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Advanced Linear Algebra

Steven Roman
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 135
[Reviewed by
Henry Ricardo
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This is a formidable volume, a compendium of linear algebra theory, classical and modern, intended for "the graduate or advanced undergraduate student." (I have not had the privilege of teaching undergraduates who could handle this text.) After a concise (30-page) treatment of set theory and basic algebraic structures, the author embarks on a two-chapter whirlwind tour of introductory linear algebra, including an optional discussion of topological vector spaces. Following this, there are several chapters of module theory, leading to structure theorems for finite-dimensional linear operators. The last parts of the "Basic Linear Algebra" section of the book are devoted to real and complex inner product spaces and the structure of normal operators.

But this is only slightly more than half of this book's contents! The last part of the volume is devoted to various "topics" such as Hilbert spaces, tensor products (including a treatment of the determinant as an antisymmetric n-linear form), affine geometry, QR and singular value decompositions, and the umbral calculus ("the first time that this subject has appeared in a true textbook").

The development of the subject is elegant, positively Bourbakian. The proofs are neat, whereas the examples are sparse and would have to be supplemented by the instructor. The exercise sets are good, with occasional hints given for the solution of trickier problems. Appropriate undergraduate prerequisite texts would seem to be Jacobson's Basic Algebra I and the classic text (still in print) of Hoffman and Kunze.

A graduate student with a typical undergraduate linear algebra course behind him or her may experience some culture shock in migrating to the text under review. There are few illustrations, no discussions of history, and no applications other than to other areas of mathematics. The book doesn't hint at the utility of the theory in modern scientific and engineering disciplines. It represents linear algebra qua algebra and does so comprehensively.

Henry Ricardo ( is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book A Modern Introduction to Differential Equations was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.

Vector Spaces

Linear Transformations

The Isomorphism Theorems

Modules I: Basic Properties

Modules II: Free and Noetherian Modules

Modules over a Principal Ideal Domain

The Structure of a Linear Operator

Eigenvalues and Eigenvectors

Real and Complex Inner Product Spaces

Structure Theory for Normal Operators

Metric Vector Spaces: The Theory of Bilinear Forms

Metric Spaces

Hilbert Spaces

Tensor Products

Positive Solutions to Linear Systems: Convexity and Separation

Affine Geometry

Operator Factorizations: QR and Singular Value

The Umbral Calculus