book should be discussed in any decent undergraduate course in ordinary differential equations (Boyce & DiPrima's ODE text makes a decent reference).
Here, you'll find that the emphasis is on learning the theoretical side of linear algebra. While there is a chapter (Chapter 3) on basic matrix algebra (wholly unnecessary in my opinion), the main use of matrices here is to express linear operators in a form more suited for computations, e.g., the determination of eigenvalues and eigenvectors. Right away, in Chapter 1, vector spaces are introduced and many familiar (some unfamiliar) examples are given. Just as in an abstract algebra course, you define a list of axioms for vector spaces (later, inner product spaces) and see what you can do with them...quite a lot, as it turns out!
To briefly outline the book: Chp 1: Vector Spaces ; Chp 2: Linear Transformations and Matrices (this is where the matrix is exposed as being a convenient representation of a linear transformation) ; Chp 3: Elementary Matrix Operations and Systems of Linear Equations (some filler content...this should have been left out...better discussed either in an "intro" course or in a numerical linear algebra class) ; Chp 4: Determinants (ok, but should have been condensed into another chapter...come on, we should know how to compute a determinant by now!); Chp 5: Diagonalization (the book really shines here...this is the most lucid treatment of the Cayley-Hamilton theorem I have ever seen); Chp 6: Inner Product Spaces (pretty good, more emphasis on linear operators as opposed to arbitrary linear transformations); and Chp 7: Canonical Forms (the highlight of course being the Jordan Canonical Form).
As I mentioned earlier, you'll learn nothing new from Chapter 3 in particular. In fact, if you had a strong enough intro course in linear algebra, the truly new material is confined to parts of Chapter 5, Chapters 6 and 7. That's partially why I only give this book 4 stars instead of 5. What the book covers, it covers quite well...but it should assume more in the way of prerequisites. Also, Chapter 7, while definitely informative, let me down somewhat. All of the material covered there can be done in greater generality (while still being very comprehensible) in the context of modules over principal ideal domains (basically, think of a vector space over a less specialized ring than a field). It turns out that you lose very little in the transition from vector spaces to modules. Also, believe it or not, it actually clarifies some of the proofs concerning rational forms, since you have much more motivation. This book does do a good job of explaining the differences between the forms: briefly, when you can diagonalize, you get the most for your money...when diagonalization is impossible, try for the Jordan form...when *that's* impossible...you can always fall back on the good old Rational Canonical form (which always exists, regardless of how the characteristic polynomial behaves).
If you look at the reviews for Hoffman and Kunze's linear algebra text, you'll find one by a mathematics professor (sometime in 2007) that is right on the money. Hoffman and Kunze is still the gold standard in theoretical linear algebra. If you're looking for "the meat" this book is missing, you'll find it there. Just be warned that Hoffman and Kunze (at least in older editions) has horrible typesetting, and it definitely takes no prisoners. This book is excellent preparation for Hoffman and Kunze, so it is well worth your time to work through it.
As far as extras goes, there are the standard appendices on material you should already know...sets, functions, fields, complex numbers, etc. Also, there is some interesting material squirrelled away in Chapters 5 and 6. Markov chains are discussed in Section 5.3 (the first look at that topic for me, and quite absorbing). In Chapter 6, expect some material from numerical linear algebra (singular value decomposition, conditioning and the Rayleigh quotient) as well as bilinear and quadratic forms (yawn...but ok) and, much to my surprise, a linear algebra spin on Einstein's Special Theory of Relativity (relax, no physics is required).
To sum up, I do not for a minute regret owning this book...in fact, I wish I had read it sooner. This is the kind of textbook that should be used in the undergraduate survey course to begin with. Leave the computational stuff to the ordinary differential equations course. In fact, being a math teacher myself, I can tell you that elementary matrix algebra is filtering down to the college algebra level now, so anyone who is a math major should already know the basics of row reduction, echelon forms, etc., before they even walk into a linear algebra class!
Ok, getting off the soapbox: buy this book, read it, love it, and remember it fondly when you have to take a graduate course in linear algebra. I found Roman's "Advanced Linear Algebra" a good text to continue with where this one leaves off.
10 of 10 people found the following review helpful.
Excellent Book
By Daniel G. Nolan
I used the 3rd Ed. in UC Berkeley's MATH 110: Linear Algebra, then used the 4th while grading homework for the same class the next year. I think the book is fairly comprehensive (though by itself not enough to prepare one for grad school), and very well-written. The exercises at the end of each section span a wide range of difficulty. The book is self-contained, except for a few basic results from the calculus (one has to know the linearity properties of derivatives and definite integrals, i.e. derivative of linear combination is linear combination of derivatives and similarly for integrals), yet does sort of assume prior knowledge of linear algebra. At UC Berkeley students have already taken MATH 54: Linear Algebra & Differential Equations, which includes a brief treatment of vector spaces, linear transformations, eigenvalues, etc. I wouldn't say this book is "not for the faint of heart," as some reviewers put it. I think it's ideally suited--essential, in fact--for entering juniors majoring in the any of the mathematical sciences. If this book is your first exposure to linear algebra, then I highly, HIGHLY recommend chapters 12 and 13 of Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition), and chapters 1-5 of Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications.
10 of 11 people found the following review helpful.
blows you away
By math student
This book blew me away! it's just great, it really explains everything (also an axiomatic approach of determinants which is very important to gain a thorough understanding of what determinants actually mean and which will help you when you are going to study multilinear algebra, exterior algebras in abstract algebra etc), the book gives you a good insight in the stucture of linear operators on a finite-dimensional vector space and provides lots of examples and useful applications (e.g. in economics and physics). There was a comment of one customer which criticized the fact that the theory doesn't offer an explanation of quotient spaces, this is true but in fact this is not important in the area of linear algebra, quotient structures should be studied in abstract algebra, when more algebraic structure has been developed so that one can really understand what quotients of algebraic structures are about. So the book is great, but I would recommend some knowledge about polynomials, fields, algebraic closure, vector space before starting to read this one.
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