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Geometric Linear Algebra 1

I-Hsiung Lin
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Henry Ricardo
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The aim of this idiosyncratic paperback volume of over 800 pages is to present the concepts of linear algebra via their geometric significance, as an antidote to approaches that have become “too algebraic and hence too abstract.” 

There are only three chapters in this book (the first of two volumes), and Chapter K (K = 1, 2, 3) is devoted to the “affine, linear, and Euclidian [sic]” structures of RK.  There are few formal definitions: Motivational discussions come first, with main results summarized and numbered afterwards.  There are more than 500 illustrations in the book.  Most sections have exercises grouped in ascending order of difficulty ( to ).  There are no applications, except to other areas of pure mathematics.  Roughly the last quarter of the book is devoted to two appendices.  Appendix B is essentially a condensed course in ordinary (i.e., algebraically-oriented) linear algebra, provided “for a striking contrast… for the sake of reference and comparison.”    This book was typed by various groups of the author’s students, some of whom are responsible for the graphs.  There are infelicities of language throughout.  In a discussion of the diagonal form of a linear transformation, the author has this strange statement:  Examples presented here might make you feel cumbersome, boring, and sick.  If so, please skip this content and go directly to Exercises or Sec. 2.7.7.  The Preface includes a religious dialogue, mostly about higher dimensions and UFOs.

The emphasis of this text is not as unusual as the author claims, except with respect to the extent of the details discussed.  Most modern linear algebra books, especially those that encourage the use of technology, try to foster geometric intuition.  The classic Linear Algebra Through Geometry by Banchoff and Wermer does a good job in far fewer pages than the text under review, while Fekete’s Real Linear Algebra provides a deeper geometric treatment in the spirit of Steenrod’s “Santa Barbara” program.   There are some interesting things in this book, but ultimately I must agree with the author’s own words: “In my opinion, this book might better be used as a reference book or a companion one to a formal course on linear algebra.”

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.