*Linear Algebra I and II* is a comprehensive, two-volume introduction to linear algebra and some of its mathematical applications. Readers are assumed to have already been exposed to the sort of matrix algebra that is traditionally covered in an undergraduate linear algebra course. These books are based on notes that were used at New York University's Courant Institute to teach two types of courses: a yearlong honors-level course for senior undergraduates wishing to strengthen their linear algebra background, and a yearlong masters-level course for mathematically mature students (perhaps from other disciplines) that wish to strengthen their mathematical skills, often with the aim of pursuing Ph.D. level work in pure or applied mathematics. The books would also serve as an excellent reference for beginning graduate students preparing for the linear algebra section of their qualifying exams.

The first volume is focused on fundamentals and discusses vector spaces, linear operators, dual spaces, determinants, and diagonalization before ending with a pair of chapters on the Spectral Theorem and Polar Decomposition. The book has a slight analytic bent and emphasizes limits, norms and function spaces to a much greater extent than do many standard texts in the area. The text also contains many examples. Some of these examples are the sort that nearly every linear algebra text covers (Let's find a basis for the subspace of four-dimensional space spanned by the following vectors.), while others connect the section's topic to other areas of mathematics (e.g. providing a linear algebraic perspective for the Lagrange Interpolation Formula, or using the Fourier Transform as an example of duality in the context of the vector space of continuous complex-valued functions on the interval). One nontraditional aspect of the books concern exercises. A nice selection of exercises appears at the end of every chapter, but also within the body of the text itself. These in text exercises do a great job of keeping the reader engaged and emphasize important proof techniques, but could cause difficulty for instructors as many theorems have proofs that cite exercises (e.g. By Theorem x.y.z and Exercise a.b, ...).

The second volume begins by covering a few topics that are more or less standard for advanced linear algebra texts: Jordan form and multilinear algebra. Afterwards is where things get interesting. Chapter 4 is a sixty page recasting of vector calculus into the language of linear algebra and covers things like differential forms, exterior derivatives and Poincare's Lemma. Chapter 5 is an eighty page introduction to matrix Lie groups that culminates with the Lie group-Lie algebra correspondence. The latter two chapters are for the most part self-contained, allowing an instructor to cover one or the other depending on their interests. The writing in this volume is exceptionally friendly, as is perhaps best illustrated by Chapter 4's discussion of the definition of tangent vectors in the context of a manifold. The definition of tangent vectors that novice readers are most likely to have encountered requires that their manifold be embedded in some larger ambient Euclidean space. But how can tangent vectors be defined if one wants to consider their manifold as "their entire universe," without some larger ambient Euclidean space in which to work? Rather than simply define tangent vectors as derivations on the local algebra of smooth functions at a point p on the manifold and be done with it, the authors instead first bring up several "more obvious" definitions and explain why each of them implicitly assumes an ambient Euclidean space in which to work. These "false starts" do an excellent job of motivating the eventual definition, and illustrate the extent to which the authors are concerned with making sure their readers understand why definitions are stated the way that they are, and more generally, why the structure of linear operators provides a useful means of solving mathematical problems.

These two volumes comprise an excellent introduction to linear algebra at the beginning graduate level. The writing is clear and friendly, examples are numerous, and connections are made between standard linear algebra topics and many other areas of mathematics. The in text exercises require a greater level of engagement on the part of the reader than do many other texts, but the payoff is certainly worth it. Greenleaf and Marques have done us a great service by making these notes available, in book form, to the mathematical community.