Overview

This then was the problem-to give an introductory course in modern algebra and geometry-and I have proceeded on the assumption that neither is complete without the other, that they are truly two sides of the same coin. Topics include lines and planes, determinants and linear equations, matrices, groups and linear transformations, and vectors and vector spaces. Additional subjects range from conics and quadrics to homogeneous coordinates and projective geometry, geometry on the sphere, and reduction of real matrices to diagonal form. Exercises appear throughout the text, with complete answers at the end. In seeking to coordinate Euclidean, projective, and non-Euclidean geometry in an elementary way with matrices, determinants, and linear transformations, the notion of a vector has been exploited to the full. There is nothing new in this book, but an attempt has been made to present ideas at a level suitable to first-year students and in a manner to arouse their interest. For these associations of ideas are the stuff from which modern mathematics and many of its applications are made. I have tried to keep the presentation as informal as possible in an attempt to arouse and maintain interest. Some of your established ideas may be challenged in Chapter 8 but this is all part of the process! The exercises have been constructed to illustrate the subject in hand and sometimes to carry the ideas a little further, but emphasis by mere repetition has been avoided. This matter of exercises is important. You should work at them contemplatively and expect to be frustrated sometimes, for this is the only way to make the ideas your own. The notion of a vector is of central significance in Euclidean geometry. As the title of this book suggests, our purpose is to develop these ideas in several different contexts. Some of these contexts are officially algebraic while others are geometric, but with this thread to guide us, we shall see their interrelations and why it is that mathematics is a living subject, changing and progressing with the introduction of new ideas.

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