Algebra and Geometry

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Algebra and Geometry
Cambridge University Press
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This book is an introduction to group theory, complex numbers and linear algebra. In an unusual approach, the first topic covered is group theory, although it is only a small part of a small toe placed in the water. Chapters covering the set of real numbers, complex numbers and three-dimensional vectors follow this. Vector spaces, systems of linear equation, matrices, and eigenvectors are the topics of the next chapters. The final chapters are "Linear maps of Euclidean space", "Groups", "Mobius transformations", "Group actions" and "Hyperbolic geometry."
The overall theme is to unify the two areas of algebra and geometry by showing that space itself can be described algebraically by repeating the use of real numbers to create multiple dimensions. Large numbers of exercises are included at the ends of the sections, making it easy to assign homework or to further study by working longer and deeper. However, no solutions to the problems are included.
While this book certainly is a good one in these areas, I am puzzled as to where it would fit into an undergraduate curriculum. Abstract algebra and linear algebra are separate courses in the curriculum and having a course covering both would certainly be an oddity. However, that criticism aside, the author does an excellent job in explaining how space can be described and transformed by first representing it as a set of vectors and then transforming those vectors.

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