Real Linear Algebra

The following is an excerpt from a review in “The Mathematical Intelligencer.” Students may know how to diagonalise a matrix, for instance, but do they really know what they are doing in terms of the associated linear transformation? The ingredient often missing in the algebraist's analytical presentation is geometric insight. Steenrod's intention of restoring the balance and putting geometrical ideas back into linear algebra was outlined in a lecture given at Santa Barbara in 1967. Its text, which serves as a guiding light in this work, is appropriately reproduced in the Preface. The central idea explored in this book is that a linear transformation abstractly defined but geometrically interpreted is everything while the associated matrix is barely worth mentioning. The author is at pains to stress that matrices should play no part in the modern development of the theory and he declares that the theory becomes "quite easy and lovely" without them. The overiding spirit of the book is to stress geometric ideas and, if necessary, at the expense of rigour. The vector spaces treated here are finite dimensional over the field of real numbers (the meaning of "real" in the title) and for the most part only R^3 is considered..