Linear Algebra. Pure and Applied Mathematics

by Peter D. Lax

This introduction to linear algebra by world-renowned mathematician Peter Lax is unique in its emphasis on the analytical aspects of the subject as well as its numerous applications. The book grew out of Dr. Lax's course notes for the linear algebra classes he teaches at New York University. Geared to graduate students as well as advanced undergraduates, it assumes only limited knowledge of linear algebra and avoids subjects already heavily treated in other textbooks. And while it discusses linear equations, matrices, determinants, and vector spaces, it also in-cludes a number of exciting topics that are not covered elsewhere, such as eigenvalues, the Hahn-Banach theorem, geometry, game theory, and numerical analysis. The first four chapters are devoted to the abstract structure of finite dimensional vector spaces. Subsequent chapters deal with determinants as a blend of geometry, algebra, and general spectral theory. Euclidean structure is used to explain the notion of selfadjoint mappings and their spectral theory. Dr. Lax moves on to the calculus of vector and matrix valued functions of a single variable—a neglected topic in most undergraduate programs—and presents matrix inequalities from a variety of perspectives. Fundamentals—including duality, linear mappings, and matrices Later chapters cover convexity and the duality theorem, describe the basics of normed linear spaces and linear maps between normed spaces, and discuss the dominant eigenvalue of matrices whose entries are positive or merely non-negative. The final chapter is devoted to numerical methods and describes Lanczos' procedure for inverting a symmetric, positive definite matrix. Eight appendices cover important topics that do not fit into the main thread of the book. Clear, concise, and superbly organized, Linear Algebra is an excellent text for advanced undergraduate and graduate courses and also serves as a handy professional reference.

Genres: Mathematics, Science, Algebra

250 pages, Hardcover

First published January 1, 1996

About the author

Peter David Lax was a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics.
Lax has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields.
In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades. Interest in the "Lax conjecture" grew as mathematicians working in several different areas recognized the importance of its implications in their field, until it was finally proven to be true in 2003.

Reviews

[Mark Moon · Rating 4 out of 5]

There's a lot of great material here (some of which is hard to find elsewhere), but the style makes it hard to read, and there are more than a few typos (and I can't find any errata online). This is the book Professor Matt Macauley uses for his Advanced Linear Algebra Class at Clemson; you can find video recordings of his lectures on YouTube.

[Ian · Rating 5 out of 5]

This is one of the very best books on linear algebra that I have ever read. It strikes a wonderful balance between theory and practice. However, it is unequivocally for a mature audience. This is not a first book in linear algebra unless you are already mathematically quite mature. For example, it makes regular use of Duality notion and abstract (but very natural and insightful) proofs. However, if you have had a course in functional analysis this book is a breeze and filled with delicious little treats and beautiful ways of approaching familiar ideas.

[David · Rating 5 out of 5]

My current favorite book on straight linear algebra (versus matrix analysis, functional analysis, or numerical linear algebra).

[Ronald Lett · Rating 4 out of 5]

An excellent presentation of the modern tools of linear algebra. One should have a good grounding in undergraduate linear algebra and modern algebra before using this text, however.