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Advanced Linear Algebra focuses on vector spaces and the maps between them that preserve their structure (linear transformations). It starts with familiar concepts and then slowly builds to deeper results. Along with including many exercises and examples, each section reviews what students need to know before studying the material.
The book first introduces vector spaces over fields as well as the fundamental concepts of linear combinations, span of vectors, linear independence, basis, and dimension. After covering linear transformations, it discusses the algebra of polynomials with coefficients in a field, concentrating on results that are consequences of the division algorithm. The author then develops the whole structure theory of a linear operator on a finite dimensional vector space from a collection of some simple results. He also explores the entire range of topics associated with inner product spaces, from the Gram–Schmidt process to the spectral theorems for normal and self-adjoint operators on an inner product space. The text goes on to rigorously describe the trace and determinant of linear operators and square matrices. The final two chapters focus on bilinear forms and tensor products and related material.
Designed for advanced undergraduate and beginning graduate students, this textbook shows students the beauty of linear algebra. It also prepares them for further study in mathematics.
- Sales Rank: #824364 in Books
- Published on: 2010-06-23
- Original language: English
- Number of items: 1
- Dimensions: 1.10" h x 6.30" w x 9.10" l, 1.75 pounds
- Binding: Hardcover
- 364 pages
Review
… The book is well written, and the examples are appropriate. … Each section contains relevant problems at the end. The ‘What You Need to Know’ feature at the beginning of each section outlining the knowledge required to grasp the material is useful. Summing Up: Recommended.
―CHOICE, January 2011
Pedagogically, a structural and general approach is taken and, topically, the material has been chosen in order to cover the material a beginning graduate student would be expected to know when taking a first course in group or field theory or functional analysis. …
―SciTech Book News, February 2011
About the Author
Bruce Cooperstein is a professor of mathematics at the University of California, Santa Cruz, USA. He was a visiting scholar at the Carnegie Foundation for the Advancement of Teaching (spring 2007) and a recipient of the Kellogg National Fellowship (1982–1985) and the Pew National Fellowship for Carnegie Scholars (1999–2000). Dr. Cooperstein has authored over fifty papers in referred mathematics journals.
Most helpful customer reviews
33 of 35 people found the following review helpful.
Written Specifically for the Second Course
By A Reader
I taught a 3-hour course in advanced linear algebra during the 2011 fall semester, using this book as the primary course text. The class included seven first-year graduate students and four seniors, all mathematics majors. All eleven students had taken a first course in linear algebra using the book Linear Algebra, 4th ed., by Friedberg, Insel and Spence. They all had prior exposure to vector spaces and linear transformations, bases and coordinate representation, elementary matrix operations, determinants, eigenvalues and eigenvectors, and a bit of inner product space theory. My comments are based on the results achieved in this course with an audience of this level of preparation.
In a perfect world, one would like the second course in linear algebra to pick up exactly where the first course left off. However, there can be a time gap of two to three years between the two courses; students---even very good students---do not retain perfect command of the material they studied in the first course. It is typically necessary to begin the advanced linear algebra course with a brief review of the core topics. However, one must avoid getting mired down in this review, or there will be inadequate time for the many advanced topics that must be covered.
This is where the first major strength of Dr. Cooperstein's book becomes apparent. The first two chapters (85 pages) provide a succinct but thorough review of vector spaces and linear transformations, thus enabling the students to quickly re-familiarize themselves with material they have seen before. Emphasis is placed on the underlying field of scalars chosen from R or C. While the pace is rapid, it is entirely appropriate for students who have already had a solid first course in linear algebra. There are several topics scattered throughout the first two chapters that are not typically covered in the first course (at least not in our first course), thus keeping the students challenged and permitting them to learn some new material while reviewing the familiar.
Beyond the first two chapters, there is more material than could reasonably be covered in a one-semester 3-hour course, thus permitting the professor to design a number of different advanced courses. Chapter 4 studies invariant subspaces of a linear operator, cyclic operators, and canonical form theory. Chapter 5 addresses inner product spaces, dual spaces, and adjoints. Chapter 6 then studies linear operators on inner product spaces, self-adjoint operators, unitary and orthogonal operators, and the spectral theorems. Chapter 8 considers bilinear forms , including symplectic spaces. The concluding Chapter 9 offers an introduction to tensor products of vector spaces and the symmetric and exterior algebras.
My students were especially appreciative of the "What You Need to Know" paragraphs that appear at the beginning of each section, clearly spelling out the prerequisites for successful study. I think this feature would be particularly helpful to one who is studying independently. It is a simple innovation that compares with Michael Sipser's "Idea of Proof" discussions in his book on the Theory of Computation.
The exercises in the book are well-chosen for a second course in the subject, both in degree of difficulty and in number. Many develop additional results or ask the student to provide proofs of selected results from the text. There are no lengthy sets of repetitive "drill" exercises, and it is possible with hard work to attempt all the problems in each section. Answers and hints to selected exercises are provided in an appendix.
My students gave the text very high marks in their course evaluations. No text is perfect, however, and the most frequent complaints about this book involved (a) unnecessarily "baroque" notation in certain sections (their word); and (b) the number of typographical errors.
In conclusion, I found this to be an excellent choice for the second course in linear algebra at my institution. The book matched up well with our students and their educational background, as described above.
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