Textbook in PDF format

The earlier chapter of this self-contained text provide a route from first principles through standard linear and quadratic algebra to geometric algebra, with Clifford's geometric algebras taking pride of place. In parallel with this is an account, also from first principles, of the elementary theory of topological spaces and of continuous and differentiable maps that leads up to the definitions of smooth manifolds and their tangent spaces and of Lie groups and Lie algebras. The calculus is presented as far as possible in basis free form to emphasize its geometrical flavour and its linear algebra content. In this second edition Dr Porteous has taken the opportunity to add a chapter on triality which extends earlier work on the Spin groups in the chapter on Clifford algebras. The details include a number of important transitive group actions and a description of one of the exceptional Lie groups, the group G
2. A number of corrections and improvements have also been made. There are many exercises throughout the book and senior undergraduates in mathematics as well as first-year graduate students will continue to find it stimulating and rewarding.
Foreword
Guide
Maps
Real and complex numbers
Linear spaces
Affine spaces
Quotient structures
Finite-dimensional spaces
Determinants
Direct sum
Orthogonal spaces
Quaternions
Correlations
Quadric Grassmannians
Clifford Algebras
The Cayley algebra
Normed linear spaces
Topological spaces
Topological groups and manifolds
Affine approximation
The inverse function theorem
Smooth manifolds
Triality
List of symbols