I didn’t give the first volume of this text a rave review, but I wanted to give the second volume a shot at redemption.
Whereas Volume 1 dealt with the “linear and affine structure” of Rn (n = 1, 2, 3), Volume 2 discusses the Euclidean structure of the same spaces, concentrating on the theory of real inner product spaces. There are only two (long) chapters! These are numbered to indicate that they continue the work of Volume 1.
Almost all results are derived and illustrated in R2 and R3 only. (There is a discussion of a general singular value decomposition for a rectangular matrix, but the proof is only outlined.) As is true for both volumes, geometric content is emphasized. There are applications to symmetry groups, relativity, hyperbolic geometry, and differential geometry. An instructor would find this a good source of examples.
In summary, there is a lot of interesting material in these books that is not covered in most introductory texts. There are many good examples and illuminating illustrations. Unfortunately, the organization is confusing and the notation is not always helpful.
Again, I use the author’s own words from Volume 1 to suggest that “this book might better be used as a reference book or a companion to a formal course on linear algebra.”
Henry Ricardo (email@example.com) has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.
Preface to Volume One
Preface to Volume Two
Part II: The Euclidean Structures of R1, R2 and R3
4. The Euclidean Plane R2
5. The Euclidean Space R3
Contents of Volume One
Errata to Volume One
Index of Notations