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This introduction to linear algebra features intuitive introductions and examples to motivate important ideas and to illustrate the use of results of theorems.
Linear Equations; Vector Spaces; Linear Transformations; Polynomials; Determinants; Elementary canonical Forms; Rational and Jordan Forms; Inner Product Spaces; Operators on Inner Product Spaces; Bilinear Forms
For all readers interested in linear algebra.
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Most Helpful Customer Reviews
85 of 86 people found the following review helpful:
5.0 out of 5 stars
The Evolution of Linear Algebra,
This review is from: Linear Algebra (2nd Edition) (Hardcover)
As a professor of mathematics, I was recently assigned a section of our undergraduate linear algebra course; the last time I taught the course was twelve years ago. While doing the obligatory search for a course text, I have been surprised to see how the first course in linear algebra for mathematicians and scientists has "evolved" since I last taught it, at least insofar as that evolution is reflected through available and popular textbooks.
In one of the more popular linear algebra texts currently on the market (I will refrain from naming it), the formal definition of a vector space does not even occur until page 198, and this is not atypical. Looking through half a dozen of the more popular texts, one finds lengthy introductory chapters on vectors in R^n and their properties, basic matrix algebra, systems of linear equations, special algorithms for computing determinants and matrix inverses in efficient time, and significant space devoted to special matrix factorizations, such as the LU factorization. I would like to point out, without passing judgment, that this has not always been the case. Over time, the undergraduate course in linear algebra for mathematicians and scientists has evidently acquired a partial resemblance to the computational, non-proof-based course in "Matrix Algebra" that used to be offered to "casual users" of this area of mathematics at nearly all major universities. Hoffman and Kunze's book was written for the undergraduate linear algebra course at MIT in the 1960s. Those of us who pursued graduate study in mathematics in the 1970s saw copies of this text, with its vivid purple stripes down the cover, on the shelves of virtually every serious graduate student. Simply put, Hoffman and Kunze was a "standard" undergraduate reference for decades, which continued to inform its readers well into graduate programs or professional careers. The author of this review did not have the good fortune to use Hoffman and Kunze in a course, but I always had a copy at hand as a reference. My first linear algebra course, taken as a sophomore in the 1970s, used a text by Robert Stoll and Edward Wong (Academic Press, 1968). In Stoll and Wong, the definition of a vector space occurs on page 4, not on page 204. There is no preliminary chapter on basic matrix algebra; these computations are discussed as they arise, in context, when one chooses a basis for a vector space and therefore places coordinates on that space. The entire organization and conceptual structure of Stoll and Wong's book is worlds apart from the texts I have been reviewing of late. The same may be said of Hoffman and Kunze, and indeed of most of the popular linear algebra books from that period of time. This is why I am a bit disturbed when I read reviews that declare Hoffman and Kunze's classic text "outdated," "irrelevant," or "impossible to read." If the younger reviewers are comparing Hoffman and Kunze to most of the popular competitors that have been published in the past five years or so, then they are comparing a remnant apple to a crate of newly harvested oranges. Against all odds, Hoffman and Kunze remains in print, 46 years after its first apperance. And this in an era when the typical college text remains in print for what seems like less than five years. There is a reason for this longevity. For serious students of mathematics and the mathematical sciences, this text remains invaluable. If one is going to be called upon to actually USE linear algbra in any substantive way (and by substantive I do not mean inverting a matrix or solving a system of two linear equations in two unknowns), then one eventually must learn about such things as dual spaces and double duals, cyclic decompositions and the Jordan canonical form, unitary operators, self-adjointness, the spectral theorem, and multilinearity and tensors. One cannot even find most of these topics in the most popular undergraduate texts currently available on the market; they appear to reach their summit when they discuss eigenvalues and eigenvectors. As a consequence, if a student in an advanced course in, say, differential geometry or differential equations is sent back to his or her linear algebra text to read about dual spaces or the Jordan canonical form, then it will be necessary to abandon the text with which he/she is familiar and refer to a more serious reference like Hoffman and Kunze. How terribly inefficient. In the spirit of fairness, I must observe that the text Linear Algebra, 4th ed., by Friedberg, Insel and Spence is a currently available undergraduate text that is comparable to Hoffman and Kunze in coverage and rigor. It is an excellent text for a first course for mathematics majors---a true anomaly among a host of weaker competitors. However, the authors may dissuade many would-be users by their declaration in the preface that their text is "especially suited for a second course in linear algebra that emphasizes abstract vector spaces, although it can be used in a first course with a strong theoretical emphasis." The second undergraduate course in linear algebra is evidently becoming increasingly common; is this because the first course has been weakened to "matrix algebra" and therefore leaves the student unprepared to cope with advanced mathematical courses? My sincere thanks go out to Prentice-Hall for keeping Hoffman and Kunze in print all these years. Linear algebra is the essential prerequisite for nearly all advanced mathematics, and it is good to see that at least one definitive reference remains available, even as market and societal forces in higher education bring about a clear, demonstrable devolution in the quality of introductory texts on the subject.
25 of 26 people found the following review helpful:
5.0 out of 5 stars
Not for the faint hearted, but worth the effort,
By
This review is from: Linear Algebra (2nd Edition) (Hardcover)
This is a fantastic book on linear algebra. Not only does it cover abstract vector spaces through the Jordan canonical form, but takes the reader through complex inner-product spaces and the Spectral Theorem as well. For the reader who has the mathematical sophistication, this is a great book. Excellent preparation for the student planning to attend graduate school.
24 of 25 people found the following review helpful:
5.0 out of 5 stars
Best Linear Algebra Book but not for lay people,
By
This review is from: Linear Algebra (2nd Edition) (Hardcover)
I got this book for my Linear Algebra class about four years ago. This is a great book if you are getting a degree in mathematics. It won't help if you are just trying to get by the class and don't like math. It is not very practical but if you are looking for a real math book on Linear Algebra this is it. It contains a wealth of theorems that only a math lover would appreciate. If you really want to learn about Linear Algebra from a rigorous mathematical point of view this is it. This book taught me so much.
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